arXiv:2111.12841v1 [math.AG] 24 Nov 2021

arX

iv:2

111.

1284

1v1

[m

ath.

AG

] 2

4 N

ov 2

021

ON THE GEOMETRY OF FORMS ON SUPERMANIFOLDS

SIMONE NOJA

Abstract. This paper provides a rigorous account on the geometry of forms on supermanifolds,with a focus on its algebraic-geometric aspects. First, we introduce the de Rham complex ofdifferential forms and we compute its cohomology. We then discuss three intrinsic definitions ofthe Berezinian sheaf of a supermanifold - as a quotient sheaf, via cohomology of the super Koszulcomplex or via cohomology of the total de Rham complex. Further, we study the properties ofthe Berezinian sheaf, showing in particular that it defines a right D-module. Then we introduceintegral forms and their complex and we compute their cohomology, by providing a suitablePoincare lemma. We show that the complex of differential forms and integral forms are quasi-isomorphic and their cohomology computes the de Rham cohomology of the reduced space ofthe supermanifold. The notion of Berezin integral is then introduced and put to the good useto prove the superanalog of Stokes’ theorem and Poincare duality, which relates differential andintegral forms on supermanifolds. Finally, a different point of view is discussed by introducingthe total tangent supermanifold and (integrable) pseudoforms in a new way. In this context, itis shown that a particular class of integrable pseudoforms having a distributional dependencesupported at a point on the fibers are isomorphic to integral forms. Within the general overviewseveral new proofs of results are scattered.

Contents

1. Introduction - A Tale of Forms and Berezinians 12. Elements of Geometry of Supermanifolds 52.1. Derivations, Differential Operators and D-modules 93. Differential Forms and de Rham Cohomology of Supermanifolds 114. The Berezinian Sheaf: Constructions and Geometry 154.1. Berezinian Sheaf as a Quotient Sheaf 164.2. Berezinian Sheaf from Koszul Complex 174.3. Berezinian Sheaf from Cohomology of Forms and Operators 205. Properties of the Berezinian Sheaf 215.1. Berezinian and Canonical Sheaf 215.2. DM -modules and Connections 225.3. Lie Derivative of Ber(M ) and Right DM -module Structure 256. Integral Forms and Spencer Cohomology of Supermanifolds 307. Berezin Integral and Stokes’ Theorem on Supermanifolds 357.1. Supersymmetry and the Berezin Integral 388. Poincare Duality on Supermanifolds 419. Different Perspectives: Forms and Integration on Total Space 439.1. Special Class of Integrable Pseudoforms: Distributions on the Fibers 47Appendix A. Nilpotent Operators in Superalgebra 53Appendix B. Right D-modules and Canonical Sheaf 54References 55

1. Introduction - A Tale of Forms and Berezinians

Supergeometry is the study of supermanifolds, i.e. manifolds characterized by sheaves of Z2-graded(commutative) algebras, called superalgebras, whose “functions” might commute or anticommute,depending on their even or odd Z2-degree [50, 51, 53]. Supermanifolds can also be endowed with atighter structure, roughly speaking a symmetry that exchange even and odd directions [77]. This

1

http://arxiv.org/abs/2111.12841v1

2 SIMONE NOJA

provides a geometric realization of what physicists call supersymmetry transformations in the con-text of modern quantum field theories, thus making supergeometry into the right mathematicalenvironment to set and study these physical theories - one such being superstring theory [28].In some sense, the characterizing Z2-graded commutativity of supergeometry is the lowest possibledegree of non-commutativity. For this reason several constructions from ordinary purely commu-tative algebra and geometry can be easily generalized to a supergeometric setting. But this is notreally the point. Indeed, supergeometry features new notions that resist such a trivial extension,challenging instead our geometric intuition.One such notion is of global nature, and related to the holomorphic theory. Indeed, complex holo-morphic supermanifolds can be non-projected or non-split, meaning that they cannot be directlyreconstructed from their underlying purely commutative manifolds [38, 53]. As such, they aregenuinely new geometric objects, having “a life of their own”. Interest into these non-split / non-projected geometries has remarkably grown within the last years [8, 9, 15], prompted by Donagiand Witten’ paper [30], where it is showed that the supermoduli space of super Riemann surfacesis indeed non-projected for genus at least 5.But issues are also of local nature. Indeed, whereas on an ordinary manifold differential formsanticommute, thus forcing the de Rham complex to terminate at the dimension of the manifold,on a supermanifold differentials of odd functions do instead commute, so that the de Rham com-plex is not bounded [28, 53]. This easy fact has very far reaching consequences. Poincare duality- as known from ordinary commutative geometry - breakdowns, and there exists a new complex,which is “dual” to the de Rham complex. This is the so-called complex of integral forms, wherethe Berezinian bundle - a characteristic supergeometric construction which controls integration onsupermanifolds - sits and plays the role the canonical bundle plays in the ordinary de Rham com-plex [11]. Accordingly, integration theory on supermanifolds is quite peculiar and highly non-trivial[53, 74]. Most notably, once again, its subtleties are related to important physical questions, regard-ing both the foundations of supersymmetric theories as manifestly invariant theory on superspaces[17] and actual computations of quantities of physical interests, such as scattering amplitudes insuperstring theory [78].This paper is mostly concerned with the algebraic-geometric aspects of this rich and beautifultheory of forms (and integration) on supermanifolds. Before we start, though, as to provide thereader with some perspective and context - and also to make justice to the researchers who hascontributed the most to build and shape this branch of mathematics -, we will briefly go throughand comment the historical development of the theory.

The concept of differential forms on supermanifolds originated from the astounding creativity ofthe work Felix Berezin - the “mastermind of super-mathematics” as in [65] - and his collaboratorsaround 1970 [10], years before the mathematical formalization of the notion of supermanifold waseven available [50, 51]. As in the ordinary theory, differential forms on supermanifolds can be struc-tured into a complex, the de Rham complex, which - differently from a purely commutative setting- is not bounded from above. Also, allow for integration on reduced, or bosonic, submanifoldsof a supermanifold by their pull-back, but they do not control integration on supermanifolds. Ameaningful notion of integration on a space with odd, or fermionic, directions indeed requires thenotion of Berezinian bundle (after Berezin) - probably one of the most peculiar construction in thetheory of supermanifolds -, which plays the same role the determinant or canonical bundle play onan ordinary manifold. The fact that the Berezinian bundle does not appear in (the generalizationof) de Rham complex on supermanifolds - and, as such, its sections are not differential forms -marks a significative departure from the ordinary integration theory and its relation to differentialforms. Later on, in 1977, in the pioneering [11, 12] Bernstein and Leites introduced the complex ofintegral (or, perhaps more appropriately, integrable) forms, which is bounded from above (but notfrom below) and whose “top” bundle is given by the Berezinian of the supermanifold. Whereas thecomplex of differential forms controls integration on ordinary bosonic submanifolds of a superman-ifold, the complex of integral forms controls integration on sub-supermanifolds of the same odd(or fermionic) dimension of the ambient supermanifold. In this integration procedure fermionicvariables are somehow “frozen” and integrated over in the Berezin sense. Bernstein and Leites ex-ploited further the characteristic geometry of supermanifolds where even and odd variables coexistas to introduced also pseudodifferential forms [12, 13]. These newly-defined type of forms generalize

ON THE GEOMETRY OF FORMS ON SUPERMANIFOLDS 3

the notion of differential forms allowing for more general, non-polynomial, dependences on the even1-forms - while nihilpotency constraints the odd 1-forms to have a polynomial dependence. In par-ticular pseudodifferential forms can be integrable, provided that they vanish fast enough at infinity:it is this particular kind of pseudoforms that allow for integration on sub-supermanifolds of anycodimension, thus supplementing integral forms in the integration theory on supermanifolds, asthey can only be integrated on codimension k|0 sub-supermanifolds. Nonetheless, differently thandiffererential and integral forms, pseudodifferential forms do not carry any grading. This problemhas been address in the Eighties and early Nineties by Voronov and Zorich [68–70] and later onby Voronov [73]. Building on earlier work by Gaiduk, Khudaverdian and Schwarz [36], these au-thors were able to develop a theory of forms on supermanifolds graded by superdimension, whichare now referred to as Voronov-Zorich forms. Remarkably, Voronov-Zorich forms admit differentdescriptions: a quite surprising one being via Lagrangians related to parametrized supersurfaces[68]. In more recent years, this intriguing point of view has been further push forward by Voronov[72, 73], who provided an extension of the de Rham complex with forms carrying a negative de-gree, something possible only due to the peculiar geometry of forms on supermanifolds. Quiteremarkably, integral forms, (integrable) pseudodifferential forms and Voronov-Zorich forms are allrelated via integral transformations. For example, it is possibile to define an integral transform[69], called odd Fourier transform, which maps isomorphically integral forms to a particular classof pseudodifferential forms having a Dirac delta distributional dependence on the even 1-forms,which are thus supported at a single point.Foundational problems in the theory of supermanifolds - and in particular in the theory of formson supermanifolds - have been addressed during the Eighties also by other groups of researchers.Manin, together with the students at his school of algebraic geometry, worked extensively onproblems related to complex supermanifolds (and their deformations) in relation to the back thennew-born superstring theory, which was polarizing the attention of the high-energy physics commu-nity. The fundamental intrinsic definition of the Berezinian bundle as arising from the homologyof a very non-trivial generalization of the Koszul complex appeared first in [58], where Ogievetskyand Penkov, students of Manin at that time, introduced Serre duality for projective supermani-folds, showing that the Berezinian bundle provides the right super-analog of the dualizing sheafin (commutative) algebraic geometry. Subsequentely, this important construction has been brieflyreported by Manin in his beautiful book [53] - but the paper [58] was inadvertently not acknowl-edged, thus causing a bit of confusion regarding the attribution of the result [59]. Quite recently aself-contained thorough discussion of the super Koszul complex and its relation with the Berezinianhas been given in [57]. Further crucial properties of differential and integral forms on superman-ifolds was discussed by Penkov in relation to the theory of D-modules on supermanifolds in thevery beautiful paper [60]: in this work, among many other things, the author shows that theBerezinian bundle carries a natural structure of right D-module, showing once again similaritieswith the canonical bundle of an ordinary manifold. Beside the “Russian school”, other groupsof researchers worked on “supermathematics” starting from the Eighties. Bartocci, Bruzzo andHernandez Ruiperez were very active in the research on foundational problems in the theory of su-permanifolds. In particular, it is due to Hernandez Ruiperez and Munoz Masque one of the neatestintrinsic construction of the Berezin bundle for smooth supermanifolds [62], to be compared to theaforementioned contemporary construction via super Koszul complex by Ogievetsky and Penkov[58], which looks instead to the algebraic category.The theory of forms and integration on supermanifolds and superspaces has many applications inmodern physics. Supersymmetry - a building pillar of contemporary high-energy physics, whichrelates bosonic fields to fermionic fields [37] -, can only be realized geometrically at the cost ofupgrading the ambient manifold of the theory to a supermanifold. In this context, the actionof the physical theory is written in term of a Berezin integral of a section of the Berezinian ofthe supermanifold, which plays the role of the Lagrangian density of the theory, and the super-symmetry transformations are generated by particular odd vector fields. The machinery of theBerezin integral and the property of the Berezinian bundle make possible for the Lie derivative ofthe action of the physical theory to integrate to zero, thus making supersymmetry into a manifestsymmetry. In other words, to set a supersymmetric theory on a supermanifold, with its peculiarnotion of integration and Berezinian, has the same meaning as set a Lorentz-invariant theory on asemi-Riemannian Lorentzian manifold: only in this way symmetries of the physical theory becomes

4 SIMONE NOJA

geometric transformations, uncovering mathematical structures and making the symmetry of thetheory manifest.It is fair to say, though, that physics has probably not yet absorbed the elegance and subtleness ofthe theory of forms on supermanifolds: integration on superspaces is regarded as an algebraic for-mal machinery and the theory of integral forms - in its various realizations - is mostly ignored. Asa result, the full power of the formalism has not been exploited yet in its applications. There existremarkable exceptions and efforts in this direction though. In [6, 7] Belopolsky used a variationon the theme of Voronov-Zorich forms in relation to the physical problem of computing scatter-ing amplitudes in superstring perturbation theory, putting forward a supergeometric version ofthe so-called picture changing operators introduced in conformal field theory in [34, 35]. In morerecent years, Catenacci, Castellani, Grassi - together with several other collaborators - realizedthat the the so-called rheonomic principle [17], that lies at the basis of the geometric formulationof supergravity theories on supermanifolds, needed to be lifted to the integral forms complex ofthe supermanifold to make sense. This opened up to the formulation of many supergravity andsupersymmetric theories via integral forms and a new understanding of their structures and super-symmetries [18, 21, 22, 24–26]. Whereas high-energy physics and string theory communities havebeen in their greatest parts pretty insensitive to the subtleties related to forms and integrationtheory on supermanifolds, in a totally opposite fashion the development of the theory of Batalin-Vilkovisky (BV) quantization has been highly influenced by supergeometry and prompted severaladvances in the field. In particular, integral forms, in their incarnation as semi-densities on oddsymplectic supermanifolds, play a major role in BV quantization, see [55] for a supergeometric-aware detailed review of the topic. In this context have to be cited the very influential work [64]by Schwarz and the seminal contributions [43–46] by Khudaverdian and Neressian, regarding therelations between BV geometry and forms and integration on supermanifolds - in particular, it isdue to Khudaverdian the first supergeometric definition of the BV Laplacian [47, 48]. Later on,building upon Khudaverdian works, in [63] Severa provided a wonderful homological constructionof the BV Laplacian via a (quite surprising) spectral sequence related to a “deformed” de Rhamoperator by the (odd) symplectic form of the ambient odd symplectic supermanifold: notably, theconstruction resides on the cohomology of the aforementioned super Koszul complex, introducedby Ogievetsky and Penkov. Finally, in recent years the theory of integral forms and the relatedintegration theory on supermanifolds has been revitalized and drawn again to physicists’ attentionby the review [76] by Witten, which was written with an eye to applications to superstrings [78].The point of view of the author emphasizes the relation of differential and integral forms withClifford-Weyl (super)algebras and their representations. This has been recently prompted otherstudies, also in the realm pure mathematics, see for example the interesting [66] in relation withLie superalgebras.The books on the topics deserves a separate mention. First off, Voronov’s [74] provides a thoroughdiscussion on integration on supermanifolds, emphasizing the author’s construction of r|s-forms asvariation of Lagrangians and the related integration theory. This is the only dedicated book onthe topic to this day - we warn the reader, though, that this paper takes a different perspective.Further, Manin’s [53] features a deep chapter on supergeometry which, among many other things,introduces integral forms, Berezin integral and densities. The exposition leans toward an algebraicgeometric point of view, which is the one taken also the present paper: as such, [53] could beconsidered as a main reference.

The paper is structured as follows. In section 2 the basic constructions in supergeometry areintroduced. In particular, the notion of supermanifold is discussed from the point of view or locally-ringed space and certain natural sheaves are introduced. Section 3 is dedicated to differential formson supermanifolds and their (de Rham) cohomology, in particular Poincare lemma is discussed.Section 4 is dedicated to one of the most peculiar construction in supergeometry, that of Berezinian.More in details, we will define the Berezinian bundle via three constructions: as a quotient sheaf,as constructed via the super Koszul complex and as resulting from the cohomology of the so-calledtotal de Rham complex, which mediates between the first two and it is substantially new. Therelations between these constructions are commented. In section 5 we will study some properties ofthe Berezinian bundle, in particular we will see that it allows a rightD-module structure, such as thecanonical bundle of an ordinary manifold. Section 6 deals with integral forms and their (Spencer)


cohomology, in particular we will prove that the complexes of differential and integral forms arequasi-isomorphic, i.e. they compute the same cohomology, actually the de Rham cohomology of thereduced manifolds. In section 7 the Berezin integral is introduced. We will prove Stokes’ theoremfor supermanifolds (without boundaries) and, as an application, we will see how it allows manifestsupersymmetry invariance in physics. In section 8 we will see how the notion of Poincare dualitygets modified when working on supermanifolds. Finally, in section 9 we will present a differentpoint of view, by introducing pseudoforms on supermanifolds as functions, endowed with certainproperties, which are defined on the total space of the tangent bundle of a supermanifold, whosegeometry is discussed in details in a new way. We will show that for a specific class of these forms- namely those having a distributional dependence supported at zero in the fiber directions -, thereis an isomorphism with the previously defined integral forms. The approach taken in this lastsection differs from the available literature, in that - consistently with the spirit of the paper - onlyalgebraic-geometric inspired ideas and methods have been employed, in place of the traditionalanalytic approach via integral transforms.We stress that this paper does not aim to be fully encompassing - and indeed some point ofviews which we have hinted upon in this introduction are not discussed here - for these, we referfor example to [74]. Instead, we have chosen to provide a - hopefully - conceptually clear andmathematically rigorous exposition, keeping our focus on the algebraic-geometric aspects of thetheory. The most important and peculiar constructions - which are not well-known outside a publicof experts - have been spelled out in details, trying to provide a firmly founded systematization ofthe results, alongside with new comprehensive proofs which are often not available in the literature.Finally, efforts have been put to make the exposition as self contained as possible and in the hopeto provide a readable, but not overwhelming, reference to the subject.

2. Elements of Geometry of Supermanifolds

In this section we briefly recall the main definitions in the theory of supermanifolds, see for examplethe classical [50, 51] or [2, 53]. We start with one of the most fundamental concept in supergeometry,that of superspace.

Definition 2.1 (Superspace). A superspace is a pair (|M |,OM ), where |M | is a topological spaceand OM is a sheaf of Z2-graded supercommutative rings over |M |, such that the stalks OM ,x atevery point of |M | are local rings. Analogously, a superspace is a locally ringed space havingstructure sheaf given by a sheaf of Z2-graded supercommutative rings.

Note that the requirement about the stalks being local rings reduces to ask that the even componentof the stalk is a usual commutative local ring. Indeed if A = A0 ⊕A1 a super ring, then A is localif and only if its even part A0 is, see for example [67].Given two superspaces we can define a morphism between them in the usual fashion.

Definition 2.2 (Morphisms of Superspaces). Given two superspaces M and N a morphism ϕ :M → N is a pair ϕ ..= (φ, φ♯) where

(1) φ : |M | → |N | is a continuous morphism of topological spaces;(2) φ♯ : ON → φ∗OM is a morphism of sheaves of Z2-graded rings, having the properties that it

preserves the Z2-grading and that given any point x ∈ |M |, the homomorphism φ♯x : ON ,φ(x) →OM ,x is local, i.e. it preserves the (unique) maximal ideal, i.e. φ♯x(mφ(x)) ⊆ mx.

It is easy to see that superspaces together with their morphisms forms a category, we call it SSp.Before we go on, some remarks on the previous definitions are in order.

Remark 2.3. With an eye to the ordinary theory of schemes in algebraic geometry, we stress thatthe request that the morphism φ♯x : ON ,φ(x) → OM ,x preserves the maximal ideal in the secondpoint of the definition above is of particular significance in supergeometry. Indeed it is importantto notice that the structure sheaf OM of a superspace is in general not a sheaf of functions. Aslong as the structure sheaf OM of a certain space or, more in general, of a scheme, is a sheafof functions, then a section s of OM takes values in the field of fractions k(x) = OM ,x/mx thatdepends on the point x ∈ |M |, as a function x 7→ s(x) ∈ k(x), and the maximal ideal mx containsthe germs of functions that vanish at x ∈ |M |. In the case of superspaces, nilpotent sections -and thus in particular all of the odd sections - would be identically equal to zero as functions on

6 SIMONE NOJA

points, and indeed the maximal ideal mx contains the germs of all the nilpotent sections in OM ,x.In this context, the request that φ♯x : ON ,φ(x) → OM ,x is local becomes crucial, while in the case ofa genuine sheaf of functions the locality is automatically achieved. In particular, locality impliesthat a non unit element in the stalk ON ,φ(x), such as a germ of a nilpotent section, can only bemapped to another non unit element in OM ,x, such as another germ of a nilpotent section. Inother words, nilpotent elements cannot be mapped to invertible elements.Anyway, we fell like we have to advice the reader that we will often abuse the notation by keepdenoting φ instead of φ♯ the morphism of sheaves related to a superspace morphism ϕ : M → N .

Remark 2.4. It is crucial to observe that one can always construct a superspace out of two “clas-sical” data: a topological space, call it again (by abuse of notation) |M |, and a vector bundle over|M |, call it E (analogously: a locally-free sheaf of O|M |-modules). We denote O|M | the sheaf of

continuous functions (with respect to the given topology) on |M | and we put∧0 E∗ = O|M |. Then

the sheaf of sections of the bundle of exterior algebras∧•E∗ has an obvious Z2-grading (by taking

its natural Z-grading mod 2) and therefore in order to realize a superspace it is enough to takethe structure sheaf OM of the superspace to be the sheaf of O|M |-valued sections of the bundle ofexterior algebras of E . This construction is so important to bear its own name [30].

Definition 2.5 (Local Model S(|M |, E)). Given a pair (|M |, E), where |M | is a topological spaceand E is a vector bundle over |M |, we call S(|M |, E) the superspace modelled on the pair (|M |, E),where the structure sheaf is given by the O|M |-valued sections of the exterior algebra

∧• E∗.Note that we have given a somehow minimal definition of local model, indeed we have let |M | tobe no more than a topological space and as such we are only allowed to take O|M | to be the sheafof continuous functions on it. Clearly, we can also work in richer and more structured category,such as the real smooth, complex analytic or algebraic category as we will do in this paper. Thisamount to consider local models based on the pair (Mred , E), where Mred is a smooth or complexmanifold or an algebraic variety - we keep denoting its underlying topological space with |M |, asabove - with OMred

being its sheaf of smooth, holomorphic of algebraic functions and E being asmooth, holomorphic of algebraic vector bundle.We will call smooth, holomorphic or algebraic local model a local model S(Mred , E) which is con-structed from the above data. This leads to the definition of supermanifold in the appropriatecategory, which we explicitly give in the real smooth or complex analytic category.

Definition 2.6 (Real / Complex Supermanifold). A real (complex) supermanifold M of dimensionn|m is a superspace that is locally isomorphic to some smooth (holomorphic) local modelS(Mred , E),where Mred is a smooth (complex) manifold of dimension n and E is a smooth (holomorphic) vectorbundle on Mred of rank m.

In other words, if Mred is covered by an atlas Uii∈I , the structure sheaf OM = OM ,0 ⊕ OM ,1 ofthe supermanifold M is described via a collection ψUi

i∈I of local isomorphisms of sheaves

Ui 7−→ ψUi: OM ⌊Ui

∼=−→ ∧•E∗⌊Ui(2.1)

where we have denoted with∧• E∗ the sheaf of OMred

-valued sections of the exterior algebra of Econsidered with its Z2-gradation. Also, notice that a morphism of supermanifolds is nothing but amorphism of superspaces, so that one has the related category of supermanifolds, that we denotewith SMan.The special case in which there is a single global isomorphism instead of a family of local isomor-phisms deserves a name by its own.

Definition 2.7 (Split Supermanifold). We say that a supermanifold M is a split supermanifold ifit is globally isomorphic to its local model. That is, if we have a sheaf isomorphism OM

∼=∧• E∗.

Clearly, split supermanifolds are the easiest supermanifolds to deal with, as their structure sheavesare simply sheaves of OMred

-valued sections of exterior algebras, and as such they are locally-freesheaves of OMred

-modules.In order to see how real and complex supermanifold might differ from the point of view of theirglobal geometry, we need to introduce some further pieces of informations related to a superman-ifold.


Definition 2.8 (Nilpotent Sheaf). Let M be a real (complex) supermanifold with structure sheafOM . We call the nilpotent sheaf JM of M the sheaf of ideals of OM = OM ,0 ⊕OM ,1 generated byall of the nilpotent sections in OM , i.e. we put JM

..= OM ,1 ⊕O2M ,1.

It is crucial to note that modding out all of the nilpotent sections from the structure sheaf OM

of the supermanifold M we recover the structure sheaf OMredof the underlying ordinary manifold

Mred , the local model was based on.

Definition 2.9 (Reduced Space). Let M be a real (complex) supermanifold with structure sheafOM . We call the reduced space of M the smooth (complex) manifold Mred with structure sheafgiven by the quotient OMred

..= OM /JM .

Loosely speaking, the reduced manifold in Mred arises by setting all the nilpotent sections in OM

to zero. In other words, more invariantly, attached to any real or complex supermanifold there isa short exact sequence that relates the supermanifold to its reduced manifold

0 // JM// OM

ι♯// OMred

// 0. (2.2)

The surjective sheaf morphism ι♯ : OM → OMredcorresponds to the existence of an embedding

Mredι−→ M of the reduced manifold Mred inside the supermanifold M . Notice that JM = ker(ι),

where ι : OM → OMredis the surjective sheaf morphism in (2.2).

Using the nilpotent sheaf associated to a supermanifold M , say of odd dimension m we can con-struct a descending filtration of length m of OM as follows,

OM ⊃ JM ⊃ J 2M ⊃ J 3

M ⊃ . . . ⊃ J qM ⊃ Jm+1

M = 0. (2.3)

This allows us to give the following definition.

Definition 2.10 (GrOM and GrM ). Let M be a supermanifold having odd dimension m to-gether with the filtration of its structure sheaf OM as in (2.3). We define the following sheaf ofsupercommutative algebras

GrOM..=

m⊕

i=0

Gr(i)OM = OMred⊕ JM

/J 2

M⊕ . . .⊕ Jm−1

M

/Jm

M⊕ Jm

M . (2.4)

where Gr(i)OM..= J i

M

/J i+1

Mand the Z2-grading is obtained by taking the obvious Z-grading

mod 2. We call the split supermanifold associated to M the supermanifold arising from the super-space (|M |,GrOM ) and we denote it by GrM .

If the supermanifold M has odd dimension m, the quotient Gr(1) OM = JM /J 2M in (2.4) is a

locally-free sheaf of OMred-modules of rank 0|m - i.e. it is locally generated by m odd sections - and

it plays a special role so that for notational convenience we denote it FM..= JM /J 2

M , as to recallits “fermionic” behavior.The importance of FM lies in that - up to parity - it is isomorphic to E∗, the vector bundle appearingin the local model S(Mred , E) whose the supermanifold M is based upon, i.e. one has E∗ ∼= ΠFM ,where Π : ShOM

→ ShOMis the so-called parity changing or parity shifting functor, which maps a

locally-free sheaf of rank of rank p|q to one of rank q|p, by reversing its parity [28, 53]. In view ofthis one has that GrM = S(Mred ,ΠF∗M ) and the local isomorphisms characterizing the structuresheaf OM can be rewritten, over an open set U ⊂ |M |, as

OM ⌊U∼= ∧•E∗⌊U∼= S•FM ⌊U , (2.5)

where Si : ShOM→ ShOM

is the i-th supersymmetric power functor [53], so that in particular

GrOM =⊕m

i=0 SiFM .

We conclude this section by stressing out the major difference between the realm of real andcomplex supermanifolds, that lies in the fact that real supermanifold are always split and actuallyall isomorphic to GrM . This result - in a slightly different form - was first proved by MarjorieBatchelor in [4].

Theorem 2.11 (Batchelor). Let M be a smooth supermanifold. Then its structure sheaf is non-canonically isomorphic to a sheaf of exterior algebras for some smooth vector bundle E over Mred .In particular, M is split and one has M ∼= GrM .

8 SIMONE NOJA

Remark 2.12. It is important to note that the theorem states the existence of a non-canonicalisomorphism. This mean that it guarantees the existence of a certain covering of open sets togetherwith charts such that the isomorphism is realized, but it is not constructive: in other words it doesnot tell how to concretely realize such an isomorphism.

Remark 2.13. Notice also that, globally, a real supermanifold can be seen as the total space ofa sheaf of exterior algebras ∧•E∗ over Mred , whose fibers are nilpotent. In particular, this meansthat charts Uℓ, x

ℓi |θℓαℓ∈I for i = 1, . . . , n and α = 1, . . . ,m can always be found such that if

xj

i|θjα and xk

i |θkα are local coordinates in any two open sets Uj and Uk in Uℓℓ∈I having non-empty

intersection Uk ∩ Uj 6= ∅, then we will have

xki = xk

i (xj

1, . . . , xjn), θ′α =

m∑

β=1

[gk j (x)]αβθβ , (2.6)

where [gk j (x)]αβ ∈ Z1(Uk ∩ Uj , GL(q,R)) are the transition functions of the vector bundle E∗.That is, when changing charts, the even local coordinates x’s transform as the coordinates ofthe ordinary smooth manifolds Mred and the odd local coordinates θ’s transform linearly, as thegenerating sections of the vector bundle E∗.Remark 2.14. Loosely speaking, a general obstruction theory to split the structure sheaf of asupermanifold can be constructed by filtering OM as in (2.4). This was first done by Green in [38].For a supermanifold based on the local model S(Mred , E), obstructions are given by cohomologyclasses ωi lying in the cohomology groups

ω2i ∈ H1(Mred , TMred⊗ ∧2iE∗), ω2i+1 ∈ H1(Mred , E ⊗ ∧2i+1E∗) (2.7)

for i ≥ 1 and where TMredis the tangent sheaf of the reduced space Mred . The subtlety here is that

whereas the fundamental obstruction, i.e. ω2, which is controlled by the groupH1(Mred , TMred⊗∧2E∗)

is always defined, instead the higher obstructions, i.e. ωi for i ≥ 3, are only defined if all the lowerones vanish [30, 38].In the smooth category all the sheaves are fine as a consequence of the existence of smooth parti-tions of unity, then these cohomology groups are automatically zero for a real supermanifold andthere are no obstruction to split OM as GrM , providing another proof of Batchelor theorem.On the other hand things change dramatically for complex supermanifolds, where the above coho-mology groups can indeed be non-zero, thus leading to the peculiar - and very interesting! - complexsupergeometry of non-split and non-projected supermanifolds, i.e. those complex supermanifoldsthat do not admit a retraction or projection for their defining exact sequence (2.2)

0 // JM// OM ι

// OMred

π

xx❨❴

// 0, (2.8)

where π♯ : OMred→ OM is such that ι♯ π♯ = idOMred

and corresponds to a map π : M → Mred .

Notice that, in particular, the structure sheaf of a non-projected supermanifold is not a sheaf ofOMred

-algebras, and as such a non-projected supermanifold cannot be “reconstructed” easily from itsunderlying ordinary complex manifold, in the words of Donagi and Witten in [30], a non-projectedsupermanifold “has a life of its own”.In complex supergeometry the absence of a projection should not be looked at as an exotic andmostly rare phenomenon, but as a characterizing and quite common phenomenon instead. Forexample the easiest supergeometric generalization of a conic in the complex projective superspace

CP2|2 cut out by the equation

X20 +X2

1 +X22 +Θ1Θ2 = 0 ⊂ CP

2|2, (2.9)

turns out to be a 1|2-dimensional non-projected supermanifold, see for example [30] or [53], whichis characterized up to isomorphism by the triple

Cω ..= (Mred = CP1, E∗ = OCP1(−2)⊕2, H1(Mred , TMred

⊗ ∧2E∗) ∋ ω2 6= 0), (2.10)

notice that in this case ω2 ∈ H1(CP1,OCP1(−2)) ∼= C. In the light of Batchelor theorem [4], onecan see the difference in terms of the structure of transition functions between split supermanifoldsand non-projected (or non-split) supermanifolds. Indeed, covering the reduced space CP

1 by the


standard (two) open sets, and considering the related systems of local coordinates to be given by(z|θ1, θ2) and (w|ψ1, ψ2) respectively, for the above super conic Cω one finds

w =1

z+θ1θ2z3

, ψ1 =θ1z2, ψ2 =

θ2z2. (2.11)

The non-vanishing obstruction class ω2 6= 0 in the cohomology guarantees that there are no choicesof coordinates or redefinitions of charts that bring the even transition back to the form of that ofthe reduced space CP1, i.e. w = 1/z. On the contrary, for a non-projected supermanifold the eventransition functions depend crucially also from the odd part of the geometry, as one can see in the(2.11).

2.1. Derivations, Differential Operators and D-modules. Having introduced the concept ofsupermanifolds - in particular in the smooth real and complex analytic category -, we now brieflydiscuss the most important natural sheaves that can be defined on them and that will be used themost in this paper.In particular, working for instance over a complex supermanifold M , the structure sheaf OM can belooked at as a subsheaf of the sheaf of its C-endomorphisms, EndC(OM ), taking g 7→ σf (g) ..= fgfor any sections f, g ∈ OM . Likewise, the tangent sheaf TM of M - or, analogously, the sheaf ofderivations DerC(OM ) - can also be looked at as a subsheaf of EndC(OM ) defining

TM..= X ∈ EndC(OM ) : X(fg) = X(f)g + (−1)|X||f |fX(g), f, g ∈ OM . (2.12)

Since we only consider smooth supermanifolds, then TM is locally-free of rank n|m = dimC M andif xi|θα are local coordinates for a chart U ⊂ M , then a section X ∈ TM over U is given by

XU =

n⊕

i=1

OM (U) · ∂xi⊕

m⊕

α=1

OM (U) · ∂θα . (2.13)

This means that the tangent sheaf is freely locally-generated by the even and odd derivations

TM (U) = OM (U) · ∂x1 , . . . , ∂xn|∂θ1 , . . . ∂θm. (2.14)

The above point of view, aimed at relating the structure sheaf and the tangent sheaf with thesheaf of endomorphisms of OM is particularly useful when one is interested into introducing thesheaf of differential operators on M , which will play an important role in what follows. We givethe definition for a complex supermanifold, but the same can be done for a real and also algebraicsupermanifold.

Definition 2.15 (The Sheaf DM ). Let M be a complex supermanifold. We define the sheaf ofdifferential operators of M to be the subsheaf of EndC(OM ) generated by OM and TM , and wedenote it by DM .

If xa|θα is a coordinate system over an open set U , then xi|xα, ∂xi|∂θαi=1,...,n,α=1,...,m gives a

local trivialization of DM ⌊U , where xi|θα ∈ OM ⌊U and ∂xi|∂θα ∈ TM ⌊U satisfy the following defining

relations

[xi, xj ] = 0, [∂xi, ∂xj

] = 0, [xi, ∂xj] = δij , θα, θβ = 0, ∂θα , ∂θβ = 0, θα, ∂θβ = δαβ

[xi, θα] = 0, [∂xi, ∂θα ] = 0, [xi, ∂θα ] = 0, [θα, ∂xi

] = 0, (2.15)

where [·, ·] denotes a commutator and ·, · denotes an anticommutator. Notice that locally, theserelations define the Weyl superalgebra DCn|m , so that posing

U 7−→ DM (U) ..= DU is a differential operator on OM (U), (2.16)

then one has

DU =⊕

ℓ∈Nn,ε∈Zm2

OM ⌊U∂ℓx∂εθ where

∂ℓx

..= ∂ℓ1x1∂ℓ2x2

· · · ∂ℓnxn,

∂εθ..= ∂ε1θ1∂

ε2θ2

· · · ∂εnθn(2.17)

In particular we define deg(DU ) ..= max(|ℓ| + |ε|), if |ℓ| =∑ni=1 ℓi and |ε| =∑m

α=1 εα and we calldeg(DU ) the degree of the differential operator DU ∈ DM ⌊U .The above local description of equation (2.17) leads to a natural filtration of DM . We define

F iDM (U) ..= DU ∈ DM (U) : deg(DU⌊V ) ≤ i for all V ⊆ U . (2.18)

10 SIMONE NOJA

Notice that the above filtration is increasing, i.e. F iDM ⊆ F i+1DM , and exhaustive, i.e. ∪iFiDM =

DM . More in general one has the following relations

F iDM · F jDM ⊆ F i+jDM , [F iDM , FjDM ] ⊆ F i+j−1DM , (2.19)

and we define the associated graded module gr•F (DM ) with respect to the above filtration:

gr•F (DM ) ..=

∞⊕

i=0

griF (DM ) =

∞⊕

i=0

F iDM /F i−1DM , (2.20)

where we have defined griF (DM ) ..= F iDM /Fi−1DM .

Remark 2.16. It has to be stressed that, in particular, F 1DM is a Lie sub-superalgebra of DM ,since indeed one has that [·, ·] : F 1DM × F 1DM → F 1DM . Further, F

0DM is a Lie ideal of this Liesuperalgebra: indeed if f ∈ F 0DM , then [f,G] ∈ F 0DM for any G ∈ F 1DM . It thus makes senseto consider the quotient of F 1DM by F 0DM , and it is not hard to realize that

TM∼= F 1DM /F 0DM , (2.21)

which in turn leads to

gr•F (DM ) = S•OMTM . (2.22)

Remark 2.17. One could think about this in analogy with the Poincare-Birkhoff-Witt (PBW)theorem for the universal enveloping algebra U(g) of a certain Lie (super)algebra g. Indeed, givena filtration as above, the quotient operation does not just reduce to the leading terms, but it doesremarkably more: it sets all the commutators in griF (DM ) to zero. Indeed, if two elements F,G

are such that their product is in griF (DM ), then their commutator is in gri−1F (DM ) so that it isset to zero in the quotient. In other words, elements surviving the quotient are those that do notcome from commutators: in particular, all of the elements commute and this leads naturally tothe symmetric (super)algebras, where all of the (super)commutators vanish. Notice that, similarly,gr•U(g) ∼= S•(g), which is the meaning of PBW theorem. In general, we might think about thefollowing analogies between a Lie (super)algebra and its universal enveloping algebra (togetherwith its PBW filtration) and the derivations on a (super)manifolds, and the differential operatorson M :

TM ! g (2.23)

DM ! U(g) (2.24)

Remark 2.18. Finally, it is worth to observe that the associated graded module gr•F (DM ) is naturallyisomorphic to π∗OT∗M , where π : T∗M → M is the cotangent bundle of the supermanifold M . Thisexpresses the fact that “functions” on a certain space are given by the symmetric algebra over thedual space. We will use this fact explicitly in the last section of this paper.

Remark 2.19. As made clear by the previous discussion, the sheaf DM is a sheaf of non-commutativealgebras, as it follows from the non-trivial relations [xi, ∂xj

] = δij and θα, ∂θβ = δαβ . Actually,the sheaf of differential operators over a certain (super)manifold, defined as above, is the proto-typical example of sheaf of non-commutative algebras. Clearly, this leads to the fact that whenan action involving DM is concerned, then it is important to distinguish between left and rightactions, as in the following definition.

Definition 2.20 (DM -Modules). Let M be a complex supermanifold and let E be a sheaf over M .We say that E is a sheaf of left/right DM -modules, or simply a left/right DM -module, if E(U) isendowed with a left/right DM (U)-module structure for any open set U , which is compatible withthe restriction morphisms of the sheaf.

It is to be observed that a left DM -module can not at all be endowed with the structure of rightDM -module and viceversa. In this paper we will indeed come across left DM -module which are notright DM -modules and viceversa.


3. Differential Forms and de Rham Cohomology of Supermanifolds

Given a smooth real supermanifold M of dimension n|m with structure sheaf OM and hav-ing introduced its tangent sheaf TM = DerR(OM ), it is immediate to consider its dual sheafHomOM

(TM ,OM ) ∼= T ∗M . This is a locally-free sheaf of OM -module of rank n|m, and one has theusual duality pairing T ∗M ⊗ TM → OM defined as dya(∂yb

) = δab if ya ..= xi|θα. Notice that forthis to be a perfect pairing one needs to assign a Z2-parity to the generators in a way such that|dxi| = 0 and |dθα| = 1. When reducing to the underlying real manifold Mred , with an eye to theassociated de Rham algebra of differential forms with respect to the wedge product, this wouldlead to the awkward situation of having a basis of commuting dx’s.In order to restore the correspondence with the usual de Rham theory on the underlying reducedspace, it is therefore customary to define the sheaf 1-forms Ω1

M on a supermanifold M to be the par-ity shifted of T ∗M , i.e. we take Ω1

M..= HomOM

(ΠTM ,OM ). This is a locally-free sheaf of OM -modulesof rank m|n and if xi|θα are local coordinates over a certain open set U , then one has

Ω1M (U) = OM (U) · dθ1, . . . , dθm | dx1, . . . , dxn , (3.1)

where now the parity is assigned such that |dθα| = 0 and |dxi| = 1 for any α = 1, . . . ,m andi = 1, . . . , n. It is important to stress that now Ω1

M has a (perfect) duality pairing with the parityshifted tangent sheaf ΠTM instead of with the tangent sheaf. Again ΠTM is a locally-free sheaf ofOM -modules of rank m|n and if xi|θα are local coordinates over an open set U as above, one has

ΠTM (U) = OM (U) · π∂θ1 , . . . , π∂θm |π∂x1 , . . . , π∂xn, (3.2)

where |π∂xi| = |∂xi

| + 1 = 1 and |π∂θα | = |∂θα | + 1 = 0 for any i = 1, . . . , n and α = 1, . . . ,m.With this convention, if ya = xi|θα is a local system of coordinates, the duality pairing 〈·, ·〉 :Ω1

M ⊗ ΠTM → OM reads

〈dya, π∂yb〉 = δab. (3.3)

Notice that in what follows we will often simply write dya(π∂yb) = δab as indeed dya and π∂yb

aregenerating sections of Hom(ΠTM ,OM ) and ΠTM respectively on a certain open set. Once giventhese basic definitions and fixed our conventions, we introduce the following [53, 60].

Definition 3.1 (de Rham Superalgebra). Let M be a real supermanifold of dimension n|m withstructure sheaf OM . We call the de Rham algebra of M the sheaf of OM -superalgebras given by

Ω•M..=

∞⊕

k=0

SkOM

Ω1M , (3.4)

where Sk : ShOM→ ShOM

is the k-supersymmetric functor and Ω1M is the sheaf of 1-forms of M

defined as above, where the Z2-grading is induced by that of Ω1M .

Remark 3.2. Notice that the above algebra is readily made into a Z-graded algebra by assigningdeg(OM ) = 0 and deg(Ω1

M ) = 1: this is the obvious degree induced by the (local) polynomialsuperalgebra

S•Ω1M (U) = OM (U) · [dθ1, . . . , dθm|dx1, . . . , dxn]. (3.5)

The sections of the locally-free OM -submodule SkΩ1M of degree k are called k-forms, as in the

ordinary commutative setting.

Remark 3.3. It is crucial to observe the difference with respect to the usual de Rham or exterioralgebra on an ordinary manifold X , whose non-zero sections can be of degree n = dimX at most,due to anti-commutativity of the exterior product. Over a supermanifold, instead, a system of localgenerator obeys the supercommutation relation of the supersymmetric algebra, i.e. if ya = xi|θαthen [dya, dyb]± = 0 [28, 53]. This leads in particular, for any α, β = 1, . . . ,m and i, j = 1, . . . , nto

[dxi, dxj ]+ ..= dxidxj + dxjdxi dxidxj = −dxjdxi. (3.6)

[dθα, dθβ ]− ..= dθαdθβ − dθβdθα = 0 dθαdθβ = dθβdθα, (3.7)

where, for notational convenience, we have left the supersymmetric product of two elements in Ω•Munderstood. While the first of these relations are just the characterizing relations of an exterior

12 SIMONE NOJA

algebra over an ordinary manifold, thus stating the anticommutativity of two 1-forms, the second ofthese relations, instead, implies for example that dθnα 6= 0 for any n ≥ 1 and for any α = 1, . . . ,m.It follows that, provided that the supermanifold has odd dimension greater or equal than 1, thereare non-zero form an any degree.We now introduce the following odd homomorphism acting on the de Rham superalgebra.

Definition 3.4 (de Rham differential). Let M be a real supermanifold of dimension n|m withstructure sheafOM . Given a section ω of the de Rham superalgebra Ω•M represented as ω = dyI⊗fIin a certain trivilization ya = xi|θα for some multi-index I with fI ∈ OM , we define the de Rhamdifferential d : Ω•M → Ω•M to be the odd derivation d ..=

∑a dya ⊗ ∂ya

. More precisely we have

d : Ω•M// Ω•M

dyI ⊗ fI // d(dyI ⊗ fI) =

∑a(−1)|ya||dy

I |dyadyI ⊗ ∂ya

(fI).

(3.8)

Remark 3.5. First of all it is easy to see that the d is well-defined, i.e. it is globally defined -this follows easily from the transformation properties of the dya’s and the ∂ya

’s -, it is odd since|dya| = |∂ya

|+1 and it is a derivation on the de Rham superalgebra, i.e. it satisfies the Z2-gradedLeibniz rule in the form

d(ωη) = (dω)ω + (−1)|ω|ω(dη), (3.9)

for ω and η any two forms in the de Rham superalgebra, and where |ω| is the parity of ω.Let indeed ω and η be represented as ω = dyI ⊗ fI and η = dyJ ⊗ gJ , for some multi-indices I andJ , then we have

d(ωη) = d((dyI ⊗ fI)(dyJ ⊗ gJ)) = (−1)|dy

J ||fI |d(dyIdyJ ⊗ fIgJ)

=∑

a

(−1)|dyJ ||fI |+|ya|(|dy

I |+|dyJ |)dyadyIdyJ ⊗ ∂ya

(fIgJ)

=∑

a

(−1)|dyJ ||fI |+|ya|(|dy

I |+|dyJ |)dyadyIdyJ ⊗

(∂ya

(fI)gJ + (−1)|ya||fI |fI∂ya(gJ )

)

=∑

a

(−1)|ya||dyI |(dyadyI ⊗ ∂ya

(fI))(dyJ ⊗ gJ)+

+∑

a

(−1)|xa||dyJ |+(|ya|+|ya|+1)(|fI |+|dy

I |)(dyI ⊗ fI)(dxadyJ ⊗ ∂ya

(gJ)) (3.10)

=∑

a

(−1)|ya||dyI |(dyadyI ⊗ ∂ya

(fI))(dyJ ⊗ gJ)+

+∑

a

(−1)|ya||dyJ |+|d||ω|(dyI ⊗ fI)(dxady

J ⊗ ∂xa(gJ))

= (dω)η + (−1)|ω|ω(dη), (3.11)

where we have used that ∂yais a (super)derivation of the structure sheaf OM and that |d| = 1. We

are now in the position to prove the following lemma.

Lemma 3.6. The pair (Ω•M , d) defines a differential graded superalgebra (DGsA).

Proof. We have already shown that d is a derivation of the de Rham superalgebra. We are left toprove that d is nilpotent, i.e. d2 = 0. To this end, we simply observe that (up to a constant)

d2 =∑

a,b

(dya ⊗ ∂ya)(dyb ⊗ ∂yb

) +∑

a,b

(dyb ⊗ ∂yb)(dya ⊗ ∂ya

)

=∑

a,b

(−1)|ya||dxb|dyadyb ⊗ ∂ya∂yb

+∑

i,j

(−1)|yb||dxa|(dybdya ⊗ ∂yb∂ya

)

=∑

a,b

((−1)|ya||yb|+|ya| + (−1)|ya||yb|+|ya|+1

)dyadyb ⊗ ∂ya

∂yb

= 0, (3.12)

where we have used that |dya| = |ya|+ 1 for any a even and odd.


Notice that, with a slight abuse of notation, we are confusing a DGsA with a sheaf of DGsA’s.The previous lemma justifies the following definition.

Definition 3.7 (de Rham Complex of M ). We call the differential graded superalgebra (Ω•M , d)the de Rham complex of M .

0 // OMd

// Ω1M

d// . . .

d// Ωn

M

d// . . . (3.13)

Remark 3.8. Once again notice that the de Rham complex of a supermanifold is not boundedfrom above ultimately due to the characteristic odd dimensions of the supermanifold itself, whichreverberate on the presence of commuting even differentials.

We are now interested into proving the main result concerning cohomology of the de Rham complexon a supermanifold, namely a generalization of the ordinary Poincare lemma.

Theorem 3.9 (Poincare Lemma for Differential Forms). Let M be a real supermanifold and let(Ω•M , d) be the de Rham complex of M . Then one has

Hkd (Ω

•M ) ∼=

RM k = 00 k > 0.

(3.14)

where RM is the sheaf of locally constant function on M . In particular:

(1) The de Rham complex is a (right) resolution of the sheaf RM ,

0 // RM// Ω•M . (3.15)

(2) Any closed form is locally exact on a real supermanifold.

Proof. We start observing that a form of zero degree is section f ∈ OM and, as in the ordinarysetting it is immediate to see that the request df = 0 forces f to be locally constant, so that onefinds indeed H0

d(Ω•M ) ∼= RM .

We now let ω ∈ ΩkM for k ≥ 1 and we show that for any k ≥ 1 there exists a homotopy hk : Ωk

M →Ωk−1

M for the differential d, i.e. a map such that

hk+1 dk + dk−1 hk = idΩk , (3.16)

where we have specified the degree involved for the sake of clarity, and where the maps go as follows

· · · // Ωk−1M

// ΩkM

id

hk

④④④④④④④④

dk

// Ωk+1M

hk+1④④④④④④④④

// · · ·

· · · // Ωk−1M

dk−1

// ΩkM

// Ωk+1M

// · · · .

(3.17)

Let us consider the following homotopy between the identity and the zero map in M :

G : [0, 1]× M // M

(t, ya ..= x1, . . . , xp|θ1, . . . , θq) // tya ..= (tx1, . . . , txp|tθ1, . . . tθq).(3.18)

This induces a map at the level of the de Rham complex via its pull-back

G∗ : ΩkM

// Ωk[0,1]×M

ω // G∗ω.

(3.19)

Now, writing the map G as a family of maps parametrized by t ∈ [0, 1], i.e. Gt : M → M , we canrewrite the pull-back above as a family of pull-back maps G∗t : Ωk

M → ΩkM .

We define the homotopy hk to be the map

ΩkM ∋ ω 7−→

∫ t=1

t=0

dt (ι∂tG∗t (ω)) ∈ Ωk−1

M . (3.20)

14 SIMONE NOJA

where ι∂tis the contraction with respect to the vector field ∂t on the interval [0, 1]. Now we

compute

(hk+1 dk + dk−1 hk

)(ω) =

∫ 1

0

dt(ι∂tG∗t (d

kω))+

∫ 1

0

dt(dk−1ι∂t

G∗t (ω))

=

∫ 1

0

dt(ι∂tdk + dk−1ι∂t

)G∗t (ω)

=

∫ 1

0

dtL∂tG∗t (ω)

=

∫ 1

0

dt ∂tG∗t (ω)

= G∗1(ω)−G∗0(ω). (3.21)

We now observe that, by definition, G∗0ω = 0 and G∗1ω = ω, so that(hk+1 dk + dk−1 hk

)(ω) = ω

for any ω ∈ Ωk>0M

. In particular, one has that Hk>0d (Ω•M ) = 0 for k > 0, and the cohomology is

concentrated in degree zero.

Remark 3.10. This result says something remarkable, but in some sense predictable: we cannotexpect new topological invariants arising from the odd part of the geometry of a supermanifold.Instead, the topology is fully encoded in the reduced manifold Mred . Intuitively, this is to beascribed to the characterizing nilpotentency of the odd part of geometry, which is not “strongenough” to modify rather rough invariants related to a geometric space, such as the topologicalones. In some circles this goes under the slogan “fermions are very small” - whatever it means.

Definition 3.11 (de Rham Cohomology of M ). Let M be a real supermanifold. We define the deRham cohomology of M to be the cohomology of the global sections of the (sheaf of) differentiallygraded superalgebras (Ω•M , d), i.e.

HkdR (M ) ..= Hk

d (H0δ(Ω•M )), (3.22)

where H0δ(Ω•M ) is 0-Cech cohomology group of Ω•M , i.e. the global sections or Ω•M .

Remark 3.12. Note that this is the usual definition of de Rham cohomology of a real manifoldupon considering an ordinary manifold instead of a supermanifold M . In particular, we have thefollowing easy consequence of the previous Poincare lemma 3.9.

Theorem 3.13 (Quasi-Isomorphism I). Let M be a real supermanifold and let Mred be its reducedmanifold. Then the de Rham complex of M is quasi-isomorphic to the de Rham complex of Mred .In particular, one has that

H•dR (M ) ∼= H•dR (Mred ). (3.23)

Proof. The theorem is an easy consequence of the Cech-to-de Rham spectral sequence for thedouble complex (Ω•M , δ, d), where δ is the Cech differential and d is the de Rham differential, see[14]. More in detail, the generalized Mayer-Vietoris short exact sequence (hence the existenceof a partition of unity) and Poincare lemma yield H•dR (Mred ) ∼= H•(|Mred |,RM ) in the ordinary

setting and, analogously, H•dR (M ) ∼= H•(|Mred |,RM ) in the supergeometric setting respectively,

where H•(|Mred |,RM ) is the Cech cohomology of the sheaf of locally-constant functions RM . Itfollows that

H•dR (M ) ∼= H•(|Mred |,RM ) ∼= H•dR (Mred ), (3.24)

thus concluding the proof.

In particular, one has a generalization of the ordinary Poincare Lemma to the “local model” realsupermanifold Rp|q.

Theorem 3.14 (Poincare Lemma for Rp|q). The de Rham cohomology of the supermanifold Rp|q

is concentrated in degree zero, i.e.

HkdR (R

p|q) ∼=

R k = 00 k > 0.

(3.25)


Proof. Follows immediately from the above 3.9 and 3.13.

Remark 3.15. Analogous results hold true for the compactly suppported de Rham cohomology ona supermanifold. More in details one finds that

H•dR ,c(M ) ∼= H•dR ,c(Mred ), (3.26)

so that in particular, one gets the compactly supported Poincare lemma for Rp|q, which reads

HkdR ,c(R

p|q) ∼=

R k = p0 k 6= p.

(3.27)

As can be imagined, a representative is given by the lift of the top form on Mred on the superman-ifold, i.e. dz1 . . . dzpBc(z1, . . . zp), where Bc is a bump function.

Remark 3.16. The above theorem 3.13 guarantees that even if the de Rham complex of a su-permanifold M of dimension n|m is not bounded from above, its de Rham cohomology groupsHk

dR (M ) can only be non-zero up to degree n, i.e. up to the degree which equals the even dimen-sion of the supermanifold or analogously the dimension of its reduced manifold Mred . All the otherhigher degree do not contributes to the de Rham cohomology or M . In other words, the de Rhamcohomology of the supermanifold can be non-zero only in the framed part of the complex

0 // Ω0M (M ) // Ω1

M (M ) // . . . // Ωn−1M (M ) // Ωn

M (M ) // Ωn+1M (M ) // . . .(3.28)

where we have denoted ΩkM (M ) the global sections of the sheaf Ωk

M for any k.

Remark 3.17. Finally, it is to be noted that the fact that the de Rham complex of a supermanifoldis not bounded from above implies - among things - that there is no tensor density playing therole of the top exterior bundle ΩdimX

X over an ordinary manifold X . This is the starting point ofthe quite unique integration theory on supermanifold, whose main character is introduced in thefollowing section.

4. The Berezinian Sheaf: Constructions and Geometry

In this section we introduce one of the crucial and most peculiar notion arising in supergeometry,that of Berezinian sheaf of a supermanifold, which we will denote as Ber(M ). An “operative” firstdefinition can be given by characterizing Ber(M ) in terms of its transition functions.

Definition 4.1 (Berezinian - via Transition Functions). Let M be a real or complex superman-ifold of dimension n|m and let (Ui, xUi

|θUi)i∈I be an atlas of charts covering M . Then we

define the Berezinian sheaf Ber(M ) of M as the locally-free sheaf of right OM -modules of rankδ0,n+m|δ0,n+m+1, whose local generators DUi

(xUi, θUi

)i∈I transforms as

DUj⌊Ui∩Uj(xUj

|θUj) = DUi⌊Ui∩Uj

(xUi|θUi

)Ber(J ac(ϕij)) (4.1)

with

Ber(J ac(ϕij)) = det(A−BD−1C) det(D)−1, (4.2)

where J ac(ϕij) is the super Jacobian of the change of coordinates

ϕij : Ui⌊Ui∩Uj// Uj⌊Ui∩Uj

xUi|θUi

// ϕij,0(x|θ) = xUj|ϕij,1(x|θ) = θUj

,

and where we have posed(A BC D

)..=

(∂xϕij,0 ∂θϕij,0

∂xϕij,1 ∂θϕij,1

)= J ac(ϕij). (4.3)

Beside its importance in relation to the quite unique and highly non-trivial integration theory onsupermanifolds [74, 76], the Berezinian sheaf deserves a special attention on its own. In whatfollows we will present and review three of its constructions, all of them inspired to algebraicgeometric methods.The first one, due to Hernandez Ruiperez and Munoz Masque [62], is a beautiful realization of theBerezinian sheaf of a real smooth supermanifold as a certain quotient of natural sheaves, whichhas the merit of being relatively easy and to make apparent its relation with integration theory.

16 SIMONE NOJA

The second construction that we will present has the complex analytic or algebraic category insight: in this context the Berezinian sheaf emerges from the cohomology of a supergeometricgeneralization of the ordinary Koszul complex. As mentioned in the introduction, the original ideais due to Ogievetsky and Penkov, and it appeared first in [58]. Later on, building upon [58], thesuper Koszul complex has been briefly sketched by Manin in [53]. Quite recently a self-containedand encompassing discussion of the topic has been given by the author and Re in [57].

4.1. Berezinian Sheaf as a Quotient Sheaf. Working on a real smooth supermanifold of di-mension n|m, the construction of the Berezinian sheaf of M as a quotient sheaf is obtained starting

from the sheaf ΩnM ,c⊗OM

D(m)M of differential operatorsD(m)

M of degreem onOM taking values in (the

sheaf of) compactly supported differential forms ΩnM ,c of degree n. We denote this (locally-free) sheaf

of OM -modules as D(m)M

(ΩnM ,c) for short, and its elements will be written as ω ⊗ F ∈ D(m)

M(Ωn

M ,c).

Notice that, locally over an open set U with coordinates ya = xi|θα, for i = 1, . . . , n andα = 1, . . . ,m a system of generators over OM is given by

D(m)M (Ωn

M ,c)(U) =

dyj1 . . . dyjn︸︷︷︸

n

⊗ ∂

∂yk1

. . .∂

∂ykm︸︷︷︸m

· OM (U) ..=

dyIj ⊗

∂

∂yIk

· OM (U) (4.4)

where Ij and Ik are two multi-indices such that |Ij | = n and |Ik| = m and where D(m)M (Ωn

M ) hasbeen considered with the structure of right OM -module.

The key object in the construction is a pretty subtle sub-sheaf of D(m)M (Ωn

M ,c). If ι : Mred → M is the

embedding of the reduced manifold into the supermanifold M as in (2.8), we have a correspondingpull-back map ι∗ : Ω•M ,c → Ω•Mred ,c

on differential forms. We introduce a sheaf of OM -modules KM

as the sub-sheaf of sections of D(m)M (Ωn

M ,c) having the following property

KM (U) ..=ω ⊗ F ∈ D(m)

M(Ωn

M ,c)(U) : ∃ η ∈ Ωn−1Mred ,c

(U) : ι∗(ω ⊗ F (f)) = dη ∀f ∈ OM ,c

, (4.5)

where U is an open set and f ∈ OM ,c(U) and η ∈ Ωn−1Mred ,c

(U) are a compactly supported function and

a compactly supported form respectively. Notice that the above is well-defined as F (f) ∈ OM ,c .Then one can prove the following theorem, see [62].

Theorem 4.2 (Berezinian as a Quotient). Let M be a real supermanifold of dimension n|m and

let D(m)M

(ΩnM ,c) and KM be defined as above. Then the sheafification of the quotient pre-sheaf

D(m)M

(ΩnM ,c)/KM is a locally-free sheaf of (right) OM -module of rank δ0,n+m|δ0,n+m+1, whose gen-

erator reads on an open set U with coordinates xi|θα for i = 1, . . . , n and α = 1, . . . ,m

D(m)M (Ωn

M ,c)(U)/KM (U) ∼=

[dx1 . . . dxn ⊗ ∂

∂θ1. . .

∂

∂θm

]· OM (U), (4.6)

where the square bracket stays for the class of the form-valued differential operator modulo KM .In particular, the above quotient is naturally isomorphic to the Berezinian sheaf of the supermani-fold, i.e.

Ber(M ) ∼= D(m)M (Ωn

M ,c) /KM . (4.7)

Proof. We consider ω ⊗ F ∈ D(m)M

(ΩnM ,c) and we work in a coordinate system xi|θα over an open

set U , such that D(m)M (Ωn

M ,c) has a basis as above in (4.4). Let us consider the following instances.If it appears a term of the form dθnα for any α and any n ≥ 1 in ω, then the corresponding ω ⊗ Fgoes to zero under ι∗, and as such it is in KM . So this force ω to be of the kind dx1 . . . dxn.Now consider ω ⊗ F to be of the kind ω ⊗ F = dx1 . . . dxn ⊗ ∂xI

∂θJ for some multi-indices Iand J such that |I| + |J | = m. If I 6= 0 then ω ⊗ F ∈ KM (U), indeed consider for exampledx1 . . . dxn⊗∂θ2 . . . ∂θm∂x1 : the crucial case is that of f of the form f(x|θ) = gc(x)θ2 . . . θm, with gc

a compact supported function on U to get ω⊗F (f) = dx1 . . . dxn∂x1gc(x) = d(x1dx2 . . . dxngc(x)),which implies ω⊗F ∈ KM . This is enough to prove that the classD(x|θ) ..= [dx1 . . . dxn⊗∂θ1 . . . ∂θm ]defines a generator for the above quotient sheaf. Also, this can indeed be identified with theBerezinian sheaf, upon checking that the class D(x|θ) transform indeed as in (4.2) under a change of


local coordinates. This is a local check: a careful computation is postponed to the next subsection,in a slightly different context.

Remark 4.3. As said above, the previous intrinsic construction of the Berezinian as the sheafifica-tion of a quotient pre-sheaf of differential operators valued into differential forms has the unques-tionable merit of being relatively easy and, at the same time, as explained at the end of the secondsection of [62], it makes the relation with the Berezinian sheaf and the related integration theoryapparent. A minor drawback of the construction is that it only holds true for real supermanifolds,as the existence of compactly supported function is crucial to the above proof.

4.2. Berezinian Sheaf from Koszul Complex. Homological algebra comes in help to providean intrinsic construction of the Berezinian sheaf on complex or algebraic supermanifolds, wherecompactly supported functions are not available and therefore the above quotient constructionbreakdowns. As hinted in the introduction to this section, the idea of a suitable generalization toa supergeometric setting of the Koszul complex originally appeared in [58] and was subsequentlyin [53]. Very recently an encompassing construction has been given in [57].We let E be any locally-free sheaf of rank p|q on a supermanifold M . We define

R ..=⊕

k≥0

REk with REk ..= SkE (4.8)

RΠ ..=⊕

k≥0

RΠEk with RΠE

k..= SkΠE , (4.9)

and in turn we consider the following sheaf of OM -superalgebras given by the tensor product

KE• ..=⊕

k≥0

K−k = R⊗OM

⊕

k≥0

RΠk = R⊗OM

RΠ. (4.10)

Further, let us consider two basis of local generators for E and ΠE respectively given by vi|χαand a πχα|πvi. Using these we can define the following operator acting on KE• :

δ : KE• = R⊗OMRΠ // KE• = R⊗OM

RΠ

r ⊗ rΠ // δ(r ⊗ rπ) ..=

(∑pi=1 vi ⊗ ∂πvi +

∑qj=1 χj ⊗ ∂πχj

)(r ⊗ rΠ),

(4.11)

where r ∈ R and rΠ ∈ RΠ. Since the derivations can be seen as the dual bases to the bases of Eand ΠE it is not hard to see that the above is globally well-defined and independent of the choiceof local bases. Further, δ is homogenous of degree −1 with respect to the Z-gradation of K• seenas a sheaf of R-modules and it is odd with respect to the Z2-graded structure on KE• : more inparticular it is not hard to show that it is nilpotent, i.e. δ δ = 0, so that the pair (KE• , δ) definesa differentially graded sheaf of R-algebras. We can thus give the following definition.

Definition 4.4 (Super Koszul Complex). Let M be a real, complex or algebraic supermanifold.Given any locally-free sheaf of OM -modules E , we call the pair (KE• , δ) defined as above the superKoszul complex associated to E :

· · · δ// R⊗ SkΠV δ

// · · · δ// R⊗ S2ΠE δ

// R⊗ΠE δ// R // 0. (4.12)

One of the main result of [57] is concerned with the homology of this complex of sheaves.

Theorem 4.5 (Homology of the Super Koszul Complex). Let M be a real, complex or algebraicsupermanifold with structure sheaf OM , let E be a locally-free sheaf of OM -modules on M and let(KE• , δ) be the super Koszul complex associated to E , defined as above. Then the super Koszulcomplex (KE• , δ) is an exact resolution of OM endowed with the structure of sheaf of R-modules,i.e. the homology Hi(KE• , δ) is concentrated in degree 0,

Hi

((KE• , δ)

) ∼=

OM i = 00 i 6= 0.

(4.13)

Remark 4.6. With reference to the above result, notice that OM is indeed a R-module thanks tothe following short exact sequence of sheaves

0 // IR // R // OM// 0, (4.14)

18 SIMONE NOJA

where IR ..=⊕

k≥1 SkE is the sheaf of ideals of R generated by E ⊂ R so that OM

∼= R /IRR .

Remark 4.7. The Koszul resolution of theorem 4.5 allows to compute other derived functors ina supergeometric context. In particular, given the Koszul super complex KE• as above, one canintroduce the dual construction via the functor HomR(−,R), which yields the pair (KE∗• , δ∗) ..=(HomR(KE• ,R),HomR(δ,R)). Defining

RΠ∗k

..=⊕

k≥0

Rπ∗i with RΠ∗

k..= SkΠE∗ (4.15)

the sheaf of OM -superalgebras generated by ∂πχα|∂πvi for i = 1, . . . , p and α = 1, . . . , q in ΠE∗,

it is easy to see that

KE∗• ..=⊕

k≥0

KE∗k = R⊗OM

⊕

k≥0

RΠ∗k = R⊗OM

RΠ∗. (4.16)

The fundamental observation is that KE∗• is acted by an operator δ∗, whose definition is formallyidentical to that of δ given above in (4.11). But here δ∗ has to be looked at as a multiplicationoperator by the element

∑pi=1 vi⊗∂πvi +

∑qj=1 χj ⊗∂πχj

in KE∗• . This is enough to guarantee thatδ∗ δ∗ = 0, as the element corresponding to δ∗ is odd, so that we can introduced the following.

Definition 4.8 (Dual of the Super Koszul Complex). Let M be a real, complex or algebraicsupermanifold. Given any locally-free sheaf of OM -modules E , we call the pair (KE∗• , δ∗) defined asabove the dual of the super Koszul complex associated to E :

0 // R δ∗// R⊗ΠE∗ δ∗

// R⊗ S2ΠE∗ δ∗// . . .

δ∗// R⊗ SkΠE∗ δ∗

// . . . (4.17)

Now, we aim at computing the (co)homology of the dual of the super Koszul complex.Recalling that by definition we have KE∗• ..= HomE(KE• ,R), then

ExtiR(OM ,R) = Hi((KE∗• , δ∗)

). (4.18)

The homology (sheaf) of dual of the Koszul complex is computed in the following theorem, see[57].

Theorem 4.9 (Homology of the dual of the Super Koszul Complex). Let M be a real, complex oralgebraic supermanifold with structure sheaf OM , let E be a locally-free sheaf of OM -modules on M

and let (KE• , δ) be the dual of the super Koszul complex associated to E , defined as above. Then itshomology is concentrated in degree p and locally-generated over OM by the class

ExtpR(OM ,R) ∼= [χ1 . . . χq ⊗ ∂πv1 . . . ∂πvp ] · OM (4.19)

where χ1 . . . χq ∈ SqE and ∂πv1 . . . ∂πvp ∈ SpΠE∗.

Proof. It is immediate to observe that the element D ..= χ1 . . . χq ⊗ ∂πv1 . . . ∂πvp ∈ SqE ⊗ S

pΠE∗belong to the kernel of δ∗. We can then observe that locally R ⊗OM

RΠ∗ is generated overOM by the elements (v1, . . . vp, ∂πχ1 , . . . , ∂πχq

|∂πv1 , . . . , ∂πvp , χ1, . . . , χq). Posing N ..= p + q,

we can redefine the generators as (s1, . . . , sN ) ..=(v1, . . . vp, ∂πχ1 , . . . , ∂πχq

)and (ψ1, . . . , ψN ) ..=(

∂πv1 , . . . , ∂πvp , χ1, . . . , χq

), so that in particular one has that δ∗ =

∑Ni=1 uiψi and D =

∏Nj=1 ψi

and R⊗RΠ∗ becomes a sheaf of exterior algebras

(R⊗RΠ∗

)(U) ∼=

•∧

O(U)[s1,...,sN ]

(ψ1, . . . , ψN ) (4.20)

the over the commutative ring O(U)[s1, . . . , sn]. This is the dual of the ordinary commutativeKoszul complex, whose cohomology is concentrated in top-degree, in this case N , and generatedover OM by the element D = ψ1 . . . ψN , see [31].

Remark 4.10. The crucial point is now that the class singled out by the dual of the Koszul complextransforms by the multiplication by the Berezinian of an automorphism of the sheaf E . Moreprecisely, we prove the following result.


Theorem 4.11. Let M be a real, complex or algebraic supermanifold with structure sheaf OM , letE be a locally-free sheaf of OM -modules on M and let φ ∈ Aut(E) be an automorphism of E. Thenthe induced automorphism ϕ ∈ Aut(ExtpR(OM ,R)) is given by the multiplication by the inverse ofthe Berezinian of the automorphism, i.e.

ϕ : ExtpR(OM ,R) // ExtpR(OM ,R)

D // D · Ber(ϕ)−1(4.21)

Proof. Fixing a local system of generator for E given by v1, . . . , vp|χ1, . . . , χq, then ϕ ∈ Aut(E)is represented by a matrix [M ] ∈ GL(p|q,OM (U))

[ϕ]αβ =

(A BC D

)=

(ahi bhjcki dkj

)(4.22)

with A,B even and B,C odd submatrices. Now, if ϕi for i = 1, 2 are automorphisms of E ,contravariant functoriality of the construction, see [57], implies that the product of two matrices[ϕ1] · [ϕ2] corresponds to the product ϕ[ϕ2] ·ϕ[ϕ1], which acts an an automorphisms of ExtpR(OM ,R).If follows that we can use the standard decomposition

(A BC D

)=

(1 BD−1

0 1

)(A−BD−1C 0

0 D

)(1 0

D−1C 1

)(4.23)

and considering separately the cases.

(1) : [ϕ] =

(A 00 D

), (2) : [ϕ] =

(1 0∗ 1

), (3) : [ϕ] =

(1 ∗0 1

). (4.24)

It is easy to see that only in the first case we have an induced transformation of the homology

class D = [χ1 . . . χq ⊗ ∂πv1 . . . ∂πvp ] given by the multiplication by det (D) · det (A)−1. In the tworemaining cases the homology class D is invariant. It follows from the above decomposition thatϕM(ϕ) = det(D) det(A−BD−1C)−1, which is indeed Ber([ϕ])−1 as claimed.

Remark 4.12. Notice that what plays a crucial role in the previous proof is the fact that D is acohomology class, so that parts of the transformation on the representative yield exact elements,which do not contribute to the induced automorphism.

Remark 4.13. Clearly, it is enough to consider E∗ ..= Hom(E ,OM ) instead of E and the relatedhomology of the dual of the super Koszul complex as to obtain the expected transformation

ϕE∗ : ExtpS•E∗(OM , S

•E∗) // ExtpS•E∗(OM , S

•E∗)D∗ // D∗ · Ber(ϕE )

(4.25)

where ϕE is an automorphism of E . This suggests that given a locally-free sheaf E on a real,complex or algebraic supermanifold, one can see the previous construction as a defining one forthe notion of Berezinian sheaf of E , by posing

Ber(E) ..= ExtpS•E∗(OM , S

•E∗) (4.26)

In particular, as to make contact with the previous subsection, we give the following definitionwith agrees also with that of Manin [53] and Witten [76].

Definition 4.14 (Berezinian of a Supermanifold). Let M be a real, complex or algebraic super-manifold with structure sheaf OM . We call the Berezinian sheaf of M and we denote it by Ber(M )the locally-free sheaf of OM -modules of rank δ0,n+m|δ1,n+m defined by

Ber(M ) ..= Ber(Ω1M ), (4.27)

where Ber(Ω1M ) = Extp

S•(Ω1M)∗(OM , S

•(Ω1M )∗).

Remark 4.15. Notice that, as above, the Berezinian sheaf of M is locally generated by the class[dx1 . . . dxp⊗∂θ1 . . . ∂θq ] and that it can be equivalently defined as Ber(M ) ..= Ber(TM ), since if A is

an automorphism, then Ber(Ast) = Ber(A), Ber(A−1) = Ber(A)−1 and Ber(ΠA) = Ber(A)−1.

20 SIMONE NOJA

4.3. Berezinian Sheaf from Cohomology of Forms and Operators. The last constructionof the Berezinian sheaf that we are to discuss stands somewhat in between the previous ones, asit employs the sheaf DM to “deform” in a non-commutative fashion the previous Koszul complexconstruction [16]. In particular we consider the following tensor product of sheaves.

Definition 4.16 (Universal de Rham Sheaf). Let M be a real or complex supermanifold. We callthe tensor product sheaf DR M

..= Ω•M ⊗OMDM the universal de Rham sheaf of M .

Remark 4.17. It is easy to see that DR M is both Z-graded and Z2-graded. Further, it is a sheaf ofleft Ω•M -modules - and hence also left OM -modules - and a sheaf of right DM -modules. Notice, bythe way, that the structure of right OM -module induced by DM does not coincide with that of leftOM -module, since DM is non-commutative.

Remark 4.18. There is an obvious operator acting on DR (M ), whose action is given by

D : DR M// DR M

ω ⊗ F // D (ω ⊗ F ) ..=∑

a(−1)|ω||xa|dxaω ⊗ ∂xa F,

(4.28)

for any ω ⊗ F ∈ DR M . It is not hard to prove that D is globally well-defined - as the dx’s and the∂xa

’s transform dually - and that for any f ∈ OM one has that indeed D(ωf ⊗ F ) = D(ω ⊗ fF ).Further, it is immediate to observe that the operator D is nilpotent, since it can be see as themultiplication by the odd element

∑a dxa ⊗ ∂xa

∈ DR M , i.e.

D(ω ⊗ F ) ..=

(∑

a

dxa ⊗ ∂xa

)· (ω ⊗ F ). (4.29)

We thus have that the pair (DR •M ,D) defines a complex of sheaves whose Z-grading is induced bythe one of Ω•M . The cohomology of this complex provides another construction of the Bereziniansheaf of M .

Theorem 4.19 (Cohomology of DR •M ). Let M be a real or complex supermanifold of dimensionp|q. Then the homology of the complex (DR •M ,D) is naturally isomorphic to the Berezinian sheafof M , i.e.

Hp ((DR •M ,D)) ∼= Ber(M ) (4.30)

and Hi ((DR •M ,D)) ∼= 0 for any i 6= p.

Proof. We need to construct a homotopy for the operator D . We work in a local chart (U, xa) sothat the sheaf DR M

..= Ω•M ⊗OMDM is given by the sheaf of vector spaces generated by monomials

having the form ω ⊗ F , with ω = dxI and F = ∂Jf for multi-indices I and J and some functionf ∈ OM ⌊U . We claim that the homotopy is given by the (local) operator

H (ω ⊗ F ) ..=∑

a

(−1)|xa|(|ω|+|∂J |+1)ιπ∂xa

dxI ⊗ [∂J , xa]f. (4.31)

where the derivation ιπ∂xa

..= ∂dxais the contraction with respect to the coordinate field π∂a, so

that ιπ∂xa(dxb) = δab. A lengthy but not too hard computation gives

(H D + DH )(ω ⊗ F ) = (p+ q + deg0(ω) + deg0(∂J )− deg1(ω)− deg1(∂J)) (ω ⊗ F ), (4.32)

where deg0 and deg1 are the degree with respect to the even and odd generators. We have thatdeg1(ω) ≤ p and deg1(∂J ) ≤ q, therefore the homotopy fails if and only if deg0(ω) = 0 = deg0(∂J )and deg1(ω) = p, deg1(∂

J) = q : the monomial ω ⊗ F is given by dz1 . . . dzp ⊗ ∂θ1 . . . ∂θqf forf ∈ OM ⌊U . This element is clearly in the kernel of D and it generates the Berezinian sheaf Ber(M )as OM -module.

Remark 4.20. Notice, once again, that also this construction holds true in the smooth and holo-morphic category, but also in the algebraic category.


5. Properties of the Berezinian Sheaf

Having defined the Berezinian sheaf in the previous section, we are now interested into studyingits properties. In the first subsection, in theorem 5.3, we will show how the Berezinian sheaf of asupermanifold M is related to the canonical sheaf of its underlying manifold Mred : this will provecrucial in the definition of the Berezin integral. Further, in the second subsection, we will showthat the Berezinian sheaf is a right DM -module. This theory has been first developed by Penkovin the marvelous [60], where Serre duality - see also [58] - and Mebkhout duality for complexsupermanifolds are also proved. Subsequently, results in this direction appeared also in the book[53], however the relation with DM -module theory is left somewhat hidden. In this section we makethis connection apparent, by spelling out all of its details and stressing differences and similaritieswith the ordinary commutative theory. Indeed, the presence of a right DM -module structure onthe Berezinian sheaf is another striking analogy between the Berezinian sheaf and its commutativecounterpart, the canonical sheaf on an ordinary manifold X , which carries as well the structure ofright DX -module. More precisely we will see that in a similar fashion as in ordinary commutativetheory, the right DM -module structure on Ber(M ) is related to the action of the Lie derivative onit.

Remark 5.1. As it should be clear from the previous section, it has to be noticed that the Bereziniansheaf is not a sheaf of differential forms, and as such it does not appear in the de Rham complexΩ•M of M . It follows that it is not trivial to define a notion of Lie derivative acting on sections ofBer(M ). Indeed, the so-called Cartan formula LXω = d, ιX(ω) which holds true for differentialforms ω ∈ Ωi

X for any i = 0, . . . , dimX and can be readily generalized to the de Rham complex ofa supermanifold, does not apply to the Berezinian.

5.1. Berezinian and Canonical Sheaf. In this first subsection we prove an easy, yet very im-portant isomorphism, which establishes a crucial relation between the Berezinian sheaf Ber(M )ofa supermanifold M of dimension p|q and the canonical sheaf Ωp

Mredof the reduced space of Mred .

We start by proving the following ancillary result, see for example [53].

Lemma 5.2. Let M be a real or complex supermanifold. Then we have the following isomorphismof sheaves of OMred

-modules

FM∼= Π

(Ω1

M

/JM Ω1

M

)0

(5.1)

where the subscript 0 refers to the Z2-grading of the quotient sheaf Ω1M /JM Ω1

M .

Proof. First of all, notice that locally, a basis of FM = JM/J 2

Mis given by θα modJ 2

M for

α = 1, . . . , q where q is the odd dimension of M . Further, locally, a basis of(Ω1

M /JM Ω1M

)1

read dθαmodJM Ω1M , again for α = 1, . . . , q where m is the odd dimension of M . We claim that

the isomorphism reads θj modJ 2M 7→ dθj modJM Ω1

M , and we prove that it is well-defined andindependent of the charts. Indeed, let x′a = zi|θ′α be another local system of coordinates, then thetransformation for the transformation of FM = JM /J 2

M one has that θ′α ≡ ∑β fαβ(x)θβ modJ 2M .

It follows that

dθ′α =∑

b

dxb∂θ′α∂xb

+∑

β

dθβ∂θ′α∂θβ

=∑

b

dxb∂

∂xb

(∑

γ

fαγ(x)θγ modJ 2M

)+∑

β

dθβ∂

∂θβ

(∑

γ

fαγ(x)θγ modJ 2M

)

=∑

b,γ

dxb∂fαγ(x)

∂xbθγ modJ 2

M +∑

β

dθβfαβ(x)modJ 2M

≡∑

β

dθβ mod(JM Ω1

M

)fαβ(x), (5.2)

since∑

b,γ dxb∂fαγ (x)

∂xbθγ modJ 2

M ≡ 0modJM Ω1M . Reversing the parity of the local generators dθβ

concludes the proof.

Using the above lemma we prove the result we claimed at the beginning of the subsection.

22 SIMONE NOJA

Theorem 5.3. Let M be a real or complex supermanifold and let JM ⊂ OM be its nilpotent sheaf.Then there is a canonical isomorphism of sheaves of OMred

∼= OM /JM -modules

ϕ : J qM Ber(M )

∼=−→ ΩpMred

, (5.3)

where Ber(M ) is the Berezinian sheaf of M and ΩpMred

is the canonical sheaf of Mred . In local

coordinates xa = zi|θα the above isomorphism reads

D(x)θ1 . . . θqf∼=7−→ dz1,red . . . dzp,redfred, (5.4)

where D(x) ∈ Ber(M ), θ1 . . . θp ∈ J qM , dz1,red ∧ . . . ∧ dzp,redfred ∈ Ωp

Mredand where f ∈ OM and

fred = f modJM ∈ OMred.

Proof. First of all we observe that J qMBer(M ) is obviously a sheaf of OMred

-modules, as J qM

∼=J q

M/J q+1

Mis a sheaf of OMred

-modules. Also, it is clear that if a generating section θα ∈ JM /J 2M is

such that θ′α ≡∑β fαβ(z)θβ modJ 2M then

J qM

∋ θ1 . . . θq = det(fαβ)θ1 . . . θq = det(fαβ)redθ1 . . . θq, (5.5)

Now, considering local coordinates xa = zi|θα, for an index a running on both even and oddcoordinates, for a generic change of coordinates x′a = z′i(x)|θ′α(x) one has that

D(x′)θ′1 . . . θ′q = D(x)Ber

(∂x′

∂x

)

red

det(fαβ)red θ1 . . . θq

= D(x) det

(∂z′

∂z

)

red

det

(∂θ′

∂θ

)−1

red

det(fαβ)red θ1 . . . θq. (5.6)

It follows from the previous lemma 5.2 that det(∂θθ′)red = det(fαβ)red , which concludes the theorem

since det(∂zz′)red is indeed the transformation of a generating section of the canonical sheaf Ωp

Mred

of the reduced manifold.

Remark 5.4. The above theorem holds true in exactly the same forms considering the compactlysupported case instead: this will prove crucial to define a meaningful notion of integral on super-manifolds, as we shall see later on in this paper.

5.2. DM -modules and Connections. It is an easy yet fundamental result of D-module theorythat giving a D-module structure on a sheaf corresponds to define a flat connection on it. Nonethe-less, as observed above, attention must be paid due to the non-commutativity of the sheaf D sothat one has to distinguish between left and right action and thus left and right D-module struc-tures: accordingly, we will introduce left and right connections on sheaves and the related notionof flatness.Let us start from left D-modules. In this case the needed left action is induced by a left connection:this is nothing but the standard notion of affine, or Koszul connection on a sheaf introduced inordinary differential geometry. In particular we will work over a real or complex supermanifold,and thus the base field will be either K = R or K = C. The notations employed mostly follow [53],in particular for any f ∈ OM and X ∈ TM we denote the commutator [X, f ] = X(f) as

[X, f ] ..= X f − (−1)|X||f |f X, (5.7)

as to stress that we are considering the operator product in DM .

Definition 5.5 (Left Connection on E). Let M be a real or complex supermanifold with structuresheaf OM and let E be a sheaf of OM -modules. Then we say that a left connection on E is aK-bilinear morphism ∆L : F 1DM ⊗K E → E such that the following are satisfied for any f ∈ OM ,X ∈ TM and e ∈ E :

(1) ∆L(f ⊗ e) = fe,(2) ∆L(f X ⊗ e) = f∆L(X ⊗ e),(3) ∆L(X f ⊗ e) = ∆L(X ⊗ fe).

In particular, we say that the left connection ∆L is flat if for any e ∈ E and X,Y ∈ TM it satisfies

∆L([X,Y ]⊗ e) = ∆L(X ⊗∆L(Y ⊗ e))− (−1)|X||Y |∆L(Y ⊗∆L(X ⊗ e)). (5.8)


Remark 5.6. It is to be noted that the previous definition is adapted as to serve the needs ofD-module theory. In particular, notice that since F 1DM = OM ⊕ TM generates DM , the definitionhas been extended as to include also the action of the structure sheaf OM ∈ F 1DM on E in thefirst point. The second point is the usual OM -linearity in the first entry of the connection and thethird point is the Leibniz rule, since

∆L(X f ⊗ e) = ∆L

((X(f) + (−1)|f ||X|f X

)⊗ e)

= ∆L(X(f)⊗ e) + (−1)|f ||X|∆L(f X ⊗ e)

= X(f)e+ (−1)|f ||X|f∆L(X ⊗ e). (5.9)

Finally, the condition of flatness, or integrability for a connection can be rephrased by saying thatthe connection commutes with the operation of commutator, i.e.

∆L([X,Y ]⊗−) = [∆L(X ⊗−),∆L(Y ⊗−)], (5.10)

for any X,Y ∈ TM .We can now state the following crucial result.

Theorem 5.7 (Left DM -Modules & OM -Modules). Let M be a real or complex supermanifold andlet E be a sheaf on M . Then E is a sheaf of left DM -module on M if and only if E is a sheaf ofOM -modules endowed with a flat left connection.

Remark 5.8. The proof of the above theorem is obvious and it simply amounts to check thatassociativity of the left DM -action on E is reproduced by the flat connection. We will prove insteadthe analogous in case of right DM -modules, which is equally simple but less ordinary, and mightcause some confusion.

Remark 5.9. Notice also that the above theorem does not require the sheaf E of OM -modules tobe locally-free of finite rank, therefore one can look at a DM -module as a generalization of a vectorbundle endowed with a a flat connection.The next corollary gives an obvious example of sheaf of left DM -modules.

Corollary 5.10 (Structure Sheaf OM ). Let M be a real or complex supermanifold and let OM beits structure sheaf. Then OM is a sheaf of left DM -modules.

Proof. Obviously, OM can be endowed with a flat left connection, which is nothing but the exteriorderivative, seen as a map d : TM ⊗ OM → OM via the X ⊗ f 7→ (df)(X) and where the actionOM ⊗OM → OM is given by the superalgebra structure of OM . Flatness is obvious.

Let us now pass to the case of right DM -module. In order to prove the analogous result of theorem5.7 for right DM -modules we need to introduce a different kind of connection, which is to be relatedto a right action. As in [53], employing the same notation as above we have the following.

Definition 5.11 (Right Connection on E). Let M be a real or complex supermanifold with struc-ture sheaf OM and let E be a sheaf of OM -modules. Then we say that a right connection on E is aK-bilinear morphism ∆R : E ⊗K F

1DM → E such that the following are satisfied for any f ∈ OM ,X ∈ TM and e ∈ E :

(1) ∆R (e⊗ f) = ef ;(2) ∆R (e⊗X f) = ∆R (e ⊗X)f ;(3) ∆R (e⊗ f X) = ∆R (ef ⊗X),

In particular, we say that the right connection ∆R is flat if for any e ∈ E and X,Y ∈ TM it satisfies

∆R (e ⊗ [X,Y ]) = ∆R (∆R (e⊗X)⊗ Y )− (−1)|X||Y |∆R (∆R (e⊗ Y )⊗X). (5.11)

Remark 5.12. Notice that the third point in the definition is OM -linearity and that we have amodified Leibniz rule, adapted to right structures. Indeed, one has

∆R (e ⊗X f) ..= ∆R (e⊗ (X(f) + (−1)|X||f |f X))

= ∆R (e ⊗X(f)) + (−1)|X||f |∆R (e ⊗ f X)

= eX(f) + (−1)|X||f |∆R (ef ⊗X), (5.12)

24 SIMONE NOJA

so that the second property above can be rewritten as

∆R (ef ⊗X) = (−1)|X||f |(∆R (e ⊗X)f − eX(f)). (5.13)

Using right connections we can prove the following.

Theorem 5.13 (Right DM -Modules & OM -Modules). Let M be a real or complex supermanifoldand let E be a sheaf on M . Then E is a sheaf of right DM -module on M if and only if E is a sheafof OM -modules endowed with a flat right connection.

Proof. The right DM -module structure on E corresponds to a right action

σR : E × DM// E

(e, F ) // e · F ..= σR (e, F )

(5.14)

In particular, associativity reads

σR (σR (e,D), H) = σR (e,D H) (5.15)

or analogously (e ·D) ·H = e · (D H) for e ∈ E and D,H ∈ DM . Let us define σR as acting byright multiplication on functions. Then for X ∈ TM and f ∈ OM we have that associativity reads

σR (e, f X) = σR (σR (e, f), X) = σR (ef,X), (5.16)

which is last of the defining condition for a right connection. Furthermore, we have

σR (e,X(f)) = σR (e, (X f − (−1)|X||f |f X))

= σR (e,X f)− (−1)|X||f |σR (ef,X)

= σR (σR (e,X), f)− (−1)|X||f |σR (ef,X)

= σR (e,X)f − (−1)|X||f |σR (ef,X), (5.17)

where we have used linearity and associativity. On the other hand, by definition σR (e,X(f)) =eX(f), so that

σR (ef,X) = (−1)|X||f |(σR (e,X)f − eX(f)), (5.18)

which is the modified Leibniz rule and it corresponds to second defining condition of a rightconnection. Lastly,

σR (e, [X,Y ]) = σR (e,X Y − (−1)|X||Y |Y X)

= σR (σR (e,X), Y )− (−1)|X||Y |σR (σR (e, Y ), X) (5.19)

which is the requirement of flatness. Notice that, conversely, starting from a right connection, itis enough to prove that it defines an associative right action on the generating F 1DM

∼= OM ⊕TM ,with [X, f ] = X(f) to have a right action of DM .

Just like above for left DM -modules, the above theorem give an alternative characterization of rightDM -modules. Notice, also, that the notion of right connection is not at all exotic as it might soundat first: namely working over a ordinary real or complex manifold X it is easy to prove that - upto a sign - the Lie derivative defines a right connection on the canonical sheaf ωX

..= ∧dimXT ∗X ,which is therefore a sheaf of right DX -modules.

Lemma 5.14 (ωX is a Right DX -module). Let X be a real or complex manifold and let ΩdimXX be

its canonical sheaf. Then ΩdimXX is a sheaf of right DM -module.

The proof of this theorem, together with a detailed discussion about the relation between the Liederivative and the right DX -module structure of the canonical sheaf of an ordinary manifold, isdeferred to the appendix, as to keep the focus on the case of supermanifolds in the main text.

In light of the previous section and the definition of the Berezinian sheaf of a supermanifold asthe correct super-analog of the notion of canonical sheaf for an ordinary manifold, it is naturalto ask if the DM -module property of lemma 5.14 goes through to the super setting and also theBerezinian sheaf is a sheaf of right DM -modules. We will prove that this is indeed the case in thenext section.


5.3. Lie Derivative of Ber(M ) and Right DM -module Structure. Before we actually computethe action of the Lie derivative on a section of the Berezinian sheaf of a supermanifold, we startwith some remark concerning the relation between left and right structure on a supermanifold. Inparticular, let E be a locally-free sheaf of left OM -modules of rank p|q which is generated over anopen set U by a set eaa∈I of p even and q odd generators. Then E is naturally also a locally-freesheaf of right OM -modules simply taking into account the sign rule, i.e. given a local section sU ∈ Eover an open set U such that sU =

∑a f

aea for fa ∈ OM (U), then

sU =∑

a

faea =∑

a

(−1)|ea||fa|eaf

a. (5.20)

In particular, the tangent sheaf TM can be seen as a sheaf of left OM -modules. Accordingly, onehas that a vector fields acts as usual from the left, i.e.

→

X(f) ..=∑

a

Xa→

∂a(f). (5.21)

As seen above, this satisfies the Lebniz rule in the form→

X(fg) =→

X(f)g + (−1)|X||f |f→

X(g), (5.22)

for f, g ∈ OM . On the other hand it makes sense to consider a right action of a vector field on afunction, when TM is seen as a sheaf of right OM -modules, and we write

(f)←

X ..=∑

a

(f)←

∂aXa. (5.23)

This is what is usually called right derivative in theoretical physics. In this case the Leibniz rulereads

(fg)←

X = f((g)←

X) + (−1)|X||g|((f)←

X)g. (5.24)

for any f, g ∈ OM . The connection by left and right derivation is given, as expected, by the signrule, i.e.

→

X(f) = (−1)|X||f |(f)←

X, (5.25)

which proves to be in agreement with the Leibniz rule as indeed→

X(fg) = (−1)|X|(|f |+|g|)(fg)←

X. (5.26)

Having these considerations in mind, we look for an action of the Lie derivative on sections ofBerezinian sheaf of a supermanifold in a similar fashion as in (B.6) for the canonical sheaf onan ordinary manifold. We remark, though, that this is not at all trivial. Indeed, sections of theBerezinian sheaf are not differential forms, and as such the usual Cartan calculus and Cartanhomotopy formula for the Lie derivative does not apply. On a very general ground, following forexample [28] or [74], one can give a definition of Lie derivative using the flow along a vector field.

Definition 5.15 (Lie Derivative). Let M be a real or complex supermanifold, let E be a sheafcanonically associated to M and let X ∈ TM be a vector field. Then we define the Lie derivativeof a section S ∈ E as

LX(S) ..=d

dε

⌊

ε=0

[(ϕX

ε )∗S], (5.27)

where ϕXε : M → M is the flow of the vector field X and where the parity of the parameter ε is

the same as the parity of the field X .

Remark 5.16. Clearly, as in the ordinary context, this formalizes the idea of infinitesimal variationinduced on a certain section by the action of certain vector field. We now apply this definition torecover an expression for the action of the Lie derivative on sections of the Berezinian sheaf.

Theorem 5.17 (Lie Derivative of Sections of Berezinian Sheaf). Let M be a real or complexsupermanifold. Let D be a section of the Berezianian sheaf Ber(M ) and X be a vector field inTM , such that they have trivializations D = D(x)f(x) and X =

∑aX

a∂a in a certain local chart

26 SIMONE NOJA

(U, xa). Then the Lie derivative LX(D) defined as in (5.27) of D ∈ Ber(M ) along the field X ∈ TM

is given in U by

LX(D) = (−1)|X||D|D(x)∑

a

(fXa)←−

∂a , (5.28)

for a = 1, . . . n|1, . . . ,m even and odd coordinates.

Proof. We use (5.27) and directly compute the variation of the sections D = D(x)f(x) in the chartU ⊂ X for an infinitesimal diffeormorphism controlled by a parameter ε, having parity |ε| = |X |,so that it makes sense to consider the even transformation xa 7→ xa + εXa. On the one hand onehas

D(xa + εXa) = D(xa)Ber((xa + εXa)←

∂ b) = D(xa)Ber(δab + ε∂bXa). (5.29)

The Berezinian of the coordinates transformation can be rewritten as

Ber(δab + ε∂bXa) = 1 + t Str(∂bX

a) = 1 + ε∑

a

(−1)|xa||Xa|∂aXa, (5.30)

where the sign is due to the super trace of the Jacobian matrix. Equivalently, acting from the leftinstead, one has

D(xa + εXa) = D(xa)Ber(→

∂ b(xa + εXa)) = D(xa)Ber(δab + ∂b(εX

a)), (5.31)

so that this gives

D(xa + εXa) = D(xa)(1 + Str(∂b(εXa)))

= D(xa)

(1 +

∑

a

(−1)|xa|(|X|+|Xa|)(∂a(εX

a))

)

= D(xa)

(1 + ε

∑

a

(−1)|xa||xa|+|xa|(|Xa|+|xa|)∂aX

a

)

= D(xa)

(1 + ε

∑

a

(−1)|xa||Xa|∂aX

a

)(5.32)

where we have used that |xa| = |xa||xa| and that |ε| = |X |. Considering the expansion of the localfunction, one finds

D(xa + εXa)f(xa + εXa) = D(xa)

(1 + ε

∑

a

(−1)|xa||Xa|∂aXa

)(f(xa) + ε

∑

a

Xa∂af

).(5.33)

In turn, this gives

D(xa + εXa)f(xa + εXa) = D(xa)f(xa)+

+D(x)ε∑

a

((−1)|xa||Xa|(∂aX

a)f +Xa(∂af)). (5.34)

We rearrange the summands inside the parentheses as follows:

(−1)|xa||Xa|(∂aXa)f +Xa(∂af) = (−1)|xa||X

a|+|f |(|X|a+|xa|)f(∂aXa) + (−1)|X

a|(|x|a+|f |)(∂af)Xa

= (−1)|Xa|(|xa|+|f |)∂a(fX

a). (5.35)

This leads to the following expression

D(xa + εXa)f(xa + εXa) = D(xa)f(xa) +D(x)ε∑

a

(−1)|Xa|(|xa|)+|f |)∂a(fX

a)

= D(xa)f(xa) + εD(x)∑

a

(−1)|Xa|(|xa|+|f |)+|D(x)|(|Xa|+|xa|)∂a(fX

a)

= D(xa)f(xa) + ε(−1)|D(x)||X|D(x)∑

a

(−1)|Xa|(|xa|+|f |)∂a(fX

a)

(5.36)


The only summand that matters in the computation of the Lie derivative (5.27) is the one whichis linear in the parameter ε and indeed we have that

limt→0

(ΦXε )∗D −D

t= (−1)|D(x)||X|D(x)

∑

a

(−1)|Xa|(|xa|+|f |)∂a(fX

a). (5.37)

Finally, notice that the right-hand side of the above expression for the Lie derivative can be re-written as

LX(D) = (−1)|D(x)||X|D(x)∑

a

(−1)|Xa|(|xa|+|f |)+|xa|(|f |+|X

a|)(fXa)←

∂ a

= (−1)|D||X|D(x)∑

a

(fXa)←

∂ a (5.38)

taking into account that |D| = |D(x)| + |f |, thus concluding the proof.

Remark 5.18. As already observed the resemblance with the expression (B.6) for the Lie derivativeof the canonical sheaf is apparent. Nonetheless, in order to get the supergeometric analog oflemma B.3 it is useful to introduce the following re-definition of the Lie derivative as to get rid ofan inconvenient sign

LX(D) ..= (−1)|D||X|LX(D). (5.39)

In local coordinates, for sections D = D(x)f and X =∑

aXa∂a, one has the following action

LX(D) = D(x)∑

a

(fXa)←

∂ a = D(x)∑

a

(−1)|xa|(|f |+|Xa|)∂a(fX

a). (5.40)

We can thus finally prove the following theorem.

Theorem 5.19 (Ber(M ) is a Right DM -module). Let M be a real or complex supermanifold andlet ∆BerR : Ber(M )⊗K F

1DM → Ber(M ) be defined as

∆BerR (D ⊗ f) ..= Df, (5.41)

∆BerR (D ⊗X) ..= −LX(D), (5.42)

for any D ∈ Ber(M ), f ∈ OM ⊂ F 1DM and X ∈ TM ⊂ F 1DM . Then the followings hold true.

(1) ∆BerR defines a flat right connections on the Berezinian sheaf Ber(M ).

(2) In any system of coordinates ∆BerR is the unique right connection on Ber(M ) satisfying

∆BerR (D(x) ⊗ ∂a) ..= −L∂a(D(x)) = 0 ∀ a = 1, . . . , n|1, . . . ,m, (5.43)

where D(x) is a local generating section of Ber(M ) and ∂a ∈ TM is a coordinate vector field.

In particular ∆BerR endows Ber(M ) with a right DM -module structure.

Proof. We start using (5.40) to show that the axioms of a right connection holds true. The onlynon trivial verification to carry out is that ∆BerR (D ⊗ X f) = ∆BerR (D ⊗ X)f . We recall that

X f = X(f) + (−1)|X||f |f X. Upon using this, one computes

∆BerR

(D(x)g ⊗

∑

a

Xa∂a f)

= ∆BerR

(D(x)g ⊗

∑

a

Xa∂af

)+∆BerR

(D(x)g ⊗

∑

a

(−1)|f ||X|fXa∂a

)

= D(x)∑

a

gXa∂af −D(x)∑

a

(−1)|f |(|Xa|+|xa|)+|xa|(|X

a|+|f |+|g|)∂a(gfXa)

= D(x)∑

a

gXa∂af −D(x)∑

a

(−1)|xa|(|Xa|+|g|)

(∂a(gX

a)f + (−1)|xa|(|Xa|+|g|)gXa∂af

)

= −D(x)∑

a

(−1)|xa|(|Xa|+|g|)∂a(gX

a)f, (5.44)

upon observing that the first term and the last term cancel pairwise in the semi-last equality above.This gives

∆BerR (D(x)g ⊗∑

a

Xa∂a)f = −D(x)∑

a

(−1)|xa|(|Xa|+|g|)∂a(gX

a)f, (5.45)

28 SIMONE NOJA

thus proving that ∆BerR (D ⊗X f) = ∆BerR (D ⊗X)f.

Also, it follows trivially from equation (5.40) that the action of ∆BerR satisfies the third definingproperty of a right connection, while the first one is just the right multiplication.Let us now prove that the right connection defined by ∆BerR is flat, i.e. it satisfies

∆BerR (D ⊗ [X,Y ]) = ∆BerR (∆BerR (D ⊗X)⊗ Y )− (−1)|X||Y |∆BerR (∆BerR (D ⊗ Y )⊗X), (5.46)

for any pair of fields X,Y ∈ TM . On the one hand, recalling that the supercommutator [·, ·] is givenby

[X,Y ] ..=

[∑

a

Xa∂a,∑

b

Y b∂b

]=∑

a

(Xb∂bY

a − (−1)|X||Y |Y b∂bXa)∂a, (5.47)

one computes

∆R (D ⊗ [X,Y ]) =−D(x)∑

a,b

[(−1)|xa|(|f |+|X|+|Y

a|)∂a(fXb∂b(Y

a))+

− (−1)|X||Y |+|xa|(|f |+|Y |+|Xa|)∂a(fY

b∂b(Xa))], (5.48)

where we have used that |Xb∂bYa| = |X |+ |Y a| and |Y b∂bX

a| = |Y | + |Xa|. On the other hand,one has

∆BerR (∆BerR (D ⊗X)⊗ Y ) = ∆BerR

(−D(x)

∑

a

(−1)|xa|(|f |+|Xa|)∂a(fX

a)⊗∑

b

Y b∂b

)

= +D(x)∑

a,b

(−1)|xa|(|f |+|Xa|)+|xb|(|f |+|X|+|Y

b|)∂b(∂a(fXa)Y b). (5.49)

Now note that ∂a(fXa)Y b = ∂a(fX

aY b)− (−1)|xa|(|f |+|Xa|)fXa(∂aY

b), so that plugging this intothe above one

∆BerR (∆BerR (D ⊗X)⊗ Y ) = +D(x)∑

a,b

(−1)|xa|(|f |+|Xa|)+|xb|(|f |+|X|+|Y

b|)∂b∂a(fXaY b)+

−D(x)∑

a,b

(−1)|xb|(|f |+|X|+|Yb|)∂b(fX

a∂a(Yb)). (5.50)

The same holds true for the other part, that is one finds

∆BerR (∆BerR (D ⊗ Y )⊗X) = ∆BerR

(−D(x)

∑

a

(−1)|xb|(|f |+|Yb|)∂b(fY

b)⊗∑

a

Xa∂a

)

= +D(x)∑

a,b

(−1)|xb|(|f |+|Yb|)+|xa|(|f |+|Y |+|X

a|)∂a(∂b(fYb)Xa). (5.51)

It follows that, by Leibniz rule, one gets

∆BerR (∆BerR (D ⊗ Y )⊗X) = +D(x)∑

a,b

(−1)|xb|(|f |+|Yb|)+|xa|(|f |+|Y |+|X

a|)∂a∂b(fYbXa)+

−D(x)∑

a,b

(−1)|xa|(|f |+|Y |+|Xa|)∂a(fY

b∂b(Xa)). (5.52)

Let us now consider the full expression

∆BerR (∆BerR (D ⊗X)⊗ Y )− (−1)|X||Y |∆BerR (∆BerR (D ⊗ Y )⊗X) =

= −D(x)∑

a,b

((−1)|xa|(|f |+|X|+|Y

b|)∂a(fXb∂b(Y

a))− (−1)|X||Y |+|xa|(|f |+|Y |+|Xa|)∂a(fY

b∂b(Xa)))+

+D(x)∑

a,b

((−1)|xb|(|f |+|X

b|)+|xa|(|f |+|X|+|Ya|)∂a∂b(fX

bY a)+

− (−1)|X||Y |+|xb|(|f |+|Yb|)+|xa|(|f |+|Y |+|X

a|)∂a∂b(fYbXa)

).

We need the last two terms to cancel pairwise. We observe that

∂a∂b(fYbXa) = (−1)|xa||xb|+|X

a||Y b|∂b∂a(fXaY a). (5.53)


Renaming the indexes in the last addendum one finds that

D(x)∑

a,b

∂a∂b(fXbY a)

((−1)|xb|(|f |+|X

b|)+|xa|(|f |+|X|+|Ya|)+

− (−1)|X||Y |+|xa|(|f |+|Ya|)+|xb|(|f |+|Y |+|X

b|)+|xb||xa|+|Xb||Y a|

). (5.54)

This leads to consider the following equality

|xb|(|f |+ |Xb|) + |xa|(|f |+ |X |+ |Y a|) == |X |||Y |+ xb|(|f |+ |Y |+ |Xb|) + |xa|(|f |+ |Y a|) + |xb||xa|+ |Xb||Y a|, (5.55)

which is easily verified by rearranging the right-hand side. This tells that term in (5.54) is identi-cally zero, thus proving flatness of ∆BerR and concluding the proof of the first point.For the second point, uniqueness of ∆BerR follows from the fact that D(x) is a local generator forBer(M ) and that the ∂a’s give a system of local generators for TM . Also, choosing a system ofcoordinates, ∆BerR (D(x) ⊗ ∂a) = −L∂a

(D(x)) = 0 is readily verified simply applying (5.40). Thatthis remains zero in any system of coordinates x′ = x′(x) is a (lengthy) local check, which is carriedout in [53].

Remark 5.20. The above result establishes that there exists a right action of vector fields on theBerezinian sheaf of a supermanifold given - up to a sign - by the Lie derivative

Ber(M )⊗ TM// Ber(M )

D ⊗X // D ·X ..= −(−1)|D||X|LX(ϕ),

(5.56)

so that, explicitly, using (5.40), one has

D ·X = −ϕ(x)∑

a

(−1)|xa|(|Xa|+|f |)∂a(fX

a). (5.57)

It is worth noticing that the above construction can be obtained in a rather more neat and economicway in a purely algebraic fashion, by using the third construction of the Berezinian sheaf that wehave discussed in the previous section in theorem 4.19, which introduces the sheaf DM from thevery beginning. In particular, one prove the following.

Corollary 5.21 (Ber(M ) is a Right DM -Module - Cohomological Version). Let M be a real orcomplex supermanifold. Then right action

Hp ((DR •M ,D))⊗OMDM −→ Hp ((DR •M ,D)) (5.58)

is uniquely characterized by the condition D(x) · ∂a = 0 for any a even and odd, where D(x) ∈Hp ((DR •M ,D)) ∼= Ber(M ). More in particular, the action is given (up to a overall sign) by the Liederivative on Ber(M ).

Proof. With reference to Theorem 4.19, one has that in cohomology D(x) ·∂a = [dz1 . . . dzp⊗∂θ1 ⊗. . . ∂θq∂a] = 0 for any a, which characterizes the right action of DM on Ber(M ).In particular, one sees that considering a section D = dz1 . . . dzp ⊗ ∂θ1 . . . ∂θqf ∈ Ber(M ) and avector fields X =

∑aX

a∂a, one finds

D ·X = D(x)∑

a

(−1)|xa|(|Xa|+|f |) (−∂a(fXa) + ∂a · fXa)

= −D(x)∑

a

(−1)|xa|(|Xa|+|f |)∂a(fX

a), (5.59)

where we have used that D(x) · ∂xa= 0 is zero in the cohomology. This matches (5.57).

This shows the relevance of the construction of the Berezinian via the total de Rham algebraΩ• ⊗OM

DM .

Remark 5.22. Finally, it is worth to observe the similarities between these results and constructionsand those elucidated in appendix B regarding the right DX -module structure and its relation withthe Lie derivative for the canonical sheaf of an ordinary manifold X .

30 SIMONE NOJA

6. Integral Forms and Spencer Cohomology of Supermanifolds

In the previous section we have shown that, given a real or complex supermanifold M , there existsa flat right connection which endows Ber(M ) with the structure of a right DM -module. In analogywith the ordinary case of the canonical sheaf of a real or complex manifold, we want to study thestructure of the so-called Spencer complex related to Ber(M ) [52]. This is particularly relevant inthe context of supermanifolds since the Berezinian sheaf does not appear in the de Rham complex- which instead it is the case for the canonical sheaf of ordinary manifolds - thus giving rise toa genuinely new complex. Also, as we shall see shortly, sections of the Berezinian sheaf are theobjects to look at for a meaningful notion of integration over a supermanifold. The followingdefinition is related to this aspect [11, 12, 28, 53].

Definition 6.1 (Integral Forms). Let M be a real or complex supermanifold of dimension p|q. Wecall integral forms of degree p− i the sections of the sheaves

Σp−iM

..= Ber(M )⊗OMSiΠTM (6.1)

for any i ≥ 0.

Notice that, equivalently, one can define integral forms of degree n− i to be sections of the sheavesΣp−i

M..= HomOM

(ΩiM ,Ber(M )).

Remark 6.2. The degree is assigned so that sections of the Berezinian sheaf ΣpM = Ber(M ) are

top-integral forms in degree p, which equals the even dimension of the supermanifold, mimickingwhat happen for the canonical sheaf in the ordinary setting. Further, it is to be stressed thatintegral forms can have any negative degree: this is in some sense “dual” to what happen in thecase of differential forms on a supermanifolds, as seen above.

We are now interested into structuring the sheaves of integral forms into a complex: to thisend we need to introduce a suitable differential. This is where the (flat) right connection ∆BerR :Ber(M )⊗TM → Ber(M ) discussed in the previous section serves its purposes [53, 60]. In particular,we define a morphism of sheaves as follows.

δ : Σp−1 = Ber(M )⊗ΠTM// Σp = Ber(M )

D ⊗ πX // δ(D ⊗ πX) ..= (−1)|D|+|X|∆BerR (D ⊗X).

(6.2)

By the previous section this is well-defined and it does not depend on the choice of local coordinates.Also, notice that it is OM -linear since

δ(Df ⊗ πX) = (−1)|D|+|f |+|X|∆BerR (Df ⊗X) = (−1)|D|+|f |+|X|∆BerR (D ⊗ fX) = δ(D ⊗ fπX),(6.3)

by the property of the right connection ∆BerR . Moreover, since ∆BerR (−⊗X) = −LX(−), the abovecan also be rewritten as

δ(D ⊗ πX) = (−1)|D|+|X|+1LX(D) = (−1)|D|+|πX|LX(D), (6.4)

which stress the dependence from the parity of the Π-vector field πX ∈ ΠTM . Using the explicitexpression of the Lie derivative (5.57), in local coordinates, for D = D(x)f and πX =

∑aX

aπ∂athe action of δ is therefore given by

δ(D ⊗ πX) = (−1)|X|+|D|∆BerR (D ⊗X) = (−1)|X|+|D|+1LX(D)

= (−1)|X|+|D|+1D(x)∑

a

(−1)|xa|(|f |+|Xa|)∂a(fX

a), (6.5)

where we have used (5.57).

Remark 6.3. In order to extend the above morphism to integral forms of any degree, in similarfashion as in ordinary differential geometry, we introduce the contraction operator ιω : S•ΠTM →S•ΠTM where ω ∈ Ω1

M , which is a derivation of the supersymmetric algebra S•ΠTM characterizedby the properties that ιω(f) = 0 for any f ∈ OM and ιω(πX) = ω(πX) for any πX ∈ ΠTM .In particular, on the bases of Ω1

M and ΠTM we set ιdxa(π∂b) = δab. We will employ the notation


ιdxa= ∂π∂a

as to stress that ιdxaacts indeed as a derivation. In a certain trivialization, for sections

ω ∈ Ω1M and πX ∈ ΠTM one has that

ιω(πX) =∑

a

ωa∂π∂a

(∑

b

Xbπ∂b

)=∑

a

(−1)(|xa|+1)|Xa|ωaXa. (6.6)

Using this, we prove the following lemma.

Lemma 6.4 (Representation Lemma). Let M be a real or complex supermanifold. Then theaction of δ : Ber(M ) ⊗ ΠTM → Ber(M ) defined as in (6.2) can be given the following operatorrepresentation

δ =∑

a

(−1)|xa|+1L∂a⊗ ιdxa

, (6.7)

where L∂ais the Lie derivative with respect to the coordinate vector field ∂xa

acting on sections ofthe Berezinian sheaf Ber(M ) and ιdxa

is the contraction with respect to the coordinate 1-form dxaacting on section of the Π-tangent sheaf ΠTM .

Proof. Let us choose a chart with πX =∑

aXaπ∂a and D = D(x)f . Then one computes

(∑

a

(−1)|xa|+1L∂a⊗ ιdxa

)(D(x)f ⊗

∑

b

Xbπ∂b

)

=∑

a,b

L∂a(D(x)fXb)(−1)(|D|+|X

b|+1)(|xa|+1) ⊗ ιdxa(π∂b)

= D(x)∑

a

(−1)(|D|+|Xa|+1)(|xa|+1)(−1)|xa|(|D|+|X

a|)(−1)|xa|(|f |+|Xa|)∂a(fX

a)

= D(x)∑

a

(−1)|D|+|X|+1(−1)|xa|(|f |+|Xa|)∂a(fX

a), (6.8)

which matches δ(D ⊗ πX) as in (6.5).

Remark 6.5. The above lemma is convenient to extend the action of δ to integral forms of anydegree. More precisely, it can be proved that δ extends to a derivation δ : Σp−i

M → Σp−i+1M for any

i ≥ 1, where p is the even dimension of the supermanifold, in the sense that

δ(D ⊗ πX(n)πX(m)) = δ(D ⊗ πX(n))πX(m) + (−1)|πX(n)||πX(m)|δ(D ⊗ πX(m))πX(n) (6.9)

for any πX(n) ∈ SnΠTM and πX(m) ∈ SmΠTM for any m,n ≥ 1, and where we have the super-symmetric product understood for notational convenience. More in particular, using (6.7), onecomputes

δ(D ⊗ πX πY ) =

∑

a,b

(−1)(|D|+|Xb|+1)(|xa|+1)L∂a

(DXa)⊗ ιdxa(π∂b)

(∑

c

Y cπ∂c

)+

+ (−1)|πX||πX|

(∑

a,c

(−1)(|D|+|Yc|+1)(|xa|+1)L∂a

(DY a)⊗ ιdxa(π∂c)

)(∑

b

Xbπ∂b

)

=

(∑

a

(−1)(|D|+|Xa|+1)(|xa|+1)L∂a

(DXa)

)(∑

c

Y cπ∂c

)+

+ (−1)|πX||πX|

(∑

a

(−1)(|D|+|Yc|+1)(|xa|+1)L∂a

(DY a)

)(∑

b

Xbπ∂b

)

= δ(D ⊗ πX)πY + (−1)|πX||πY |δ(D ⊗ πY )πX. (6.10)

This proves that Leibniz formula in the form (6.9) holds true, so that δ : Σp−1M

→ ΣpM

is indeed aderivation. Working by induction on the degree of the integral forms one gets to the conclusion.Notice that this shows that δ : Σp−1

M→ Σp

Mindeed extends to a derivation δ : Σp−i

M→ Σp−i+1

Mfor

any i ≥ 0 and that it is globally well-defined, i.e. it does not depends on the choice of coordinates.We can thus prove the following lemma.

32 SIMONE NOJA

Lemma 6.6. The pair (Σ•M , δ) defines a differential graded supermodule (DGsM).

Proof. We are only left to prove that δ is nilpotent, i.e. δ2 = 0. First we note that, for any a evenand odd, one has that L∂a

and ιdxahave opposite parity so that δ is odd. Let us prove that L∂a

and ιdxa satisfies the same commutation relations, in particular they (super)commute pairwise.On the one hand, clearly [ιdxa , ιdxb ] = 0. On the other hand

[L∂a,L∂b

]D = L∂aL∂b

(D)− (−1)|xa||xb|L∂bL∂a

(D)

= (−1)|D||xb|+1L∂a(∆BerR (D ⊗ ∂b))− (−1)|xa||xb|(−1)|D||xa|+1L∂b

(∆BerR (D ⊗ ∂b))

= (−1)|D|(|xa|+|xb|)+|xa||xb|(∆BerR (DBerR (D ⊗ ∂b)⊗ ∂a)− (−1)|xa||xb|(∆BerR (DBerR (D ⊗ ∂a)⊗ ∂b)

)

= (−1)|D|(|xa|+|xb|)+|xa||xb|∆BerR (D ⊗ [∂a, ∂b]) = 0, (6.11)

where we have used the flatness of the right connection in the semilast equality and that [∂a, ∂b] = 0.It follows from Lemma A.1 that δ is nilpotent.

Thanks to the previous lemma 6.6 we can now give the following definition.

Definition 6.7 (Spencer Complex / Complex of Integral Forms of M ). We call the differentialgraded supermodules (Σ•M , δ) the Spencer complex of M or complex of integral forms of M

. . . // Σp−nM

δ// . . .

δ// Σp−1

M

d// Σp

M

d// 0, (6.12)

where p is the even dimension of M .

Remark 6.8. It is important to stress the following important difference between differential andintegral forms - which emerges also from the statement of the previous lemma, if compared tothe analogous lemma 3.6 for differential forms: integral forms are not structured into a sheaf ofsuperalgebras, i.e. it does not make sense to multiply two integral forms. Indeed, if on the onehand the supersymmetric product of two a Π-polyfields yields a Π-polyfields since S•ΠTM is a sheafof superalgebras, the multiplication of two sections of the Berezinian sheaf Ber(M ) is not a sectionof the Berezinian sheaf, but instead a section of Ber(M )⊗2, which never appears in the definition

of the sheaves Σp−iM

.

Remark 6.9. Also, it is crucial to observe the difference between the de Rham complex / complexof differential forms and the Spencer complex / complex of integral forms on a supermanifold M .Whereas the first one is not bounded from above, the second one is not bounded from below instead.

0 // Ω0M

// Ω1M

// . . . // ΩnM

// Ωn+1M

// . . .

. . . // Σ−1M

// Σ0M

// Σ1M

// . . . // ΣpM

// 0.

(6.13)

We now state the analog of the Poincare lemma in the context of integral forms. This appearswithout an actual proof in [53]. The following proof is adapted from the very recent [16].

Theorem 6.10 (Poincare Lemma for Integral Forms). Let M be a real supermanifold and let(Σ•M , δ) be the Spencer complex of M . Then one has

Hkδ (Σ

•M ) ∼=

RM k = 00 k > 0.

(6.14)

where RM is the sheaf of locally constant function on M . In particular, H0δ (Σ

p−•M

) is generated bythe section σ0 = D(x) θ1 . . . θq⊗π∂x1 . . . π∂xp

, where D(x) is a generating section of the Bereziniansheaf and π∂x1 . . . π∂xp

is the totally anti-symmetric Π-polyvector field in SpΠTM .

Proof. We show for any k 6= 0 a homotopy hk : Ber(M )⊗ Sp−kΠTM → Ber(M )⊗ Sp−k−1ΠTM forδ, i.e. a map such that

hk+1 δk + δk−1 hk = idBer(M )⊗Sp−kΠTM, (6.15)


where we have specified the degree and where the maps go as follows

· · · // Ber(M )⊗ Sp−k+1ΠTM

// Ber(M )⊗ Sp−kΠTM

id

hk

tt

δk// Ber(M )⊗ S

p−k−1ΠTM

hk+1tt

// · · ·

· · · // Ber(M )⊗ Sp−k+1ΠTMδk−1

// Ber(M )⊗ Sp−kΠTM// Ber(M )⊗ Sp−k−1ΠTM

// · · · .

(6.16)

Working locally on M , we define xa ..= z1, . . . , zp|θ1, . . . , θq the even an odd local coordinates andwe introduce a parameter t ∈ [0, 1]. Working as in the above Poincare lemma for differential

forms, we consider the map (t, xa)G7−→ txa, so that in turn one has f(xa)

G∗

7−→ f(txa) on sectionsof the structure sheaf OM . Writing G as a family of maps as Gt : M → M , we write in turnG∗t : OM → OM for the corresponding map on OM .We claim that the homotopy is given by

hk(D(x)f ⊗ πX) ..= (−1)|f |+|πX|D(x)∑

b

(−1)|f |(|xb|+1)

(∫ 1

0

dt tKsxbG∗t f

)⊗ π∂b(πX), (6.17)

where D(x) is a genereting section of the Berezinian, f is a section of the structure sheaf and πX isa Π-polyfield of degree k in the form πX = π∂I for some multi-index |I| = k and Ks is a constant,dependent on the integral form s = D(x)f ⊗ πX chosen which will be fixed in what follows.On the one hand we have

hk+1 δk(D(x)f ⊗ πX) = D(x)∑

a,b

(−1)(|xa|+|xb|)(|f |+|xa|)

(∫ 1

0

dt tKδsxbG∗t (∂af)

)⊗ π∂b · ∂π∂a

πX.

(6.18)

On the other hand, the action of δk−1hk is more complicated, namely made out of four summands:

δk−1 hk(D(x)f ⊗ πX) = +D(x)∑

a

∫ 1

0

dt tKsG∗t f ⊗ πX (6.19)

+D(x)∑

a

(−1)|xa|

∫ 1

0

dt tKsxb∂aG∗t f ⊗ πX (6.20)

+D(x)∑

a

(−1)|xa|+1

∫ 1

0

dt tKsG∗t f ⊗ π∂a · ∂π∂aπX (6.21)

−D(x)∑

a,b

(−1)(|f |+|xa|)(|xa|+|xb|)

∫ 1

0

dt tKsxb∂a(G∗t f)⊗ π∂b · ∂π∂a

πX.

(6.22)

For the last line (6.22) to cancel (6.18) we need Kδs = Qs + 1, by chain-rule. For the summand(6.19) it is immediate to observe

ϕ∑

a

∫ 1

0

dt tQsG∗t f ⊗ F = (p+ q)ϕ

(∫ 1

0

dt tQsG∗t f

)⊗ F. (6.23)

For the summand (6.20), assuming without loss of generality that f is homogeneous of degreedegθ(f) in the theta’s we have

D(x)∑

a

(−1)|xa|

∫ 1

0

dt tKsxb∂aG∗t f ⊗ πX = D(x)f ⊗ πX − δKs+1+degθ(f),0

(D(x)f(0) ⊗ πX)+

− (Ks + 1 + 2 degθ(f))D(x)

(∫ 1

0

dt tKsG∗t f

)⊗ πX, (6.24)

34 SIMONE NOJA

upon integration by parts. For the summand (6.21), we define degπ∂θ(πX) and degπ∂z

(πX) thedegree of πX in its even and odd monomials. We have

D(x)∑

a

(−1)|xa|+1

∫ 1

0

dt tKsG∗t f ⊗ π∂a · ∂π∂aπX =

=(degπ∂θ

(πX)− degπ∂z(πX)

)D(x)

(∫ 1

0

dt tKsG∗t f

)⊗ πX. (6.25)

Altogether one gets

(δk−1hk + hk+1 δk)(D(x)f ⊗ πX) = D(x)f ⊗ πX − δKs+1+degθ(f),0ϕf(0)⊗ πX+

+(p+ q + degπ∂θ

(πX)− degπ∂z(πX)− 2 degθ(f)−Ks − 1

)D(x)

∫ 1

0

dt tKsG∗t f ⊗ πX.

(6.26)

This implies that in order to have a homotopy Ks must be such that

Ks = p+ q + degπ∂θ(πX)− degπ∂z

(πX)− 2 degθ(f)− 1, (6.27)

so that one is led to consider

(δk−1hk + hk+1 δk)(D(x)f ⊗ πX) = D(x)f ⊗ πX+

− δ(p+q+degπ∂θ(πX)−degπ∂z

(πX)−degθ(f)),0D(x) f(0|θ) ⊗ πX. (6.28)

It is easy to see that the above homotopy fails if degπ∂θ(F ) = 0, degπ∂z

(F ) = p and degθ(f) = q, sothat one identifies the generator of the cohomology in the element σ0 ..= D(x)θ1 . . . θq⊗π∂z1 . . . π∂zp ,which is indeed obviously closed under the action of δ by inspection.

This result allows us to compute the cohomology of integral forms and, in turn, to make contactwith the cohomology of differential forms. Namely, we give the following definition.

Definition 6.11 (Spencer Cohomology of M ). Let M be a real supermanifold. Then we define theSpencer cohomology of M to be the cohomology of the global sections of the (sheaf of) differentiallygraded supermodules (Σ•M , δ), i.e.

HkSp(M ) ..= Hk

δ (H0δ(Σ•M )), (6.29)

where H0δ(Σ•M ) is 0-Cech cohomology group of Σ•M , i.e. the global sections or Σ•M .

Having proved Lemma 6.10, the Spencer cohomology of M is easily computed in exactly the samefashion as in Theorem 3.13.

Theorem 6.12 (Quasi-Isomorphism II). Let M be a real supermanifold and let Mred be its reducedmanifold. Then the Spencer complex of M is quasi-isomorphic to the de Rham complex of Mred . Inparticular, one has that

H•Sp(M ) ∼= H•dR (Mred ). (6.30)

Proof. Once again, the theorem is an easy consequence of the Cech-to-de Rham spectral sequencefor the double complex (Σ•M , δ, d), where δ is the Cech differential and δ is the integral formsdifferential. On the one hand, generalized Mayer-Vietoris short exact sequence (hence the existenceof a partition of unity) and Poincare lemma yield H•dR (Mred ) ∼= H•(|Mred |,RM ) in the ordinary

setting, on the other hand lemma 6.10 gives that H•Sp(M ) ∼= H•(|Mred |,RM ) in the supergeometric

setting, where H•(|Mred |,RM ) is the Cech cohomology of the sheaf of locally-constant functionsRM . It follows that

H•Sp(M ) ∼= H•(|Mred |,RM ) ∼= H•dR (Mred ), (6.31)

thus concluding the proof.

Just like in the case of differential forms, we have in particular the Poincare lemma for integralforms for the model supermanifold Rp|q.


Theorem 6.13 (Poincare Lemma for Rp|q). The Spencer cohomology of the supermanifold Rp|q isgiven by

HkSp(R

p|q) ∼=

R k = 00 k > 0.

(6.32)

Proof. Follows immediately from the above 6.10 and 6.12.

The above theorem 6.12 and theorem 3.13 have an obvious yet important corollary: the cohomologyof differential and integral forms compute exactly the same invariants, i.e. the de Rham cohomologyof the reduced space of the supermanifold.

Corollary 6.14 (H•Sp(M ) ∼= H•dR (M )). Let M be a real supermanifold. Then the de Rham coho-

mology of differential forms of M is naturally isomorphic to the Spencer cohomology of integralforms of M , i.e.

H•Sp(M ) ∼= H•dR (M ). (6.33)

Proof. It follows immediately from theorem 3.13 and theorem 6.12

Remark 6.15. An analogous result hold true for the compactly suppported Spencer cohomology.One finds that

H•Sp,c(M ) ∼= H•dR ,c(M ), (6.34)

where H•dR ,c(Mred ) is in turn isomorphic to H•dR ,c(Mred ). In particular, the compactly supported

Poincare lemma for integral forms on Rp|q reads

HkSp,c(R

p|q) ∼=

R k = p0 k 6= p.

(6.35)

A representative is given by D(x)θ1 . . . θqBc(z1, . . . , zp) for a compactly supported bump functionBc which integrate to one on the reduced space. Compactly supported integral forms will play acrucial role in the next section, when integration on supermanifold will be introduced.

Remark 6.16. As an addendum to the above remark 3.16, the previous corollary 6.14 says that,despite both of the complex of differential and integral forms are actually not bounded either fromabove or below, their cohomology can indeed be non-zero (and isomorphic) only in the framed partof the diagram below - where the degree of differential forms matched the degree of integral forms- from zero to the even dimension of the supermanifold.

0 // Ω0M (M ) // Ω1

M (M ) // . . . // ΩnM (M ) // Ωn+1

M(M ) // . . .

. . . // Σ−1M

(M ) // Σ0M (M ) // Σ1

M (M ) // . . . // ΣpM(M ) // 0.

(6.36)

Once again, here we have denoted ΩkM (M ) and Σk

M (M ) the global section of the sheaves of differ-ential and integral forms.

7. Berezin Integral and Stokes’ Theorem on Supermanifolds

In this section we introduce the notion of Berezin integral for real supermanifolds [10–12, 74, 76],following the philosophy and exposition given in [53], that underlies the role of integral formsof their cohomology as introduced above. In particular, we denote with Berc(M ) the Bereziniansheaf whose sections have compact support on M and with Σ•c the Spencer complex of compactlysupported integral forms on M . Further, we assume our supermanifold always has a finite goodcover. We start with the following preparatory lemma, see [53].

Lemma 7.1. Let M be a real supermanifold of dimension p|q. Then one has the following iso-morphism of sheaves of OM ,c-modules

Berc(M ) ∼= J qMBerc(M ) + δ(Σp−1

M ,c ). (7.1)

More in particular the intersection J qMBerc(M ) ∩ δ(Σp−1

M ,c ) is such that

ϕ(J qMBerc(M ) ∩ δ(Σp−1

M ,c ))∼= d(Ωp−1

M ,c ), (7.2)

where ϕ : J qMBerc(M )

∼=−→ ΩpMred ,c

is the isomorphism of Theorem 5.3.

36 SIMONE NOJA

Proof. Let Dc ∈ Berc(M ) = ΣpM ,c be a section of the Berezinian sheaf. Then, using a partition of

unity ρκκ∈I for M , we represent D as a (locally) finite sum

Dc =

n∑

i=1

D(i)c (7.3)

where supp(D(i)c ) is compact and it is contained in a certain open set U (i) locally described by the

coordinate system xa = z1, . . . , zp|θ1, . . . , θq, where the dependence of i is understood. Then one

can in turn decompose D(i)c as follows

D(i)c (x) = D(i)

c,0(x) +D(i)c,1(x)θ1 . . . θq, (7.4)

such that every monomial in the θ-expansion of Dc,0(x) contains at most q − 1 theta’s, i.e. it is ofthe form

Dǫc,0(x) = D(x)θǫ11 . . . θǫqq fǫ,c(x) (7.5)

for ǫ = (ǫ1, . . . , ǫq) with ǫi ∈ 0, 1 and |ǫ| < q and D(x) = [dz1 . . . dzp ⊗ ∂θ1 . . . ∂θq ] a generating

section. It is easy to see that any such section is in the image of δp−1 : Σp−1M ,c → Σp

M ,c . Indeed, letfor example be ǫj = 0, then, by definition of δ, one sees that up to sign

δ(D(x)θǫ11 . . . θj . . . θǫqq fǫ,c(x) ⊗ π∂πθj ) = D(x)θǫ11 . . . θ

ǫj−1

j−1 θǫj+1

j+1 . . . θǫqq fǫ,c(x). (7.6)

Notice that since every real supermanifold real is split, the θ-degree of the expansion is invariant.It follows that

∑

i

D(i)c,0 ∈ δ(Σp−1

M ,c ), (7.7)

and hence the difference Dc −∑

i D(i)c,0 lies in J q

M Berc(M ), which proves the first statement.

For the second statement, let first be Dc ∈ J qM Berc(M ) ∩ δ(Σp−1

M ,c ), with Dc = δωc for some

ωc ∈ Σp−1M ,c . Let us decompose ωc ∈ Σp−1

c as in (7.3),

ωc =∑

i

ω(i)c . (7.8)

for supp(ω(i)c ) compact and contained in some open set U (i) locally described by the coordinate

system xa = z1, . . . , zp|θ1, . . . , θq, where again we have left the dependence of i understood. Now,

without loss of generality, any ω(i)c can be taken of the form

ω(i)c = D(x)θ1 . . . θqfc(z)⊗

∑

a

π∂xa(7.9)

with fc depending on the even coordinates only, since we have seen that any monomial which doesnot have all the theta’s is the image of δ, and since δ2 = 0 it will not contribute to Dc ∈ Σp

M ,c .

Further there should be at least one of the π∂zj ’s, since otherwise either δω(i)c /∈ J q

M Berc(M ) or

δω(i)c = 0 in J q

M Berc(M ). But then, one concludes that δω(i)c should be of the form

δ(ω(i)c ) =

p∑

j=1

D(x)θ1 . . . θq∂zjfc(z), (7.10)

so that if follows

ϕ(δ(ω(i)c )) =

p∑

j=1

dz1 . . . dzp∂zjfc(z) ∈ d(ΩpMred ,c

), (7.11)

proving that ϕ(J qM Berc(M ) ∩ δ(Σp−1

M ,c ) ⊂ d(ΩpMred ,c

).

Viceversa, let us take Dc ∈ JM Berc(M ) such that ϕ(Dc) = dηred ,c , for ηred ,c ∈ Ωp−1Mred ,c

and let us

prove that Dc ∈ δ(Σp−1M ,c ). To this end, once again we decompose ωred ,c as follows

ηred ,c =∑

i

η(i)red ,c , (7.12)


as above, where now supp(η(i)red ,c) is compact and contained in some open set U (i) locally described

by the coordinate system z1, . . . , zp for Mred , so that in these coordinates one has

η(i)red ,c =

∑

k

dz1 . . . dzk . . . dzpfkc (z) (7.13)

Accordingly, one can lift η(i)red ,c to the integral form in Σp−1

M ,c given by

ω(i)c =

∑

k

D(x)θ1 . . . θqfkc (z)⊗ π∂zk (7.14)

so that up to sign one gets ϕ(δω(i)c ) = dη

(i)red ,c . Notice that the support of η

(i)red ,c is the same as the

one of ω(i)c and that, with abuse of notation, we have denoted the even local coordinates in the

same way on Mred and M . Summing over the index i, defining ωc..=∑

i ω(i)c , one has that

ϕ(Dc − δωc) = ϕ(Dc)− ϕ(δωc) = ϕ(Dc)− dηred ,c = 0, (7.15)

by hypothesis. Let us set Dc..= Dc − δωc , choose a partition of unity ρj subordinate to the above

open sets, so that it commutes with the isomorphism ϕ and posing so that posing D(i)c

..= ρiDc

one gets ϕ(D(i)c ) = 0. Then, it follows that in the above domain D(i)

c does not contain all of the

theta’s, otherwise one would get ϕ(D(i)c ) 6= 0. In turn, this implies that D(i)

c is in the image of δ,

i.e. there exists ω(i)c ∈ Σp−1

M ,c such that δ(ω(i)c ) = D(i)

c and the support of ω(i)c is at most the same

as the one of D(i)c . Posing ωc

..=∑

i ω(i)c and summing over i, recalling that Dc = Dc − δωc , we have

Dc = δ(ωc + ωc), proving that Dc ∈ δ(Σp−1M ,c ).

The previous lemma has the following immediate consequence, that we state as a theorem.

Theorem 7.2. Let M be a real supermanifold. Then the following natural isomorphism of sheavesof OMred

-modules holds true

Berc(M )/Im(δp−1) ∼= Ωp

Mred ,c

/Im(dp−1) . (7.16)

where δp−1 : Σp−1M ,c → Σp

M ,c and dp−1 : Ωp−1M ,c → Ωp

M ,c .

Proof. The isomorphism is induced by the map ϕ : J qMBerc(M ) → Ωp

M ,c and it follows imme-

diately from lemma 7.1 since Berc(M ) decomposes as Berc(M ) ∼= J qMBerc(M ) + δ(Σp−1

M ,c ) and

ϕ(J qMBerc(M ) ∩ δ(Σp−1

M ,c )∼= d(Ωp−1

M ,c ).

We can thus finally define the Berezin integral of a section of Berc(M ).

Definition 7.3 (Berezin Integral). Let M be a real supermanifold of dimension p|q such that Mred

is oriented and let xa = z1, . . . , zp|θ1, . . . , θq is a local system of coordinate on an open set U . LetDc ∈ Berc(M ) be a compactly supported section of the Berezinian sheaf which reads

Dc =∑

ǫ

D(x)θǫ11 . . . θǫqq fǫc (z1, . . . , zp), (7.17)

with ǫ = (ǫ1, . . . , ǫq) for ǫj = 0, 1 in the above local coordinates in (U, xa). Then we define theBerezin integral of Dc as the map

∫

M

: Berc(M ) −→ R (7.18)

(7.19)

given in the coordinate domain (U, xa) by∫

U

Dc..=

∫

Ured

dz1 . . . dzpf1...1c (z1, . . . , zp), (7.20)

and we extend the definition to all M by additivity via a partition of unity.

38 SIMONE NOJA

Remark 7.4. The above definition is well-given. Indeed, it does not depends on the choice of localcoordinates as a consequence of the previous lemma 7.1 and theorem 7.2. Indeed, more invariantly,the Berezin integral is the map given by composition of the isomorphism of theorem 7.2 with theordinary integral, i.e.

Berc(M )/δ(Σp−1M ,c )

ϕ// Ωp

Mred ,c/d(Ωp−1

Mred ,c)

∫// R, (7.21)

which induces the isomorphism HpdR ,c(Mred ) ∼= R in compactly supported de Rham cohomology.

More in particular, the above constructions allows to prove an analog of Stokes theorem for super-manifolds.

Theorem 7.5 (Stokes Theorem for Supermanifolds). Let M be a real supermanifold of dimensionp|q with Mred oriented and let Dc ∈ Berc(M ). Then the following are true.

(1) There exists ωc ∈ Σp−1M ,c such that Dc = δωc if and only if

∫

M

Dc = 0. (7.22)

In other words a compactly supported integral form of degree p is exact, i.e. [Dc ] ≡ 0 ∈ HpSp,c(M )

if and only if it has vanishing Berezin integral.(2) If M is connected, the Berezin integral defines an isomorphism

∫

M

: HpSp,c(M )

∼=−→ R. (7.23)

In particular, a representative of HpSp,c is given by σc

..= D(x)θ1 . . . θqBc(z1, . . . , zp), where Bc

is any bump function which integrate to one on Mred .

Proof. It is enough to use (7.21). More in particular, on the one hand we have observed that ifDc /∈ J q

M Berc(M ), i.e. does not have all the theta’s, then it is in the image of δ and, by definition,

its Berezin integral yields zero. On the other hand, if Dc ∈ J qM Berc(M ) ∩ δ(Σp−1

M ,c ), i.e. it has all

of the theta’s and it is in the image of δ, then if one has Dc = D(x)θ1 . . . θqfc(z) then fc(z) is adivergence, and Dc gets mapped to an element in d(Ωp

Mred ,c), which integrates to zero by ordinary

Stokes theorem. The second point follows immediately from (7.21) and the previous theorem7.2.

Remark 7.6. Explicitly one has the following isomorphism between local representatives

HpSp,c(M ) ∋ D(x)θ1 . . . θqBc(z1, . . . zp)

ϕ// dz1 ∧ . . . ∧ dzpBc(z1, . . . zp) ∈ Hp

dR ,c(Mred ). (7.24)

where again Bc(z1, . . . , zp) is a bump function which integrate to one over Mred . More in general,as in remark 6.15, it can be proved that the compactly supported Spencer cohomology of integralforms is isomorphic to the compactly supported de Rham cohomology, so that one has H•Sp,c(M ) ∼=H•dR ,c(Mred ).

Remark 7.7. We will not deal with the subtle case of supermanifold with boundaries and the relatedStokes’ Theorem. More on this can be found in [53] and [74, 76].

7.1. Supersymmetry and the Berezin Integral. The foremost application of the theory ofintegration on supermanifolds is related with high-energy physics, in particular with modern su-persymmetric field theories [17, 28]. Very roughly speaking, physical elementary particles aredescribed mathematically via irreducible (projective and unitary) representations of the Poincaregroup, whose Casimir invariants are in turn related to the mass and the spin (or helicity in thezero-mass case) of the particles. In this context, a supersymmetry is a physical symmetry whichrelates particles characterized by integer spins (bosons, physically describing “interactions”) toparticles characterized by half-integer spins (fermions, physically describing “matter”): such sym-metries are building pillars of the most far-reaching theories in contemporary physics, string theorybeing an example.Briefly, a physical theory on an unspecified space(time) M , where M is an ordinary manifold, is


described by an action functional A : FM → R where FM is the space of fields ϕi of the theory.One usually writes the action as an integral over M

A ..=

∫

M

L (ϕi) (7.25)

where L is the so-called Lagrangian density of the theory, which - for an ordinary space-timemanifold M - is a (compactly supported) section of the canonical sheaf of M , i.e. L ∈ Ωdim M

M ,c ,

so that the integral makes sense (we consider M to be oriented). We say that the physical theorydescribed by A is invariant under the (infinitesimal) transformation generated by a vector fieldX ∈ TX if

δXA ..=

∫

M

LXL = 0, (7.26)

where LXL is the Lie derivative of L with respect to the field X . In this case one says that thetransformation is a symmetry of the theory and the field X is the generator of the symmetry.The simplest way to make a physical theory manifestly invariant under a given transformation is toconstruct the theory in a space whose isometry group contains such a transformation: supermani-folds do this job for supersymmetry transformations. More precisely, supersymmetric field theoriescan be made into manifestly invariant theories under supersymmetry if they can be constructedas theories on particular supermanifolds called superspacetimes, which are defined as homogenoussuperspaces for the action of Poincare Lie supergroups and whose reduced manifolds are ordinaryphysical spacetimes (e.g. the Minkowski spacetime R1,d−1 in D dimensions) [29, 32, 33, 67].Superspacetimes are constructed out of three pieces of data,

(1) a real quadratic vector space (V,Q ) of dimension dim(V ) = D, with Q : V → R whoseassociated symmetric bilinear form B : V × V → R has signature (1, D − 1);

(2) a real spinorial representation S : Spin(V ) → Aut(V ) of dimension dim(S) = q of the groupSpin(V ).

(3) a Spin(V )-equivariant symmetric non-zero bilinear map γ : S × S → V .

Notice that the map γ : S × S → V always exists in the given setting [67]. In the above dataV is called (abelian) translation algebra, as it is appear as the summand in the Poincare Liealgebra Iso(V ) ..= V ⋊ so(V ) corresponding to spacetime-translation. The related Poincare Liesuperalgebra SIso(V ) = g0 ⊕ g1 is constructed out of the direct product of vector spaces

SIso(V ) = (V ⋊ so(V ))⊕ S, (7.27)

where g0..= V ⋊ so(V ) and g1

..= S. Here S is looked at as a g0-module with a trivial action

V · S ..= [v, s ] = 0 (7.28)

for any v ∈ V and s ∈ S. Defining further

[s1, s2] ..= γ(s1, s2) (7.29)

for any s1, s2 ∈ S, one obtains a Lie superalgebra, indeed [s , [s , s ]] is zero since [s , [s , s ]] = [s , γ(s , s)] =0 by (7.28). Letting S be irreducible with basis given byQa for a = 1, . . . , q and Pµ with µ = 1, . . .Dthe standard basis of V ∼= R

1,D−1, one can write - upon a suitable normalization of γ -:

[Qa, Qb] =

D∑

µ=1

γµabPµ, (7.30)

where [Qa, Qb] ..= γ(Qa, Qb) so that γµab = γµba: this is the way the crucial commutation relation ofPoincare Lie superalgebra appears in physics.The Poincare Lie supergroup SIso(V ) is obtained exponentiating this construction, and the relatedsuperspacetime is the quotient supermanifold (or coset superspace, as most frequently called inphysics) obtained by modding out the Lie group SO(V ) of Lorentz transformations [67],

M ..= SIso(V )/SO(V ) . (7.31)

Equally, one can notice that V ⊕S is also a Lie superalgebra, with g0 = V , the ordinary translationalgebra and g1 = S: this is called translation superalgebra and the superspacetime is given by its

40 SIMONE NOJA

related Lie supergroup. Notice that dim(M ) = dim(V )| dim(S) = D|q, and if (xµ|θa) and (yµ|ψa)are coordinates for M , then the group law (z|λ) ..= (x|θ) · (y|ψ) reads

zµ = xµ · yµ = xµ + yµ − 1

2

∑

a,b

γµabθaψb, λa = θa + ψa. (7.32)

Correspondingly, a set of even left invariant vector fields is given by ∂µµ=1,...,D, while a set ofodd left invariant vector fields are given by Qaa=1,...,q, with

Qa..=

∂

∂θa+

1

2

∑

µ,b

γµabθb ∂

∂xµ(7.33)

and it is not hard to verify that [Qa,Qb] =∑D

µ=1 γµab∂µ, just like the above (7.30), upon iden-

tifying Pµ..= ∂µ, as customary, since the ∂µ’s generate space-time translations. In view of this,

supersymmetry transformations are generated by the vector fields Qa.In light of the previous sections, one can generalize the definition (7.25) and (7.26) on a super-spacetime. The action of the physical theory is now given by an integral on the superspacetime

As ..=

∫

M

Ls(ϕi), (7.34)

where now the Lagrangian density is a section of the compactly supported Berezinian sheaf of thesuperspacetime M , i.e. L s ∈ Berc(M ). It can be trivialized as

Ls(ϕi) = D(x|θ)Φ(ϕi(x), θa), (7.35)

with D(x|θ) = [dx1, . . . , dxD ⊗ ∂θ1 . . . ∂θq ] a generating section for the Berezinian and where

Φ(ϕi, θa) ∈ OM ,c is a so-called superfield, containing the original physical fields ϕi ∈ OM ,c - plusauxiliary fields - in its component expansion. The component fields transform one into anotherunder supersymmetry, forming a so-called supersymmetry multiplet.

Lemma 7.8 (Supersymmetry Invariance). Let As be an action on a connected superspacetime M

as in (7.34) and let Q be any supersymmetry generator of the form (7.33). Then

δQAs =

∫

M

LQ(L s) = 0. (7.36)

In particular, the action is invariant under supersymmetry.

Proof. The result follows from the action of the Lie derivative on sections of the Berezinian sheafas in equation (5.57) and from Stokes theorem 7.5 for supermanifolds.More precisely, the part of the Lie derivative with respect to the odd coordinate vector fields ∂θa ∈Qa integrate to zero since L∂a

(L s) /∈ J qM

Berc(M ) and, as such, it is δ-exact as an integral form.

Similarly, the part of the Lie derivative with respect to the odd vector field∑

µ,b γµabθ

b ∂∂xµ ∈ Q

yields a divergence, which is an element in J qMBerc(M )∩ δ(ΣD−1

M ,c ) that integrate to zero again byStokes theorem 7.5.

Remark 7.9. In general, verifying that a theory is indeed invariant under supersymmetry is nottrivial matter, and it often requires going through delicate and lengthy calculations. The abovelemma shows that upon using an adequate and mathematically aware formalism - based on super-manifolds, integral forms and the related integration theory -, supersymmetry invariance becomesapparent and virtually no checks are required. Nonetheless, it is fair to say that writing a super-space action is in general not an easy task and indeed there exist theories of great physical interestfor which action on superspacetime is not known [29, 33].

Remark 7.10. More in general, Lagrangian densities on superspacetimes might also involve alsocovariant derivatives, i.e. differential operators acting on superfields which commutes with thesupersymmetry generators: this is the case for example of the kinetic term (field strength) ofsupersymmetric gauge theories. Geometrically, these can be constructed as right invariant vectorfields on the superspacetime M , in opposition with Q being left invariant [29, 33, 67].


8. Poincare Duality on Supermanifolds

Having available a notion of integration on supermanifolds via the Berezin integral, we can provethe analog of Poincare duality for supermanifolds. Whereas Poincare duality on ordinary manifoldsyields a perfect pairing between the Rham cohomology groups of the manifolds, we will see insteadthat on a supermanifold Poincare duality defines a perfect pairing between the cohomology of twodifferent complexes, that of differential and that of integral forms: this is rooted in the peculiar inthe peculiar geometry of forms and, in turn, integration theory on supermanifolds. We start withsome preliminary remarks.

Remark 8.1. As discussed in the first section around equation (3.3), there is an obvious pairingΠTM ×Ω1

M → OM , given by the contraction of Π-vector fields and forms on M . This can be extendedto higher supersymmetric powers of ΠTM and Ω1

M ,

〈·, ·〉 : SnΠTM × ΩmM

// Sn−mΠTM (8.1)

for any n,m ≥ 1 with n ≥ m so that associativity is met in the form 〈〈πX(n), ω1〉, ω2〉 =〈πX(n), ω1ω2〉, for any πX(n) ∈ SnΠTM and ω1, ω2 ∈ Ω•M whose sum of degrees does not exceed

n. This pairing induces a right multiplication between integral forms in Σp−nM

= Ber(M )⊗ SnΠTM

and differential forms in ΩmM , that we write as

· : Σp−nM

× ΩmM

// Σp−(n−m)M

,

(σ, ω) // σ · ω

(8.2)

for n,m ≥ 0, n ≥ m and n +m and where p is the even dimension of M , so that ΣpM = Ber(M ).

Notice that, at this stage, this does not define a structure of right Ω•M -module on integral forms,since the multiplication is defined only for certain degrees. The crucial observation is that theabove multiplication is compatible with the differential, i.e. we have a Leibniz rule in the form

δ(σ · ω) = δσ · ω + (−1)|σ|σ · dω, (8.3)

for any σ ∈ Σp−nM

and ω ∈ ΩmM and where d is the de Rham differential. This can be proved in local

coordinates by expanding the tensor πX(n) over a base of SnΠTM of supersymmetric products ofthe π∂xa

’s and use (8.1). This is lengthy computation, where the bookkeeping of the signs involved

play a crucial role: we check it in the case σ ∈ Σp−2M

and ω ∈ Ω1M , for σ =

∑a,b D(x)f⊗Xabπ∂a π∂b

and ω =∑

c gcdxc. One has

δ(σ) · ω + (−1)|σ|σ · dω =

∑

a,b

(−1)|xa|+1+|Xab|+|D(x)|+|f |+|gb|(|xb|+1)D(x)

((−1)|xa|(|f |+|Xab|)∂a(fXab)gb + fXab∂ag

b

)± (a ↔ b)

∑

a,b

(−1)|xa|+1+|Xab|+|D(x)|+|f |+|gb|(|xb|+1)+|xa|(|f |+|Xab|)D(x)∂a(fXabgb)± (a↔ b). (8.4)

On the other hand, one computes directly that

δ(σ · ω) =∑

a,b

(−1)|xa|+1+|Xab|+|D(x)|+|f |+|gb|(|xb|+1)+|xa|(|f |+|Xab|)D(x)∂a(fXabgb)± (a ↔ b), (8.5)

matching (8.4). The previous Leibniz rule (8.3) is the key to prove Poincare duality for superman-ifolds.

Theorem 8.2 (Poincare Duality for Supermanifolds). Let M be a real supermanifold of dimensionp|q with Mred oriented. Then, for any n ≥ 0, the Berezin integral defines a perfect pairing incohomology

Hp−nSp,c (M )×Hn

dR (M ) // R

([σc ], [ω]) //

∫Mσ · ω

(8.6)

In particular, there is a natural isomorphism

(Hp−nSp,c (M ))∗ ∼= Hn

dR (M ). (8.7)

42 SIMONE NOJA

Proof. First of all let us observe that the map is well-defined. Indeed if σc ∈ Σp−nM ,c and ω ∈ Ωn

M

are two representative, then σc · ω ∈ ΣpM ,c = Berc(M ), so that it can indeed be integrated in the

Berezin sense. In other words the map is given by the following composition:

Σp−nM ,c × Ωn

M// Σp

M ,c// R

(σc , ω) // σc · ω //

∫σ · ω.

(8.8)

Further, let σc and ω be closed, i.e. δσ = 0 and dω = 0. It follows immediately from the (8.3) thatσc · ω is closed, i.e. δ(σ · ω) = 0.

Now let σc be exact, i.e. σc = δηc for some ηc ∈ Σp−(n−1)M ,c . Then, using again (8.3) one has that

σc · ω = δηc · ω = δ(ηc · dω), (8.9)

proving that σc · ω is exact. If instead ω ∈ ΩnM is exact, i.e. ω = dγ for some γ ∈ Ωn−1

M , then by(8.3) one has

σc · ω = σc · dγ = (−1)|σc |δ(δσc · γ), (8.10)

proving that σc ·ω is exact. We thus have that the composition of maps (8.8) descends to cohomologyas to give

Hp−nSp,c (M )×Hn

dR (M ) // HpSp,c(M ) // R

([σc ], [ω]) // [σc · ω] = [σc ] · [ω] //

∫σ · ω.

(8.11)

Now let us work by induction on the cardinality of the good covering, namely let us start provingthe result for the pivotal case of a covering of cardinality one. This corresponds to the Poincarelemma for Rp|q, which follows straightforwardly from the Poincare lemmas for differential formsand compactly supported integral forms, i.e.

HkdR (R

p|q) ∼=

R k = 00 k ≥ 0,

HkSp,c(R

p|q) ∼=

R k = p0 k 6= p.

(8.12)

The only non-zero pairing reads

HpSp,c(R

p|q)×H0dR (R

p|q) // HpSp,c(M ) // R

([D(x)θ1 . . . θqBc ], [1]) // [D(x)θ1 . . . θqBc ] · [ω] //

∫Rp|q D(x)θ1 . . . θqBc .

(8.13)

Upon choosing Bc(z1, . . . , zp) a bump functions which integrates to one on Rp, one gets

∫

Rp|q

D(x)θ1 . . . θqBc =

∫

Rp

dz1 . . . dzpBc(z1, . . . , zp) = 1, (8.14)

concluding the proof of the isomorphism (HpSp,c(R

p|q))∗ ∼= H0dR (R

p|q).

Now, if M is covered by two open sets U and V so that with abuse of notation we can writeM = U ∪V as supermanifolds, thanks to the Mayer-Vietoris sequences for ordinary and compactlysuppported cohomology we have the following (sign-)commutative diagram [14]

. . . // Hp−k+1Sp (U)∗ ⊕Hp−k+1

Sp,c (V )∗

αk−1U+V

// Hp−k+1Sp,c (U ∩ V )∗

αk−1U∩V

// Hp−kSp,c (U ∪ V )∗ //

αkU∪V

Hp−kSp,c (U)∗ ⊕Hp−k

Sp,c (V )∗

αkU+V

// . . .

. . . // Hk−1dR (U)⊕Hk−1

dR (V ) // Hk−1dR (U ∩ V ) // Hk

dR (U ∪ V ) // HkdR (U)⊕Hk

dR (V ) // . . .

(8.15)


Notice that the above is induced by the pairing of the two Mayer-Vietoris long exact sequence, sothat one has

R

. . . Hp−k+1Sp,c (U ∩ V )oo

∫U∩V

OO

⊗

Hp−kSp,c (U ∪ V )

δoo

⊗

. . .oo

. . . Hk−1dR (U ∩ V )oo d

// HkdR (U ∪ V ) //

∫U∪V

. . .

R

(8.16)

in a way such that (sign-)commutativity means that for any σc ∈ Hp−kSp,c (U∪V ) and ω ∈ Hk−1

dR (U∩V )

one has ∫

U∩V

δσc · ω = ±∫

U∪V

σc · dω. (8.17)

By Poincare duality for Rp|q the maps αk−1U+V , α

kU+V and αk

U∩V , αkU∩V are isomorphisms. It follows

from five lemma that also αkU∩V is an isomorphism. The proof is then concluded working by

induction on the cardinality of the covering. Namely, suppose Poincare duality holds for super-manifolds admitting a covering by n opens sets at most and consider a manifold with a covering ofn+1 open sets, U1, . . . , Un+1: then by hypothesis Poincare duality holds for the supermanifolds(⋃n

i=1 Ui) ∩ Un+1, Un+1 and⋃n

i=1 Ui, so reasoning exactly as above one conclude that Poincare

duality holds true also for⋃n+1

i=1 Ui.

Remark 8.3. The same is true also switching the compact support from integral to differentialforms, i.e. we have a perfect pairing

Hp−nSp (M )×Hn

dR ,c(M ) // R

([σ], [ωc ]) //

∫Mσ · ωc .

(8.18)

Notice that in this context Poincare duality for the supermanifold Rp|q depends once again from thePoincare lemmas. The non-trivial part of the pairing is given by H0

Sp(Rp|q)×Hp

dR ,c(Rp|q) → R, and

the representative are σ = D(x)θ1 . . . θq⊗π∂z1 . . . π∂zp ∈ H0Sp(R

p|q) and ωc = dz1 . . . dzpBc(z1, . . . , zp) ∈Hn

dR ,c(Rp|q) respectively, where Bc(z1, . . . , zp) is a bump function which integrate to one on Rp. The

pairing integral then reads∫

Rp|q

σ · ωc =

∫

Rp|q

D(x)θ1 . . . θq ⊗ π∂z1 . . . π∂zp · (dz1 . . . dzp)Bc(z1, . . . , zp)

=

∫

Rp|q

D(x)θ1 . . . θqBc(z1, . . . , zp)

=

∫

Rp

dz1 . . . dzpBc(z1, . . . zp) = 1, (8.19)

where we have used that π∂z1 . . . π∂zp · (dz1 . . . dzp) = 1 and the very definition of the Berezinintegral.

9. Different Perspectives: Forms and Integration on Total Space

Given a supermanifold M , there exists a different point of view on the theory of forms and inte-gration on M , which highlights the role of the total space Tot(ΠTM ) of the (parity-shifted) tangent

bundle ΠTMπ−→ M of M . This point of view has been introduced since the early days by by Bern-

stein and Leites [12], Voronov and Zorich [68], Gaiduk, Khudaverdian and Schwarz [36]. RecentlyWitten has provided a terse review with applications to string theory in sight in [76], inspired bythe work of Belopolsky [7]. Further, Castellani, Catenacci, Grassi and collaborators have workedextensively on the formalism and its applications to supergravity and superstrings [18]-[26]. In this

44 SIMONE NOJA

section we provide a different exposition, which an eye to the global geometry of the total tangentspace supermanifold.

Definition 9.1 (Tot(ΠTM ) and Pseudoforms). Let M be a smooth supermanifold of dimensionp|q and let ΠTM be its parity-shifted tangent sheaf. We define TM ..= Tot(ΠTM ) as the p+ q|p+ q-dimensional supermanifold given as a ringed space by the pair (|TMred|,OTM ), where |TMred| ..=⊔

x∈M (ΠTM ,x)0, i.e. the total space of the even part of the (parity-shifted) tangent sheaf, and wheresections of OTM allows any dependence on the fiber coordinates.

Remark 9.2. Locally, the supermanifold TM admits the following description. Let (U, xa) be alocal chart for the p|q-dimensional manifold M , where the index a runs on even and odd localcoordinates. Then (π−1(U), xa, Xa) where Xa

..= dxa, gives a local chart for TM . We stress thatXa is now seen as a local coordinate for TM , better than a section of a vector bundle on M .Given two charts (U, xa) and (V, zb) on M with U∩V 6= ∅, we let (π−1(U), xa, Xa) and (π−1(V ), zb, Zb)be the corresponding charts on TM . Then we have the obvious transition functions for TM :

xa = za(x), Xa = Zb

(∂xa∂zb

). (9.1)

In the following, if no confusion occurs and as to conform with recent literature, we will denotethe local coordinates Xa of the supermanifold TM simply as dxa, see [76].

Remark 9.3. Let M be a real supermanifold. The above local description of the (real) supermanifoldTM via charts allows to represent local sections of the sheaf OTM over an open set π−1(U) in termsof the local coordinates xa and Xa as a function f(xa, Xa). Notice that in the smooth categoryalso generalized and trascendental functions are allowed. In particular, it makes sense to consider -for example - a trascendental dependence on the even coordinates Xa’s. For example, consideringthe 0|1-dimensional real supermanifold R0|1 described by an odd coordinate θ, it makes sense forthe 1|1-dimensional supermanifold TR0|1 to consider sections of its structure sheaf OTR0|1 of theform

f(θ, dθ) = exp (dθ) , g(θ, dθ) = log(dθ), h(θ, dθ) = sin(dθ), (9.2)

where dθ = X is the even coordinate of the 1|1 dimensional supermanifold TR0|1. In this respect,notice that asking that sections of the structure sheaf OTM have polynomial dependence on theeven Xa is equivalent to set OTM

..= Ω•M , i.e. the sections of the structure sheaf of TM are ordinarydifferential forms on M . This remarks leads to the following definition [12, 53, 76].

Definition 9.4 (Pseudodifferential Forms). Let M be a real supermanifold and let TM be definedas above. A section of OTM is said to be a pseudodifferential forms on M , or a pseudoform on M

for short.

The above definition is justified by the previous remark 9.3, which shows how ordinary differentialforms are indeed a specific subclass - actually a subalgebra - of pseudodifferential forms.

Let us now study the geometry the geometry of the cotangent sheaf Ω1TM of the supermanifold

TM . This is a locally-free sheaf of OTM -modules of rank p+ q|p+ q. The transition functions ofΩ1

TM are easily characterized, thanks to (9.1).

Lemma 9.5 (Transition Functions of Ω1M). Let TM be defined as above and let (dxa, dpa) and

(dzb, dqb) two local bases of Ω1Mon the open sets π−1(U) and π−1(V ) on M with U ∩ V 6= ∅. The

transition functions of Ω1Mread

dxa = dzb

(∂xa∂zb

)(9.3)

dXa = dZb

(∂xa∂zb

)+ (−1)|zb|+1Zb d

(∂xa∂zb

). (9.4)

Proof. The transition functions of the dxa’s are obvious. For the transition functions of the dXa’sone first observe that

dXa = dzb

(∂Xa

∂zb

)+ dZb

(∂Xa

∂Zb

). (9.5)


The first summand in (9.5) reads

dzb

(∂Xa

∂zb

)= dzb

∂

∂zb

(Zc∂xa∂zc

)= (−1)|zb|+1Zb d

(∂xa∂zb

). (9.6)

The second summands in (9.5) reads

dZb

(∂Xa

∂Zb

)= dZb

∂

∂Zb

(Zc∂xa∂zc

)= dZb

(∂xa∂zc

),

(9.7)

which concludes the proof.

The previous lemma has an immediate consequence on the global geometry of Ω1TM .

Lemma 9.6 (Ω1TM as Extension). Let TM be defined as above. Then the canonical exact sequence

0 // π∗Ω1M

// Ω1TM

// Ω1TM/M

// 0 (9.8)

induces the isomorphism of locally-free sheaves Ω1TM/M

∼= π∗T ∗M . In particular, Ω1TM is defined as

the extension of locally-free sheaves

0 // π∗Ω1M

// Ω1TM

// π∗T ∗M // 0. (9.9)

Proof. It follows immediately from the form of the transition functions given in the previous Lemma9.5, upon noticing that dxa and dXa have opposite parity, by the very definition of Xa.

Remark 9.7. On a general ground, extensions as in (9.9) are classified by the first Ext-group

Ext1(π∗T∗M , π∗Ω1

M ) ∼= H1(|M |,Hom(π∗T∗M , π∗Ω1

M )). (9.10)

In the case M is smooth and TM is the associated smooth supermanifold defined as above, theextension (9.9) is always (non-canonically) split. More in particular, one has a non-canonical iso-morphism Ω1

TM∼= π∗Ω1

M ⊕ π∗T ∗M . Locally, this means that it is always possible to find a coveringby open sets in which the second summand in equation (9.4) is zero. This result follows from thefact that in the smooth category every sheaf is fine, thus soft and acyclic, and from an applicationof Leray-Serre spectral sequence.On the other hand, in the case of a complex supermanifold M - and the related complex super-manifold TM -, the extension (9.9) might indeed be non-split. Studying the related cohomologyclass is a non-trivial and interesting problem.

In any case, both for smooth and complex supermanifold, the above characterization of Ω1M as an

extension allows to easily prove a fundamental property of TM .

Lemma 9.8 (Ber(TM ) ∼= π∗OM ). Let TM be defined as above. Then there is a canonical isomor-phism

Ber(TM ) ∼= π∗OM . (9.11)

Proof. From the extension exact sequence (9.9) and the definition of the Berezinian it follows that

Ber(TM ) ∼= π∗ (Ber(M )⊗ Ber∗(T ∗M )) . (9.12)

Since for any locally-free sheaf E one has Ber(ΠE) ∼= Ber∗(E), then in particular Ber∗(T ∗M ) ∼=Ber(ΠT ∗M ). It follows that, by definition of Berezinian sheaf of a supermanifold, one has Ber(ΠT ∗M ) ∼=Ber∗(M ). Then one has

Ber(TM ) ∼= π∗ (Ber(M )⊗ Ber∗(M )) ∼= π∗OM , (9.13)

concluding the proof.

Remark 9.9. The above result could have also been proved upon using the explicit form of thetransition functions for Ω1

TM , as given in Lemma 9.5.

46 SIMONE NOJA

Remark 9.10 (TM is a “Calabi-Yau” Supermanifold). While in general there is no natural choicefor a section of the Berezinian sheaf on the supermanifold M , instead it is a crucial consequenceof lemma 9.8 that the associated supermanifold TM comes endowed with a canonical Berezinianor canonical volume form, which is independent on the choice of coordinates on M , and thereforeon TM . When working in the complex holomorphic category one would say that the complexsupermanifold TM is a Calabi-Yau supermanifold. We will denote the canonical volume form onTM with

DTM (x,X) ∈ Ber(TM ). (9.14)

Remark 9.11 (Integration on TM ). Thanks to the existence of a canonical volume form for TM ,any function on TM can be mapped naturally to the Berezin integral on the supermanifold TM ,i.e.

OTM ∋ f(x,X) 7−→∫

TM

f(x,X)DTM (x,X) ∈ R. (9.15)

It is important to notice, though, that the above integral can be divergent, thus making the mappingill-defined. This is related to the fact the supermanifold TM is in general not compact, since thefibers of TM above every point of the p|q-dimensional supermanifold M are isomorphic to Rq|p,see also [53].An important class of functions in OTM which are in general not integrable over TM are differentialforms on M , i.e. functions on TM which have polynomial dependence on the fiber coordinates.To see this consider the example in Remark 9.3. We take the superpoint R0|1 and the related1|1-dimensional supermanifold TR0|1. The differential form P (θ, dθ) = θdθ ∈ Ω1

R0|1 is indeed anelement of O

TR0|1 having polynomial dependence on the fibers, and whose integral is divergent

since the ordinary Riemann-Lebesgue integral on the even variable dθ is clearly divergent,∫

TR0|1

D(θ|dθ) θdθ =

∫

R0|1

(D(θ)θ

∫

R1|0

D(dθ)dθ

). (9.16)

On the other hand, the function f(θ, dθ) = θe−(dθ)2

is integrable on TR0|1, and indeed one has∫

TR0|1

D(θ|dθ) θe−(dθ)2 =

∫

R0|1

(D(θ)θ

∫

R1|0

D(dθ)e−(dθ)2

)=

√π. (9.17)

This justifies the following definition, which distinguish among classes of functions on TM inrelation with their Berezin integral [11, 53, 76].

Definition 9.12 (Integrable Pseudodifferential Form). Let M be a real supermanifold and letTM be defined as above. We say that the pseudodifferential form f ∈ OTM is integrable if itsBerezin integral on TM is convergent, i.e.

∫TM

DTM f <∞. We will denote the sheaf of integrablepseudodifferential forms with Oint

TM .

Remark 9.13 (Integral over the Fibers). Let us now restrict to integrable pseudodifferential formsso that the integral (9.15) makes sense, i.e. we have a well-defined map

Oint

TM ∋ fint(x,X) 7−→∫

TM

fint(x,X)DTM (x,X) ∈ R. (9.18)

This integral can be understood as a (Berezin) integral along the fibers Fx..= π−1(x) of the fibration

TMπ−→ M , i.e.

∫

FM

≡ π∗ : Oint

TM −→ Ber(M ), (9.19)

followed by the “usual” Berezin integral on the base supermanifold M , so that one has a map

Oint

TM

π∗// Berc(M )

∫M

// R

fint // (π∗fint)

//∫

M(π∗fint).

(9.20)

Loosely speaking, defining integration on the total space TM via the composition of the abovemaps, i.e.

∫TM

..=∫

M π∗, has to be seen as a sort of factorization of the integral over the total

space as an integral in the fiber, or vertical directions followed by an integral over the base [14, 76].


Nonetheless, it is important to stress that the map π∗ corresponds to a Berezin integral. We choosea certain trivialization over an open set U in the base of a p|q-dimensional real supermanifold M

and define the corresponding even and odd coordinate x = yi|θα and X ..= dθα|dyi on the q|p-dimensional fibers over U for i = 1, . . . p and α = 1, . . . q. By anticommutativity of the dx’s afunction fint(x,X) ∈ Oint

TM (U) can be written as

fint(x,X) = dxi1 . . . dxiℓFc(xi, dθα|θα), (9.21)

for ℓ ≤ p and some compactly supported, integrable function Fc in the coordinates x’s and dθ’s.The map π∗ is thus defined so that it acts as a true Berezin integral on the odd fiber coordinatesdx’s, i.e. it yields

π∗ (fint(x,X)) =

0 ℓ < p(∫

Rq Fc(xi, dθα|θα)dµ(dθα))⊗DM (xi|θα) ℓ = p,

(9.22)

where dµ(dθa) is a (Lesbegue) measure for the real even variables dθ’s. Notice the result of theintegral is indeed a function in OM ,c(U), depending on the coordinates x’s and θ’s on the basemanifold.It can be proved that given a pair of open sets U, V in M with U∩V 6= ∅ and defining fU

int∈ Oint

TM (U)

and fVint

∈ Oint

TM (V ), then π∗(fUint) = π∗(f

Vint) in the intersection. It follows that for a certain open

covering Uii∈I and the related trivializations, one finds that π∗fUiint

i∈I glue together, yieldinga section π∗fint ∈ Berc(M ), see [14, 76].

Remark 9.14. In light of the recent physics-oriented literature, a brief remark on notation is in orderhere. In the influential review [76] Witten denotes a section of the Berezinian of a supermanifoldM as [dx|dθ] ∈ Ber(M ) for a choice of coordinate x|θ on M . Accordingly, the canonical volumeform on TM is denoted with

[dx d(dθ)|dθ d(dx)] ∈ Ber(TM ), (9.23)

as to reminds that dθ’s and the dx’s are now seen as coordinates for TM . In other words theexpressions d(dx)’s and d(dθ)’s are just the symbols corresponding to the dX ’s in the notation oflemma 9.5. As such, one should not read them as the application of the de Rham differential d ona local basis of the cotangent bundle, which would be clearly vanishing as d d = 0.

9.1. Special Class of Integrable Pseudoforms: Distributions on the Fibers. In the spiritof supersymmetric localization [61] it is convenient to focus on a particular class of pseudoformsin Oint

TM , admitting only a particular dependence of the fiber coordinates X = dθα|dxi. These arethe pseudoforms Belopolsky [5], Castellani, Catenacci and Grassi [18] and Witten [76] focus on,thought they have been introduced in the early days of the theory, see for example [12, 69].

Definition 9.15 (Delta Forms). Let M be a real supermanifold and let TM be defined as above.We call delta forms the class of integrable pseudoforms in Oint

TM whose dependence of the even fibercoordinates of TM is distributional and supported at the origin. We denote the sheaf of delta formsby Oδ

TM .

Given an open set U in M , in the local trivialization of TM over U with coordinates x = xi|θαand X = dθα|dxi, a delta form ω ∈ Oδ

TM can be written as

ωU (x,X) =∑

ǫk,ℓj

fǫk,ℓj (xi|θα)(dx1)ǫ1 . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓq (dθq), (9.24)

for ℓj ≥ 0 and ǫk = 0, 1 and where fǫ,ℓ ∈ π∗OM . The expressions δℓj (dθj) are Dirac’s deltadistributions [18, 76] - and their derivative, when ℓj > 0 - supported at the origin in the even realvariable dθj of TM . They can be seen as linear functional acting on differential forms, instead ofon ordinary functions: these kind of mathematical object are called de Rham current [39]. In thispaper they will be treated in a formal algebraic fashion.The sheaf Oδ

TM is endowed with several different structures, which interplay with the notion ofintegration on this particular class of pseudoforms. These structures are spelled out in the followingremarks.

48 SIMONE NOJA

Remark 9.16 (OM -module structure of OδTM ). We first note that the sheaf Oδ

TM is not a sheaf ofalgebras. This comes from the well-known fact from analysis that the product of two distributionsis not well-defined. On the other hand, Oδ

TM carries the structure of a sheaf of π∗OM -modules, asthe multiplication of delta forms by sections coming from the supermanifold M is well-defined.

Remark 9.17 (DΩ1M-module structure of Oδ

TM ). The sheaf of delta forms OδTM is a D-module. More

precisely, it carries the structure of Dπ∗Ω1M-module. Indeed, locally, the Clifford-Weyl superalgebra

CWq|p(R) generated by dθα, ∂dθα |dxi, ∂dxi, for i = 1, . . . , p and α = 1, . . . , q with non trivial

(super)commutation relations given by[

∂

∂dθα, dθβ

]= δαβ ,

∂

∂dxi, dxj

= δij , (9.25)

acts on sections in OδTM (U) according to the following definitions (given on monomials)

dxi·((dx1)

ǫ1 . . . (dxi)ǫi . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓq (dθq)

)

..= δ0ǫi (−1)∑i

j=1 ǫj (dx1)ǫ1 . . . (dxi) . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓq (dθq) (9.26)

∂

∂dxi·((dx1)

ǫ1 . . . (dxi)ǫi . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓq (dθq)

)

..= δ1ǫi (−1)∑i

j=1 ǫj (dx1)ǫ1 . . . dxi . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓq (dθq) (9.27)

dθα·((dx1)

ǫ1 . . . (dxp)ǫpδℓ1(dθ1) . . . δ

ℓα(dθα) . . . δℓq (dθq)

)

..= (−1)ℓαℓα(dx1)ǫ1 . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓα−1(dθα) . . . δ

ℓq (dθq) (9.28)

∂

∂dθα

((dx1)

ǫ1 . . . (dxp)ǫpδℓ1(dθ1) . . . δ

ℓα(dθα) . . . δℓq (dθq)

)

..= (dx1)ǫ1 . . . (dxp)

ǫpδℓ1(dθ1) . . . δℓα+1(dθα) . . . δ

ℓq (dθq), (9.29)

where again ǫi = 0, 1 and ℓα ≥ 0, for any i = 1, . . . , p and any α = 1, . . . , q. Notice that theaction (9.28) can be seen as arising from “integration by parts”, and in particular one defines

dθα ·((dx1)

ǫ1 ∧ . . . ∧ (dxp)ǫpδℓ1(dθ1) ∧ . . . ∧ δ0(dθα) ∧ . . . δℓq (dθq)

)= 0. (9.30)

This is can be seen as the “de Rham current” analog of usual relation that characterizes a Diracdelta distribution in functional analysis: working for simplicity over the real line R, for δx0 ∈ D ′(R),we have

δx0(f) ≡∫

R

f(x)δx0dx = f(x0), (9.31)

for any point x0 ∈ R and any test function f ∈ S (Rn) of Schwartz class. The algebraic way

to (9.31) consist into endowing the OR-module δ(ℓ)x0 ℓ≥0 of Dirac delta and its derivatives with aDR-module structure, by defining the D-action by

d

dx· δ(ℓ)x0

= δ(ℓ+1)x0

, (x− a) · δ(ℓ)a = (−1)ℓℓ δ(ℓ−1)a , (x− a) · δ(0)a = 0. (9.32)

In light of these, the previous (9.26)-(9.29) should not be surprising, as they are exactly the samerelations, but given in the context of forms (or de Rham current) on a supermanifold.

Remark 9.18 (Z2-grading of OδTM ). There is a twist in the description of the Z2-grading of the sheaf

OδTM . Indeed, whereas the dθ’s are even, the δ(dθ)’s and their derivatives are defined to be odd, so

that their product has parity q. It follows that a generic monomial in the expression (9.24) for ωU

above has parity given by q + |f | +∑pk=1 ǫk, where f is assumed homogeneous in its Z2-degree.

The reason behind the odd parity of the delta’s can be seen by looking at the following integral[76]

I =

∫

TR0|2

θ1θ2δ(dθ1)δ(dθ2)D(dθ|θ) (9.33)

Thanks to the delta’s, when applying π∗ as to integrate along the fibers dθ’s, the integral is localizedto the ordinary Berezin integral of θ1θ2 over the superpoint R0|2, which yields 1. Exchangingθ1 ↔ θ2, would yield an integral equal to −1 instead, unless the transformation of δ(dθ1)δ(dθ2)


would correct it with −1. Formally, we thus say the delta’s anticommute, e.g. δ(dθ1)δ(dθ2) =−δ(dθ2)δ(dθ1).Remark 9.19 (Z-grading of Oδ

TM ). A Z-gradation can also be introduced on OδTM . Considering a

section ω ∈ OδTM trivialized as in (9.24), its Z-degree is given by

degZ(ω) =

p∑

k=1

ǫk −q∑

j=1

ℓj , (9.34)

which implies that the Z-degree of a delta form is such that −∞ < degZ(ω) < p. This is stableunder change of coordinates, and therefore the definition of the Z-degree is well-posed, as we shall

see shortly. We will call the sub-sheaf the degree-k delta forms Oδ(k)TM .

Notice that delta forms in degree k ≤ p can be generated via the DΩ1M-module structure introduced

above starting from the (unique, up to a multiplication by a section of OM ) delta form in degreep, given by

ω(p)(x, dθ|θ, dx) = dx1 . . . dxpδ(dθ1) . . . δ(dθq), (9.35)

in a certain choice of coordinates. Then, sections ω(k) ∈ Oδ(k)TM of degree k < p are obtained by

acting with differential operators of order k on ω(p). These are locally constructed from the ∂dθ’sand ∂dx’s. In particular, we will have that locally

∂|I|

∂dθJ∂dxK· ω(p)(x, dθ|θ, dx) ∈ Oδ(p−|I|)

TM, (9.36)

where I, J and K are multi-indices such that I = (J,K) so that |I| = |J |+ |K|. Notice that K issuch that |K| ≤ p, since the ∂dx’s are anticommuting, while J can be of any order |J | ≥ 0. On theother hand, given a form in degree n < p, a differential forms of degree k acts via the DΩ1

M-action

defined above by raising the degree of the delta form by k. This means that

dθIdxK · ω(n)(x, dθ|θ, dx) ∈ Oδ(n+|I|)TM

, (9.37)

if again |I| = |J |+ |K|, for some multi-indices I = (J,K). Clearly, the (local) DΩ1M-action can also

result in annihilating the delta form, both in the case of (9.36) and (9.37), as it is clear from the(9.26)-(9.29).Notice that in the above picture delta forms can be seen locally as element of a Fock space, which

is constructed via the DΩ1M-action starting from a pivot ω(p) ∈ Oδ(p)

TM “state”, see [76].

Further, it is to be observed that, given the above definition, delta forms exist in the very samedegrees as integral forms in Σn

M , where again −∞ < n < p. This does not happen by chance,indeed it is possible to prove that there exists an isomorphism between integral and delta forms.Before we see this, a remark on the transformation properties of the delta’s is in order.

Remark 9.20 (Transformations Properties of the Delta’s). It has to be stressed that an expressioninvolving any number of delta’s which is lesser than the odd dimension q of base supermanifoldM , e.g. δ(dθ1) . . . δ(dθq−1) does not make sense, as its transformation properties are not well-defined. To see this in an informal way, let us consider a generic 1|2 dimensional supermanifoldand look at the transformation properties of a single delta δ(dθ) under a change or coordinatesdθ′ = αdθ + βdψ + γdx, for α, β, γ 6= 0, α, β even and γ odd. First, we observe that since dx is“infinitesimal” (as it is nilpotent) we can formally expand about it in the following fashion

δ(dθ′) = δ (αdθ + βdψ + γdx) = δ (αdθ + βdψ) + γdxδ′((αdθ + βdψ) . (9.38)

Now the problem is to make sense out of the expression δ (αdθ + βdψ) and its derivative. Keepworking formally, focusing on the first summand, one could get to the following expression

δ (αdθ + βdψ) = δ

(α

(dθ +

β

αdψ

))=

1

α

(δ(dθ) +

b

adψδ′(dθ) + . . .

)

=

∞∑

k=0

βk

k!αk+1(dψ)kδk(dθ), (9.39)

which would suggest that, if δ(dθ) was as a section, the corresponding sheaf would not be locally-free of finite rank. But clearly, this is just the tip of the iceberg, as there are more inconsistencies:

50 SIMONE NOJA

in the first place dθ and dψ are honest even variables on TM , so that the expression αdθ andβdψ are not at all nilpotent nor infinitesimal. Further, also forgetting about this, one might havechosen to expand about dψ instead of dθ.On the other hand, by definition, all of the q delta’s (or their derivatives) are required to appearin sections of Oδ

TM . We shall see in the next theorem that this requirements leads to well-definedtransformation properties.

Theorem 9.21 (Delta Forms are Isomorphic to Integral Forms). Let M be a real supermanifold

of dimension p|q. Then the sheaf Oδ(k)TM

of delta forms of degree k is isomorphic to the sheaf ΣkM

of integral forms of degree k for any k ≤ p.

Proof. To prove the statement is enough to verify that the transformation properties of the gener-ating sections do coincide. Let us start from degree p, corresponding to Σp

M= Ber(M ). Without

loss of generality, we can restrict ourselves to consider coordinate transformations ϕ of M of the

split type, i.e. x′i = fi(x) and θ′α =∑

β gαβ(x)θβ . The only degree p delta form in Oδ(p)TM

is givenin a certain trivialization by

ω(p) = dx1 . . . dxpδ(dθ1) . . . δ(dθq). (9.40)

The part dx1 . . . dxp contributes with the determinant of the Jacobian of the change of coordinatesx′i = fi(x), while the Dirac-delta part contributes in the following way

δ(dθ′1) . . . δ(dθ′q) = δ

q∑

β=1

(∂θβθ′1)dθβ +

p∑

i=1

(∂xiθ′1)dxi

. . . δ

q∑

β=1

(∂θβθ′q)dθβ +

p∑

i=1

(∂xiθ′q)dxi

.

The part proportional to dx’s does not contribute by (9.38), due to the presence of dx1 . . . dxp, sothat one is left with

δ(dθ′1) . . . δ(dθ′q) = δ

q∑

β=1

(∂θβθ′1)dθβ

. . . δ

q∑

β=1

(∂θβθ′q)dθβ

=

(det

(∂θ′α∂θβ

))−1δ(dθ′1) . . . δ(dθ

′q) (9.41)

upon using the properties of the Dirac’s delta distributions. Putting the pieces together, we find

ω′(p) = det

(∂x′i∂xj

)det

(∂θ′α∂θβ

)−1ω(p) = Ber(J ac(ϕ))ω(p). (9.42)

This settle the degree p case. For degree lower than p it is enough to observe that, following remark

9.19, delta forms in Oδ(p−k)TM

are obtained via the DΩ1M-action of differential operators of order k

on the above section ω(p) ∈ ΣpM

∼= Oδ(p)TM

. In particular for degree p − 1, working locally, we have

an action of ∂dx’s and ∂dθ’s: these can be as linear maps acting on π∗Ω1M , hence they belong to

(π∗Ω1M )∗ ∼= π∗ΠTM . It follows that the transformation of a section ω(p−1) ∈ Oδ(p−1)

TM will be givenby

ω′(p−1) = J ac(ϕ)Π ⊗ Ber(J ac(ϕ)))ω(p), (9.43)

where J ac(ϕ)Π is the parity-transpose of the Jacobian of the change of coordinates, which identifies

the transition functions of the locally-free sheaf ΠTM . This yields the isomorphism Oδ(p−1)TM

∼=Ber(M ) ⊗ ΠTM = Σp−1

M: explicitly ∂dxi

7→ π∂xiand ∂dθα 7→ π∂θα , for any i = 1, . . . , p and

α = 1, . . . , q. In the very same fashion, higher degree differential operators are identified with

sections on SkΠTM , thus showing that Oδ(p−k)TM

∼= Ber(M )⊗ SkΠTM = Σp−kM

.

Remark 9.22 (Integration theory). It follows from the previous theorem that integration on su-

permanifolds via integral forms ΣkM parallels integration theory via delta forms Oδ(k)

TMon tangent

supermanifolds. While this is clear in the case of Berezinian, which accounts for integration on thefull supermanifold, it might be helpful to consider a codimension 1|0 example. To this end, let usconsider the supermanifold R1|2, with a system of coordinates given by x|θ1, θ2. The integral form


σ0 = D(x|θ1, θ2)θ1θ2 ⊗ π∂x ∈ H0Sp(M ) can be integrated over a codimension 1|0 sub-supermanifold

of M . In particular, we consider

N ..= x|θ1, θ2 ∈ R1|2 : x = 0 ⊂ R

1|2. (9.44)

The integral of σ0 over N is defined via the Poincare dual of N and using the pairing (8.8) betweenintegral and differential forms, i.e.

∫

N

σ0 ..=

∫

M

σ0 · ωN , (9.45)

where ωN ∈ H1dR ,c(M ) is given by ωN = δ(x)dx, with δ(x) a Dirac delta distribution centered in

zero. Plugging these into (9.45) one gets

∫

N⊂R1|2

σ0 =

∫

R1|2

D(x|θ1θ2)θ1θ2 ⊗ π∂x · δ(x)dx =

∫

R1|2

D(x|θ1, θ2)θ1θ2δ(x) = 1, (9.46)

where we have used the duality pairing between π∂x and dx, given by dx(π∂x) = 1.In a similar fashion, via delta forms, one first observe that σ0 corresponds to the delta formθ1θ2δ(dθ1)δ(dθ2). Now, instead of duality, the integral uses the Ω•M -module structure (or DΩ1

M-

structure), multiplying σ0 ∈ Oδ(0)TM

in the delta representation by the Poincare dual form ωN of N ,

which yields a delta form in Oδ(1)TM

. More precisely, one has

∫

N⊂R1|2

σ0 ..=

∫

TR1|2

D(x, dθ1, dθ2|θ1, θ2, dx)θ1θ2δ(dθ1)δ(dθ2)δ(x)dx

=

∫

R1|2

D(x|θ1, θ2)θ1θ2δ(x) = 1. (9.47)

Notice that the integral of σ0 on N only depends on the (co)homology on N . Let us indeed considerthe sub supermanifold

N ..= x|θ1, θ2 ∈ R1|2 : x+ θ1θ2 = 0 ⊂ R

1|2, (9.48)

which is in the same homology class of N . We have ωN= δ(x)dx+ dθ1θ2 − θ1dθ2 and it is easy to

see that∫

N

σ0 =

∫

M

σ0 · ωN=

∫

M

σ0 · (ωN + dη) =

∫

M

σ0 · ωN =

∫

N

σ0, (9.49)

where dη = d(θ1θ2) = dθ1θ2 − θ1dθ2. Notice that this consideration is totally general - and notlimited to the present example, as it relies on Stokes theorem 7.5 - indeed any summand of thekind σ0 · dη is exact and does not contribute to the Berezin integral - and Poincare duality 8.2.

Remark 9.23 (Delta Forms and Pseudoforms). Delta forms have been defined by requiring thatall of the coordinate dθ’s have distributional dependence of Dirac delta type. It was this veryrequirement that allowed to prove theorem 9.21 above, thus showing that the formalism of deltaforms is equivalent to that of integral forms that have previously defined.Nonetheless, the above requirement can be relaxed to a less stringent one, allowing, for example, a“mixed setting”, in which some of the dθ’s have distributional dependence (hence of the kind of adelta-integral form) and the remaining have a polynomial dependence (hence of a kind of differentialform) [76]. Even if there are important mathematical problems related to this framework - as weshall see -, this particular kind of (generally non-integrable) pseudoforms is that which is consideredin superstring perturbation theory [76, 78]. In this context the number of localized variable dθ’s isreferred to as picture number p of the (pseudo)form. In particular, forms having picture numberp = 0 are differential forms and forms having maximal picture number p = q (which equal the odddimension q of the supermanifold) are integral forms. An example of pseudoform having middledimensional - i.e. non minimal and non maximal - picture number can be given considering again

52 SIMONE NOJA

the easy case of R1|2. The most general pseudoform of picture p = 1 is given by

ω(x, dθ|θ, dx) =∞∑

k1,k2=0

fk1,k2(x|θ)(dθ1)k1δ(k2)(dθ2) +∞∑

ℓ1,ℓ2=0

gℓ1,ℓ2(x|θ)(dθ2)k2δ(k1)(dθ1)

+

∞∑

i1,i2=0

hi1,i2(x|θ)dx(dθ1)i1δ(i2)(dθ2) +∞∑

i1,i2=0

ci1,i2(x|θ)dx(dθ2)j2δ(j1)(dθ1),

(9.50)

where f, g, h, c ∈ OR1|2 for any choice of indices. The degree of these kind forms is defined asto agree with the definition given for differential and integral forms: a k-derivative of any delta’scounts −k. It is thus easy to see that there exists pseudoforms of middle picture 1 < p < q at anydegree, whereas differential forms have non-negative degree and we have defined integral formsso that they have degree lower or equal than the even dimension of the supermanifold. Further,working in the same way as above, one sees that any module of pseudoforms of a certain middle

dimensional picture 1 < p < q at a fixed degree k ∈ (−∞,+∞) - we call it Ωk,p

M - has an infinitenumber of generators, hence cannot be described as a vector bundles or locally-free sheaves ofOM -modules of finite rank. For example, pseudoforms of picture 1 and degree k on R1|2 would begenerated by

Ωk,1R1|2 = OR1|2 · (dθ1)ℓ1δ(ℓ2)(dθ2), dx(dθ1)j1+1δj2(dθ2), 1 ↔ 2, (9.51)

for any ℓ1 − ℓ2 = k and j1 − j2 = k − 1. On the other hand, the crucial problem which preventsfrom having a well-given mathematical definition of these modules of pseudoforms having a middledimensional picture 0 < p < q is rooted in the ill-defined transformation of a single delta (and its

derivative) δ(ki)(dθi). Indeed, as explained above in the discussion around equation (9.39), a singledelta δ(dθ) does not define a section of any a vector bundle on M and in particular it does notsurvive a change of coordinate. As it stands, a single δ(dθ) - and more in general an expressionconsisting of a non-maximal number of delta’s - is no more than a symbol: only delta forms -where all of the q dθ’s have distributional dependence of the Dirac delta type - do indeed yield awell-defined section of a vector bundle as seen in theorem 9.21.It is to be noted, though, that when restricted to a specific immersed sub-supermanifold of theright codimension in M , pseudoforms of non-maximal picture are well-behaved, as they define theintegral forms of the sub-supermanifold. As such, pseudoforms of non-maximal picture 0 < p < qare seen in relation to integration over general sub-supermanifolds of codimension k|q− p, for anyk = 0, . . . , p, depending on the degree of the pseudoform. It would be interesting to elucidate therelations between pseudoforms of non-maximal picture p and the d-densities defined by Manin in[53], chapter 4, paragraph 7.


Appendix A. Nilpotent Operators in Superalgebra

In this appendix we report an easy yet very useful result that gives a criterion to establish thenilpotency of an operator. This appears as lemma 3 in [53], chapter 3, section 4 - the reader isadvised of little confusing misprint in the proof. Let us consider the following generic setting: letS and T be A-modules, for A a supercommutative ring and let σa : S → S and τa : T → T twofamilies of homogeneous homomorphisms for a finite set of indices a = 1, . . . , n.We say that σa and σb commute if

[σa, σb] ..= σaσb − (−1)|σa||σb|σbσa = 0 (A.1)

We say that σa and σb anticommute if

σa, σb ..= σaσb + (−1)|σa||σb|σbσa = 0 (A.2)

Also we assume the following:

(1) |σa|+ |τa| does not depends on a;(2) the pairs (σa, σb) and (τa, τb) either commute or anticommute.

Then, it make sense to define the following operator

d ..=∑

a σa ⊗ τa : S ⊗A T // S ⊗A T

f = s⊗ t // d(f) ..=∑

a(−1)|τa||s|σa(s)⊗ τa(t).

We note that it has a well-defined parity (even or odd), since we have assumed that the parity of|σa|+ |τa| does not depend on a. We have the following lemma.

Lemma A.1. Let d ..=∑

a σa ⊗ τa be as above, then if either one of the following is satisfied

(1) d is even and the pair (σa, σb) and (τa, τb) have opposite commutation rules(2) d is odd and the pair (σa, σb) and (τa, τb) have the same commutation rules

then d d = 0.

Proof. Let us consider the expression for d2 ..= d d. One finds the sum

d2 ∋ (σa ⊗ τa)(σb ⊗ τb) + (σb ⊗ τb)(σa ⊗ τa) (A.3)

for some a, b. Commuting one has

(σa ⊗ τa)(σb ⊗ τb) + (σb ⊗ τb)(σa ⊗ τa)

= ((−1)|σb||τa| + (−1)|σa||τb|+|σa||σb|+|τa||τb|ǫσǫτ )(σaσb ⊗ τaτb) (A.4)

where ǫσ = ±1 and ǫτ = ±1 depending on the commutation relations of σ and τ .Say d is even. Then this implies that |σa| = |τa| and |σa| = |τa| and ǫσǫτ = −1 since one pair ofmorphisms commutes and the other anticommutes. Then one finds that

(−1)|σa||σb| − (−1)|σa||σb|+|σa||σb|+|σa||σb| = (−1)|σa||σb| − (−1)|σa||σb| = 0. (A.5)

Say d is odd. Then this implies that |σa| = |τa| + 1 and |σb| = |τb| + 1 and ǫσǫτ = +1 since thetwo pairs of morphisms both commutes or anticommutes. Then one finds

(−1)|σb||σa|+|σb| + (−1)|σa||σb|+|σa|+|σa||σb|+|σa||σb|+|σa|+|σb|+1

= (−1)|σb||σa|+|σb| + (−1)|σa||σb|+|σb|+1

= (−1)|σb||σa|+|σb| − (−1)|σa||σb|+|σb| = 0 (A.6)

Therefore, in both cases the sum in the parenthesis in (A.4) vanishes.

Notice that, even if it is given in the context of superalgebra, the above lemma A.1 applies to abroad range of geometrical constructions - for example, the ordinary de Rham differential can beimmediately proved to be nilpotent as a consequence of lemma A.1 as σa = dxa and τa = ∂a.

54 SIMONE NOJA

Appendix B. Right D-modules and Canonical Sheaf

In this appendix we prove lemma (5.14) and we comment further on the relations between the Liederivative and the DX -module structure of the canonical sheaf of an ordinary manifold X . Theinterested reader is invited to compare and appreciate the similarities of these constructions withthose of section 5.3, where the right DM -module structure on the Berezinian sheaf is discussed.

Lemma B.1 (ωX is a Right DX -module). Let X be a real or complex manifold and let ΩdimXX be

its canonical sheaf. Then ΩdimXX is a sheaf of right DM -module.

Proof. By the previous theorem 5.13, it is enough to show that we can define a flat right connectionon ωX . For sections ωtop ∈ ΩdimX

X and f ∈ OX and X ∈ TX we give the following definition

∆R (ωtop ⊗ f) ..= ωtopf

∆R (ωtop ⊗X) ..= −LX(ωtop). (B.1)

First off, observe that

∆R (ωtop ⊗ f X) = LfX(ωtop) = d ιfXωtop = d(fιXω

top)

= df ∧ ιX(ωtop) + fd(ιX(ωtop)) (B.2)

On the other hand, one has

∆R (ωtopf ⊗X) = LX(ωtopf) = ιX d(ωtopf) + d ιX(ωtopf)

= ιX(df ∧ ωtop + f ∧ dωtop) + d (ιX(ωtop)f)

= df ∧ ιX(ωtop) + fd(ιX(ωtop)) (B.3)

since df ∧ωtop = 0 = dωtop, so that ∆R (ωtop⊗f X) = ∆R (ω

topf ⊗X), which is the third definingproperty of a right connection. Further, we have that

∆R (ωtop ⊗X)f = −LX(ωtop)f, (B.4)

but also

∆R (ωtop ⊗X f) = −LfX(ωtop) + ωtopX(f) = −LX(ωtopf) + ωtopX(f)

= −LX(ωtop)f − ωtopX(f) + ωtopX(f) = −LX(ωtop)f, (B.5)

so that we have indeed ∆R (ωtop ⊗ X f) = ∆R (ω

top ⊗ X)f, which proves the second definingproperty for a right connection. Finally, it is an obvious property of the Lie derivative thatL[X,Y ] = [LX ,LY ], which settles flatness.

Remark B.2. Notice that it is crucial for the above to hold true that ωtop is really a sectionof the canonical sheaf. In other words, Ωi

X is not a right DM -module unless i = dimX . Also,notice that working locally in a chart U ⊂ X with local coordinates x1, . . . , xn such that a sectionof the canonical sheaf ΩdimX

X over U reads ωtopU = ω(x)f for some functions f ∈ OX(U) and

ω(x) = dx1 ∧ . . . ∧ dxn and considering a vector fields over U such that XU =∑

iXi∂xi

, then oneeasily has that

LX(ωtop) = ω(x)

n∑

i=1

∂xi(X if), (B.6)

indeed

LX(ωtop) = LX(ω(x))f + ω(x)LX(f) = dιX(ω(x))f + ω(x)X(f)

= ω(x)∂xi(X i)f + ω(x)X i∂xi

(f) = ω(x)∂xi(X if). (B.7)

Starting from the above (B.6), it can be seen that there exists a unique right connection on ΩdimXX

satisfying the condition ∆R (ω(x)⊗∂xi) for all i = 1, . . . , n in any coordinate system. In particular,

the following holds true.

Lemma B.3. Let X be a real or complex manifold and of dimension n let ΩnX be its canonical sheaf.

Then there exists a unique right connection on ΩnX such that

∆R (ω(x)⊗ ∂xi) = 0 (B.8)


for any i = 1, . . . , n and for all system of local coordinates (U, x1, . . . , xn), with ω(x) = dx1∧. . .∧dxna generating section of Ωn

X and ∂xi is a coordinate vector field over U .

Proof. It is immediate using (B.6) to see that indeed ∆R (ω(x) ⊗ ∂xi) = −L∂xi

(ω(x)) = 0.

Uniqueness follows from the fact that ∂xii=1,...,n is a system of generators for TM and ω(x) =

dx1 ∧ . . . ∧ dxn is a generator for ΩnX . It is an exercise to check that changing coordinates to

x′i = x′i(x) one still gets ∆R (ω(x′)⊗ ∂x′

i) = 0 for any i, thus concluding the proof.

Remark B.4. Notice that the above Lemma B.3 can be rephrased in terms of DX -module theoryby saying that the right DX - module structure on Ωn

X is uniquely characterized by the right action

ω(x) · ∂xi= 0. (B.9)

for any i = 1, . . . , dimX. This is to be related to theorem 5.19 and corollary 5.21

References

[1] M. A. Baranov, A. S. Schwarz, Cohomology of Supermanifolds, Funct. Anal. Appl. 18, 3 (1984)69-70 (in russian)

[2] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds, Reidel(1991)

[3] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, V. G. Pestov, Foundations of SupermanifoldsTheory: the Axiomatic Approach, Diff. Geom. Appl. 3 (1993) 135-155

[4] M. Batchelor, The Structure of Supermanifolds, Trans. Am. Math. Soc. 253 (1979) 329-338[5] A. Belopolsky, De Rham cohomology of the supermanifolds and BRST cohomology, Phys. Lett.

B 403 (1997) 47-50[6] A. Belopolsky, Picture Changing Operators in Supergeometry and Superstring Theory,

arXiv:9706033[7] A. Belopolsky, New Geometrical Approach to Superstrings, hep-th/9703183 (1997)[8] K. Bettadapura, Higher Obstructions of Complex Supermanifolds, SIGMA 14 (2018), 094[9] K. Bettadapura, Obstructed Thickenings of Supermanifolds, J. Geom. Phys. 139 (2019) 25-49[10] F.A. Berezin, Introduction to Superanalysis, D. Reidel Publishing (1987)[11] J. Bernstein, D. Leites, Integral forms and Stokes formula on supermanifolds, Funct. Anal.

Appl. 11, 1, 55-56 (1977).[12] J. Bernstein, D. Leites, How to integrate differential forms on supermanifolds, Funct. Anal.

Appl. 11, 3, 70-71 (1977).[13] J.N. Bernstein, D.A. Leites, Invariant Differential Operators and Irreducible Representation

of Lie Algebras of Vector Fields, Serdica, Bulg. Math. Journ. 7 (1981) 320-334[14] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer (1982)[15] S. Cacciatori, S. Noja, R. Re, Non Projected Calabi-Yau Supermanifolds over P2, Math. Res.

Lett. 26 (4) (2019) 1027-1058[16] S.L. Cacciatori, S. Noja, R. Re, The Unifying Double Complex on Supermanifolds,

arXiv:2004.10906[17] L. Castellani, R. D’Auria, P. Fre, Supergravity and Superstring: a Geometric Perspective, Vol

1, World Scientific (1991)[18] R. Catenacci, M. Debernardi, P.A. Grassi, D. Matessi, Cech and de Rham Cohomology of

Integral Forms, J. Geom. Phys. 62, 4 (2012) 890-902[19] L. Castellani, R. Catenacci, P.A. Grassi, The Integral Form of Supergravity, JHEP 10 (2016)

049[20] L. Castellani, R. Catenacci, P.A. Grassi, Super Quantum Mechanics in the Integral Forms

Formalism, Ann. Henri Poincare 19 (5), 1385-1417 (2018)[21] R. Catenacci, P.A. Grassi, S. Noja, A∞-Algebra from Supermanifolds, Ann. Henri Poincare

20 (12) 4163–4195 (2019)[22] R. Catenacci, P.A. Grassi, S. Noja, Superstring Field Theory, Superforms and Supergeometry,

J. Geom. Phys., 148 103559 (2020)[23] R. Catenacci, C. Cremonini, P.A. Grassi, S. Noja, On Forms, Cohomology, and BV Laplacians

in Odd Symplectic Geometry, Lett. Math. Phys. 111 (2), (2021) 1-32[24] R. Catenacci, C. Cremonini, P.A. Grassi, S. Noja, Cohomology of Lie Superalgebras: Forms,

Integral Forms, and Coset Superspaces, arXiv:2012.05246

http://arxiv.org/abs/hep-th/9703183

http://arxiv.org/abs/2004.10906


56 SIMONE NOJA

[25] C.A. Cremonini, P.A. Grassi, Pictures from Super Chern-Simons Theory, JHEP 03 (2020) 043[26] C.A. Cremonini, P.A. Grassi, S. Penati, Supersymmetric Wilson Loops via Integral Forms,

JHEP (2020) 161[27] C.A. Cremonini, P.A. Grassi, Cohomology of Lie Superalgebras: Forms, Pseudoforms, and

Integral Forms, arXiv:2106.11786[28] P. Deligne, J. Morgan, Notes on Supersymmetry (following Joseph Bernstein), in Quantum

Field Theory and Strings: a Course for Mathematicians, Vol 1, AMS (1999)[29] P. Deligne, D. S. Freed, Supersolutions, in Quantum Field Theory and Strings: a Course for

Mathematicians, Vol 1, AMS (1999)[30] R. Donagi, E. Witten, Supermoduli Space is Not Projected, Symp. Pure Math. 90 (2015) 19-72[31] D. Eisenbud, Commutative Algerbra - with a view toward Algebraic Geometry, Springer GTM

(1995)[32] R. Fiorese, M. A. Lledo, The Minkowski and Conformal Superspaces: The Classical and

Quantum Descriptions, World Scientific (2015)[33] D. S. Freed, Five Lectures on Supersymmetry, AMS (1999)[34] D. Fridan, E. Martinec, S. Shenker, Covariant Quantization of Superstrings, Phys. Lett. B 60

(1985) 55[35] D. Fridan, E. Martinec, S. Shenker, Conformal Invariance, Supersymmetry and String Theory,

Nucl. Phys. B (1986) 93[36] A.V. Gaiduk, H.M. Khudaverdian, A.S. Schwarz, Integration over Surfaces in Superspace,

Theor. Math. Phys. 52 (1982)[37] K. Gawedzki, Supersymmetry - Mathematics of Supergeometry, Ann. Inst. H. Poincare Sect.

A (N.S.) 27 (1977) 335-366[38] P. Green, On Holomorphic Graded Manifolds, Proc. Am. Math. Soc. 85, 4 (1982)[39] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley (1978)[40] R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, Perverse Sheaves and Representation Theory,

Birkhauser (2008)[41] M. Kapranov, Supergeometry in Mathematics and Physics, in New Spaces in Physics, M. Anel,

G. Catren eds, CUP (2021)[42] H.M. Khudaverdian, R.L. Mkrtchian, Integral Invariants of Buttin Bracket, Lett. Math. Phys.

18, 229-231 (1989)[43] H.M. Khudaverdian, Geometry of superspace with Even and Odd Brackets, J. Math. Phys 32,

1938-1941 (1991)[44] H.M. Khudaverdian, A.P. Nersessian, On Geometry of Batalin-Vilkovisky Formalism, Mod.

Phys. Lett. A 8 (25), 2377-2385 (1993)[45] H.M. Khudaverdian, A.P. Nersessian, Batalin-Vilkovisky Formalism and Integration Theory

on Manifolds, J. Math.Phys 37, 3713-3724 (1996)[46] H.M. Khudaverdian, Odd Invariant Semidenstiy and Divergence-like Operators on Odd Sym-

plectic Superspace, Comm. Math. Phys. 198, 591-606 (1998)[47] H. M. Khudaverdian, Laplacians in Odd Symplectic Geometry, Contemp. Math. 315, 199-212

(2002)[48] H. M. Khudaverdian, Semidensities on Odd Symplectic Supermanifolds, Commun. Math.

Phys., 247, 353-390 (2004)[49] H. M. Khudaverdian, Th. Th. Voronov, Differential forms and odd symplectic geometry, Ge-

ometry, Topology and Mathematical Physics. S. P. Novikov seminar: 2006-2007, V. M. Buch-staber and I. M. Krichever, eds. AMS Translations, Ser. 2, Vol. 224, Amer. Math. Soc.,Providence, RI, 2008, 159-171

[50] B. Konstant, Graded Manifolds, Graded Lie Theory, and Prequantization, in Differential ge-ometrical methods in mathematical physics, K. Bleuler and A. Reetz (eds.) 177-306, LectureNotes in Mathematics 570, Springer-Verlag (1977)

[51] D. A. Leites, Introduction to the Theory of Supermanifolds (Russian), Uspekhi Mat. Nauk.35 (1980) 3-57, English translation in Russian Math. Surveys 35 (1980) 1-64

[52] P. Maisonobe, C. Sabbah, Aspect of the Theory of D-modules, Lecture Notes - Kaiserslautern 2002, available athttp://www.math.polytechnique.fr/cmat/sabbah/livres/kaiserslautern.pdf

[53] Yu. I. Manin, Gauge Fields and Complex Geometry, Springer-Verlag, (1988)


http://www.math.polytechnique.fr/cmat/sabbah/livres/kaiserslautern.pdf


[54] Yu. I. Manin, I. B. Penkov, A. A. Voronov, Elements of Supergeometry, J. Soviet Math. 51(1990) 2069-2083

[55] P. Mnev, Quantum Field Theory: Batalin-Vilkovisky Formalism and its Applications, AMS(2019)

[56] S. Cacciatori, S. Noja, R. Re, The Unifying Double Complex on Supermanifolds,arXiv:2004.10906

[57] S. Noja, R. Re, A Note on Super Koszul Complex and the Berezinian, Ann. Mat. Pura Appl.(2021), https://doi.org/10.1007/s10231-021-01121-6

[58] O.V. Ogievetskii, I.B. Penkov, Serre Duality for Projective Supermanifolds, Funct. Anal. itsAppl. 18 68-70 (1984)

[59] O.V. Ogievetskii, private communication[60] I. B. Penkov, D-Modules on Supermanifolds, Invent. Math. 71, 501-512, (1983)[61] V. Pestun, M. Zabzine, eds., Localization techniques in quantum field theories, J. Phys. A:

Math. Theor., 50 (44) 440301 (2016)[62] D. Hernandez Ruiperez, J. Munoz Masque, Construction Intrinsique du faisceau de Berezin

d’une variete graduee, C. R. Acad. Sc. Paris 301 915-918 (1985)[63] P. Severa, On the Origin of the BV Operator on Odd Symplectic Supermanifolds, Lett. Math.

Phys. 78 55-59 (2006)[64] A. S. Schwarz, Geometry of Batalin-Vilkovisky Quantization, Comm. Math. Phys. 155 (1993)

249-260;[65] M. Shifman (edited by), Felix Berezin: Life and Death of the Mastermind of Supermathemat-

ics, World Scientific (2007)[66] Y. Su, R. Zhang, Mixed Cohomology of Lie Superalgebras, J. Alg. 549 (2020) 1-29[67] V.S. Varadarajan, Supersymmetry for Mathematician: an Introduction, Courant Lecture

Notes, AMS (2004)[68] T. Voronov, A. Zorich, Complexes of forms on a supermanifold, Funct. Anal. Appl. 20 (2),

58-59 (1986)[69] T. Voronov, A. Zorich, Integral Transformations of Pseudodifferential Forms, Russian Math.

Surv. 41 (6), 221-222 (1986)[70] T. Voronov, A. Zorich, Integration on vector bundles, Funct. Anal. Appl. 22 (2), 94 103 (1988)[71] T. Voronov, Supermanifold forms and integration. A dual theory, In: Solitons, Geometry, and

Topology: On the Crossroad, AMS Translations, ser. 2, 179, Providence, RI, pp. 153-172(1997)

[72] T. Voronov, Quantization of Forms on the Cotangent Bundle. Comm. Math. Phys. 205, 315-336 (1999)

[73] T. Voronov, Dual forms on supermanifolds and Cartan calculus, Comm. Math. Phys. 228,1-16 (2002)

[74] Th. Th. Voronov, Geometric Integration Theory on Supermanifolds, Cambridge Scientific Pub-lisher (2014)

[75] C. Voisin, Hodge Theory and Complex Algebraic Geometry, Cambridge Studies in AdvancedMathematics (2002)

[76] E. Witten, Notes on Supermanifolds and Integration, Pure Appl. Math. Q., 15 (1) 3–56 (2019)[77] E. Witten, Notes on Super Riemann Surfaces and Their Moduli, Pure Appl. Math. Q., 15 (1)

57–211 (2019)[78] E. Witten, Superstring Perturbation Theory via Super Riemann Surfaces: an Overview, Pure

Appl. Math. Q., 15 (1) 517–607 (2019)

Universitat Heidelberg

Current address: Im Neuenheimer Feld 205Email address: [email protected]


https://doi.org/10.1007/s10231-021-01121-6

arXiv:2111.12841v1 [math.AG] 24 Nov 2021

Documents

Transcript of arXiv:2111.12841v1 [math.AG] 24 Nov 2021