Nuclear Physics B363 (1991) 185-206 North-Holland

GRAVITATIONAL ANYONS, CHERN-SIMONS-WI’ITEN GRAVITY AND THE GRAVITATIONAL AHARONOV-BOHM EFFECT

Miguel E. ORTIZ*

The Blackett Laborutory, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 282, UK

Received 23 February 1991

The anyonic behaviour of massive point particles in the (2 + D-dimensional theory of topologically massive gravity, as first suggested by Deser, is further investigated. At the linearised level, we recall that the exotic statistics of the particles may be understood in terms of the gravitational Aharonov-Bohm effect discussed by several authors in the 1960’s. This approach suggests that massive spinning point particles in ordinary Einstein gravity may also exhibit exotic statistics, and that it is the Einstein term which takes on the role carried out by the Chern-Simons term in the standard U(1) anyon model. We investigate this further in the Chern-Simons-Witten formulation of 2 + 1 gravity, where the Einstein term is indeed a Chem-Simons term. In this context the exotic statistics may be understood in terms of a non-abelian Aharonov-Bohm effect, giving a description in terms of the fully non-linear theory of gravity, and making explicit the relationship between gravitational anyons and Chern-Simons anyons of (2 + I)-dimensional U(1) gauge theory. The analogue of magnetic flux in this context is seen to be both torsion and Riemann curvature. We finally show the correspondence between the statistics phase obtained in this approach and that obtained by Deser and McCarthy in the linearised approximation of topologically massive gravity.

1. Introduction

The possibility of exotic spin and statistics in 2 + 1 dimensions is well under- stood in terms of the topology of the configuration space Q of n identical particles [l, 21. As a consequence there is no fundamental restriction to bosons and fermions in 2 + 1 dimensions, and one must also consider anyons, particles with inherent spin and statistics which are neither bosonic nor fermionic. It is perhaps more interesting to consider physical theories in 2 + 1 dimensions in which bosons (or fermions) acquire exotic spin and statistics through an external mechanism. In these theories, the mechanism gives different quantum mechanical weights to trajectories which are topologically distinct, and as a result exotic spin and statistics are said to be “dynamically realised”, rather than being fundamental.

* SERC Research Fellow.

0550-3213/91/$03.50 0 1991 - Elsevier Science Publishers B.V. All rights reserved

186 M.E. Ortiz / Gral:itatiotral atlyons

Perhaps the most interesting mechanism found to date is that in which exotic statistics arise from the coupling of point particles to electrodynamics with a Chern-Simons action [3,4]. The statistics in this model are explained by a 2-dimensional Aharonov-Bohm effect, since the Chern-Simons coupling has the effect of inducing a magnetic flux point at the position of each charged particle. This behaviour is all the more interesting as the picture is unchanged (for long range interactions) by the addition of the usual Maxwell term in the electromag- netic action. This model of dynamically realised anyons is now thought to play a role in high T, superconductivity [5].

Although several other dynamical mechanisms for inducing exotic statistics have been proposed [6,7], the discovery by Deser that it is sufficient to couple massive point particles to topologically massive gravity (where a topological term known as the gravitational Chern-Simons term is added to the usual Einstein action) is striking. The first evidence of this anyonic behaviour comes from the similarity between the action of a charged point particle in an electromagnetic field and the linearised action of a massive point particle in a gravitational field [8]. This similarity allows one to define the gravitational analogue of a magnetic flux point, at least at the linearised level, giving rise to what several authors have described as the “gravitational Aharonov-Bohm effect” [9-121 (similar ideas have been applied more recently in, for example refs. [13,14]). The special role of the topologically massive theory of gravity in Deser’s model is explained by the fact that the gravitational field of a massive point particle in that theory is precisely the flux point analogue. Hence the gravitational Aharonov-Bohm effect appears to turn massive point particles into anyons, by a mechanism similar to that observed in Chern-Simons electrodynamics.

Although in a subsequent calculation, Deser and McCarthy have calculated the statistics phase in a more rigorous linearised framework, the status of gravitational anyons remains unclear for the following reasons. Firstly, the gravitational Aharonov-Bohm effect is a result of the linearised approximation to gravity. It is by no means clear that the results obtained in this approximation should survive a transition to the full theory, nor is it clear that the linearised approximation is an appropriate one to make in the first instance [15]. Secondly, an analysis in terms of an Aharonov-Bohm effect is incomplete without identifying the gravitational analogues of magnetic flux and electric charge which play such a crucial role; although the gravitational Aharonov-Bohm effect has been much discussed in the literature, this question appears to be as yet unanswered. It will be our objective in this paper to shed light on both of these questions.

The essential observation we make below is that the above questions may be addressed by looking at the gravitational anyon model in an appropriate context. This context, we argue below, is the Chern-Simons-Witten (CSW) formulation of (2 + l)-dimensional Einstein gravity, where the Einstein action is replaced by an equivalent ISO(2,l) Chern-Simons action (not to be confused with the gravita-

M.E. Ortiz / Gra~drational anyotrs 187

tional Chern-Simons term), and gravity is thus re-expressed as a ISO(2,l) gauge theory. There are two features of Deser’s gravitational anyon model in the Aharonov-Bohm interpretation which suggest that the CSW formalism is appro- priate. The first is the observation that a massive point particle with spin in Einstein gravity gives rise to the same asymptotic gravitational field as a massive spinless point particle in topologically massive gravity [16,17]. This suggests that the same mechanism which makes massive point particles have exotic statistics in topologically massive gravity should induce exotic statistics for massive spinning particles in Einstein gravity. The Einstein action in 2 + 1 dimensions appears to act in a very similar way to the abelian Chern-Simons term, giving rise to “flux points” at the position of each spinning particle. The second feature directly suggests the use of the CSW formulation rather than the usual metric formulation of Einstein gravity. This is the suggestion that torsion may play the role of magnetic flux in the gravitational Aharonov-Bohm effect. The component of the linearised metric giving rise to the Aharonov-Bohm effect is proportional to the spin of the particle, and in 2 + 1 dimensions it is well known that a spinning point particle gives rise to a metric which has a torsion singularity at the origin. We therefore need to include torsion as part of the curvature of our theory of gravity, which may be achieved by considering gravity as an ISO(2,1> gauge theory. In 2 + 1 dimensions this leads precisely to the CSW theory of gravity, an ISO(2,l) gauge theory with a Chern-Simons action.

Particle scattering in the CSW theory has been considered by various authors [lg-201. In general, the problem of scattering of particles with non-abelian charge is considerably more complicated than the abelian case, since non-abelian charge is vector valued and labels internal degrees of freedom. It is this non-abelian charge which is identified with the momentum and angular momentum of the particles, with the Casimir invariants labelling the mass and intrinsic spin of each particle. The charges also determine how the particles couple to the gauge (gravitational) field. The trajectory of the particle in space-time, as we shall illustrate below, contributes only through its topology. In a quantum mechanical treatment of point particles coupled to the ISO(2,l) gauge field, the charges become operators from which the Wilson loop observables are constructed, and give rise to representations of ISO(2,l) fixed by the mass and spin of each particle.

To date, most authors [18,20] have restricted their attention to particles carrying no spin. For the case of a test particle scattering off a fixed source, there is an equivalence between their results and earlier work on point particle scattering in (2 + l)-dimensional Einstein gravity by ‘t Hooft and others [21]. In the spinless case it is clear from ref. [20] that topologically distinct trajectories in space-time carry the same topological phase, and hence the particles are not anyons. An analysis of spinning particles in CSW theory has been carried out by de Sousa Gerbert [19]. In this case, he argues that a topological phase must be included, although the origin and consequences of this phase are not discussed in detail.

188 ME. Ortiz / Gravitational anyons

In sect. 2 we begin by giving a brief review of both Einstein and topologically massive gravity in 2 + 1 dimensions, and in particular of the gravitational field of point particles in both theories. We then go on to discuss the gravitational Aharonov-Bohm effect, and how this gives rise to gravitational anyons. Sect. 3 deals with the scattering of point particles in CSW theory. We review the CSW theory, and examine the topological consequences of the interaction term in the hamiltonian for massive spinning point particles to bring out the analogy with charged particles in Chern-Simons electrodynamics. In sect. 4 we discuss the relationship between the results of sect. 3 and those of Deser and McCarthy for their gravitational anyon model, and give a more detailed explanation of the role of topologically massive gravity in their model.

2. (2 + lbdimensional gravity and linearised gravitational anyons

2.1. EINSTEIN GRAVITY

It is well known that general relativity in 2 + 1 dimensions is trivial in the sense that Einstein’s equations

Gpy = hGT,,

imply that the Riemann tensor is non-zero only at the location of matter, and as a result that the theory has no local degrees of freedom [17,22,23]. This means that the newtonian limit of general relativity is lost in 2 + 1 dimensions and we are left with an apparently uninteresting theory. However, by coupling point particles to gravity, we obtain some non-trivial space-times with what will turn out to be extremely interesting properties.

A massive spinless point particle with energy momentum tensor

T,, = mC3*( r) , T,‘=O and ‘;.i=O, (1)

coupled to Einstein gravity gives rise to a conical space-time [24]

ds* = dt* - r-8Gm(dr2 + r* de*) m > 0, (2)

which is clearly flat everywhere except at the position of the particle. At the origin, there is a curvature singularity given by

R = 8rrGm6*( r) .

This space-time is familiar from the theory of cosmic strings, where it appears as the transverse section of the metric of a static, straight string. As a result, its properties have been extensively studied. Despite being flat everywhere, it gives

ME. 0rti.z / Gravitational anyons 189

rise to double imaging and a classical Aharonov-Bohm effect due to a non-trivial holonomy around curves encircling the origin, a result of the non-trivial global properties of the space-time (for a discussion of the gravitational effects of cosmic strings as cones, see ref. [25] and references therein).

The co-ordinates in which the conical metric is given in (2) are well-defined on lR2 - {O} in the sense that 0 = 0 is identified with 0 = 2~. We may of course cast the metric in a Minkowski form

ds2 = dt2 - dz2 - d R2 - R2 de2,

by making a change of co-ordinates. However, a consequence of this change of co-ordinates is that 0 = 0 is identified with 0 = 27r(l - 4Gm). In these now co-ordinates the Levi-Civita connection vanished and the non-trivial nature of the space-time (holonomy) is contained in the boundary condition. This change of co-ordinates is reminiscent of the change of gauge in the Aharonov-Bohm effect which removes the gauge field and puts the non-trivial quantum interaction in the boundary conditions obeyed by the wave function [l]. In our treatment of the gravitational interaction produced by the conical metric and the generalisations we shall discuss below, we shall always work in co-ordinates in which the gravitational interaction arises from a non-zero connection rather than from non-trivial bound- ary conditions to make explicit the origin of any interactions.

The conical solution generalises exactly to a static multi-centre equivalent [17]

ds2 = dt2 - nlr -riIFSGfni(dr2 + r2 de’). i

(3)

for particles of mass mi located at ri, subject to the constraint that the masses give rise to a total deficit angle of less than 21r. The existence of these solutions illustrates the absence of a classical gravitational interaction between two point particles in 2 + 1 dimensions.

In the case of a point particle with spin, with energy momentum tensor

T,, = mS2( r) ) c.= -$&3jS2(r) and qj=O, (4)

the metric [17]

ds2 = (dt + 4Ga d0)2 - r-8G”‘(dr2 + r2 de’), (5)

is no longer static. It has not only a curvature singularity at the origin, correspond- ing to a non-trivial holonomy around the origin, but also a torsion singularity, indicated by the non-closure parallelograms encircling the origin. This feature is most simply understood in terms of flat space co-ordinates where the metric is

190 M.E. Ortiz / Gra~~itntional anyoru

minkowskian. Again changing co-ordinates so that the metric becomes

ds2 = dT2 - dz2 - dR2 - R2 dB2,

leads to unusual boundary conditions. We must now identify the point (0, T> with (0 + 2rr(l- 4Gm, T + 87rG~). The metric can thus be seen to have a time helical structure, which illustrates the presence of torsion. In this spinning case there are also multi-centre solutions [26].

Although classically the existence of multi-centre solutions to the Einstein equations indicates that there is no gravitational interaction between particles, there is an interaction between particles at a quantum mechanical level, due to the global properties of the gravitational field. A detailed treatment of this quantum scattering is easy in 2 + 1 dimensions since, as mentioned above, the gravitational interaction may be transferred to the boundary conditions. An analysis of point particle scattering of this type has been carried out by various authors [21,27,28].

2.2. TOPOLOGICALLY MASSIVE GRAVITY

In the topologically massive theory of gravity [291, an SO(2,l) Chern-Simons term is added to the standard Einstein action, so that

I=I,+ (l/P)IGCS. (6)

The new term arises from considering the spin connection 6~‘~ as an SO(2,l) gauge potential and constructing the corresponding Chern-Simons term,

I GCS = /( M3 uab A duba + $,Y’/, A 6~“~ A w’,) . (7)

However, the spin connection is itself a function of the dreibein ea and one cannot therefore consider the new term in the action as a Chern-Simons term in the usual sense. In particular, this term is third order in derivatives whereas a Chern-Simons term is first order. One property of the gravitational Chern-Simons term, common also to Chern-Simons terms in gauge theories, is that the term breaks parity invariance.

As a result of the new term, gravity acquires one local degree of freedom, and the Einstein equations are replaced by the Einstein-Cotton equations

R,, + W-W,, =O.

Here C,, is the Cotton tensor [30] and is defined by

cju = EpcQ3 (V”R& + $i; VPR) . (9

M. E. Ortk / Gravitational anyoru 191

The Cotton tensor [30] is trace and divergence free and vanishes identically if the space-time is conformally flat, making it the three-dimensional analogue of the (otherwise undefined) Weyl tensor. The new Einstein-Cotton equations no longer constrain the curvature to vanish in the absence of matter, and indeed a series of non-trivial vacuum solutions are known [31-341.

The sources we shall consider are the same as those discussed above for Einstein gravity, namely point masses with and without spin. The corresponding energy momentum tensor is (4),

T,, = m6*( r) , r,i = - $xiidji3*( r) and qj = 0,

where the spin 2Gma may vanish. Deser [16] and Linet [35] have both studied eqs. (8) in the linearised approxima-

tion for a point source of the above type. They show that the weak-field metric for spinning point particles takes the form,

h,, = 8n-G( m + pu)Y,

hoi= (8rG/p)(m +pa)Eijdj(C- Y),

where glLy = rl,, + h,, and C(r) and Y(r) are defined by

-PC(x) =62(x), C(r) = -(1/27r)lnr,

(-V*+p*)Y(x) =6*(x), Y(r) =(1/27r)K,(pr).

Here KJpr) is a generalised Bessel function which behaves like -ln(pr) at the origin and like e -I*“/* at infinity. We see that the solutions have the asymp- totic form

ds*w (dt+ 4G[a+m/p])2-r-8G”‘(dr2+r2df32)

which is that of the pure Einstein metric (5) for a particle with mass m and spin (2Gm*/p.) + 2G ma. Hence the effect of the Chern-Simons interaction is to induce an extra spin 2Gm*/p on the space-time of the particle. In particular, the solution with m + pa = 0 is asymptotically conical in the linearised approximation.

In the case of a source with vanishing spin, the metric is not static, but stationary. In this sense the gravitational field carries angular momentum. This feature is a result of the parity non-invariance of the gravitational Chern-Simons term, and is common to Chern-Simons terms in gauge theories. For example, in the self-dual Chern-Simons soliton of Jackiw and Weinberg [36], the scalar field is

192 ME. Ortiz / Grauitational anyons

in a “static” configuration, but the electromagnetic field has an induced parity- breaking spin due to the Chem-Simons interaction.

It is interesting to note at this stage that a full non-linear solution to the Einstein-Cotton eqs. (8) with a distributional matter source has only been found in the special case m + pa = 0 where the cone is an exact solution [37]. Away from this limit, it has recently been shown by ClCment [151 that no exact solutions with the asymptotics given by Deser and Linet exist. One would however expect to find solutions for a physical (non-distributional) source with spin such as the Chem-Simons soliton [36].

2.3. THE GRAVITATIONAL AHARONOV-BOHM EFFECX

Before describing the gravitational anyon model of Deser and McCarthy, let us recall the existence of a gravitational analogue of the Aharonov-Bohm effect which was first pointed out by several authors in the 1960’s [g-12]. This effect arises from considering the quantum mechanics of a gravitating point particle in a gravitational field, and is not restricted to two space dimensions.

It is well known that the presence of localised curvature can have effects on geodesic motion and parallel transport in regions where the curvature vanishes. The best known example of this is provided by the aforementioned non-trivial holonomy of the Levi-Civita connection around an idealised cosmic string with a conical space-time. This behaviour is a type of Aharonov-Bohm effect, but is purely classical in character. There is an even more direct relationship between the quantum Aharonov-Bohm effect and gravity if we look at the lagrangian for a non-relativistic test particle in a weak gravitational field.

The lagrangian for a point particle in a gravitational field is given by

The corresponding hamiltonian, in the limit of small velocities, and taking

g ,,v = rl,,v + h,,

in the weak field approximation, becomes [8]

H=(1/2m)(p-A)*+V.

This is just like the hamiltonian for a charged non-relativistic particle in an electromagnetic field, but now Ai = mhoi, V= - imh,, and the charge e has been replaced by the mass m of the particle.’ Hence, in the particular case where h, = 0, the (non-dynamical) phase change for a closed particle trajectory is given

ME. Ortiz / Gravitational anyons 193

This result is particularly interesting in 2 + 1 dimensions, where a direct ana- logue of the Aharonov-Bohm solenoid is provided by the metric for a spinning point particle (51, or by the asymptotic form of the metric for a massive spinless point particle in topologically massive gravity, which has a non-vanishing ha,. and a vanishing h,*. The spin of the particle is the source of the “flux of the solenoid”, and the metric is of course flat everywhere outside the source. Thus all the results for the standard Aharonov-Bohm effect carry over to this model.

From the starting point of the gravitational Aharonov-Bohm effect, it is straightforward to see how to dynamically realise exotic statistics by coupling massive point particles to topologically massive gravity. At the location of each particle the gravitational field resembles a “flux point”,

h,, = 4Gm/p,

and therefore interchange of two particles gives rise to an Aharonov-Bohm phase

which may be interpreted as corresponding to exotic statistics. The analogy between the gravitational and abelian Chern-Simons realisations is very close, with mass giving rise to a “flux point” of spin in the gravitational field just as in the standard anyon model electric charge gives rise to a magnetic flux point.

This is the essence of Deser’s gravitational anyon model [161, although in the subsequent treatment given by Deser and McCarthy [38], a more rigorous deriva- tion of the spin and statistics phase of the particles is given. According to their calculation, the statistics phase is

which is half of that obtained directly from the gravitational Aharonov-Bohm effect. Exactly the same occurs in the abelian anyon model, where the factor of 2 is explained by an additional contribution to the statistics phase arising from the change in the field configurations which takes place under an interchange [4]. It is natural to assume that there is a similar explanation in the gravitational case, although the calculation is considerably less straightforward.

*This metric has a natural generalisation in 3 + 1 dimensions where it has been interpreted by de Sousa and Jackiw as the metric for a spinning cosmic string [28].

194 M.E. 0th / Gravitational anyons

An immediate observation arising from this presentation of gravitational anyons is that the realisation of exotic statistics through interaction with the gravitational field need not be restricted to topologically massive gravity. At first sight, the presence of the gravitational Chern-Simons term in the gravitational action might appear necessary (after all a Chern-Simons term is needed to implement exotic statistics in U(1) gauge theory). However, the essential requirement in any realisa- tion of anyons relying on an Aharonov-Bohm effect is that a “charged” particle gives rise to a “flux point”. In the gravitational model, a flux point, corresponding to a non-vanishing hai, is produced by a point particle in pure Einstein theory provided that the particle has a “charge” which is now not its mass but rather a non-zero spin. The only effect of the gravitational Chern-Simons term is to give rise to an Aharonov-Bohm effect and exotic statistics for massive spinless particles. If we are willing to sacrifice this admittedly attractive feature of the gravitational anyon model, then we may proceed in a pure Einstein theory. To summarise, it is evident from this analysis that massive, spinning point particles in Einstein gravity are as much gravitational anyons as massive particles in topologically massive gravity. The special role of the Einstein term in the gravita- tional anyon model was already suggested in Deser’s original paper, where he pointed out that the Einstein term dominates at large distance from the sources, determining their long-range interactions.

The second question which one is led to ask by the (non-relativistic, weak field) gravitational Aharonov-Bohm effect, is how this effect survives a transition to a non-linear treatment of the gravitational field, where there should be a fully covariant description. In this case, we would expect the flux arising from the potential h,, to have an invariant interpretation in terms of curvature as in electromagnetism. If we seek to identify this gravitational analogue of magnetic flux, a clue is provided by the form of the metric (5) for a spinning point particle. As remarked above, this metric has not only a curvature singularity at the origin (which we expect at this stage not to play a part in the Aharonov-Bohm effect since there is no effect around a conical space-time) but also a torsion singularity. This torsion singularity is proportional to the spin u of the source, and would therefore seem to play an important part. However, at the linearised level it does not seem to be possible to identify

s6 hOi dri

with torsion, suggesting that the description is incomplete. Since it seems that the natural theory in which to view gravitational anyons is one which incorporates torsion, we are led naturally to consider an ISO(2,l) gauge theory formulation of gravity, in which both the curvature and the torsion appear as components of the curvature.

M.E. Ortiz / Gravitational anyons 195

Both the above observations - that it is the Einstein action that induces the “flux points” and which should be thought of as a Chern-Simons type term, and that we should consider an ISO(2,l) formulation of gravity - lead us to look at the CSW theory of gravity, where one may hope to find a coherent picture of gravitational anyons. The scattering of point particles in this theory has already been considered by various authors [l&20], and we shall make use of some of their results to better understand gravitational anyons and the gravitational Aharonov-Bohm effect.

3. Chern-Simons-Witten theory

3.1. GRAVITY AS AN ISO(2,l) GAUGE THEORY IN 2 + 1 DIMENSIONS

In order to write down general relativity in a form which most closely resembles a gauge theory, it is appropriate to employ the first-order formalism where the vielbein and the spin connection are independent variables. In n dimensions, a gauge connection with gauge group ISO(n - 1,l) can be constructed as

A, = ea,, P, + coaDp Jo,,

where P, and Job are the natural generators of ISO(n - l,l> (for a general discussion, see some of the articles contained in ref. [39]). With this gauge connection, the curvature can be seen to take the form

Fwy = T”p,,P, + RablrvJob

where T” and Rab are the torsion and curvature form constructed from e” and w ab respectively.

In 2 + 1 dimensions, we may define J” = ~~~~~~~~ and the P, and Jo satisfy the algebra

[J,,J,] =EabcJC,

[pn,p,] =o, making explicit the semi-direct product structure of ISO(2,l).

Defining 0~‘~ = - ~&abC~bcP~ the ISO(2,l) connection becomes

A, = eal, P, + co’,, J, .

Under a gauge transformation u = u“T,, A changes by SA = -Du = - du -

196 M.E. 0rti.z / Gravitational anyons

[A, u]. Writing u as u = p”P, + K’IJ~, the components of the connection change as

6e” = - dp” - E”bcebKC - .?RbcWbpc (11)

(12)

The parameter K’ is related to Lorentz transformations, whilst p” is related to diffeomorphisms [40]. From this expression it is apparent that a general gauge transformation with non-zero pa mixes the dreibein and the spin connection, confirming that what one might like to think of as the metric,

is not a gauge invariant quantity. The existence of a non-degenerate inner product in 2 + 1 dimensions [40]

defined by

(13)

allows us to write down an action for the gauge field A,. Witten’s crucial observation is that the Chern-Simons action

I csw - - -(1/16pG)/( AdA + $4 A A A A), (14)

gives rise to equations of motion which make this gauge theory equivalent to general relativity, with diffeomorphisms equivalent on-shell to ISO(2,l) gauge transformations.

The field equations for the Chern-Simons action are simply

which are equivalent to

*F=O,

Ta=O and Rab=O.

Therefore the space of classical solutions is equivalent to that of general relativity in 2 + 1 dimensions, with the exception that in the Chern-Simons formalism, eaP need not be invertible. The phase spaces of the Einstein and Witten theories may be identified as a tangent bundle over the moduli space of flat SO(2,l) connec- tions modulo gauge transformations [40,41-J, and quantisation is then straightfor- ward since the bundle is finite-dimensional.

ME. Ortiz / Gravitational anyons

3.2. POINT PARTICLE SCATTERING

191

Although it is not possible to couple arbitrary matter fields to CSW gravity without breaking the equivalence with general relativity, it is possible to probe the behaviour of matter in (2 + Ddimensional gravity by coupling point particles to the theory. These point particles may be thought of in the first instance as sources of non-abelian charge, which one would expect to couple to the non-abelian gauge field A, = eal,Pa + w’~J~ in the usual way. Non-abelian charges have been dis- cussed by Wong [421, and they differ from those in an abelian gauge theory in that whilst abelian charge is a scalar, non-abelian charge is a vector

Z(T) =j’(T)P, +~‘(T)J,

taking values in the Lie algebra of the gauge group G (here T is a parameter along the world-line of the particle); the only gauge invariant quantities are the Casimir invariants of this non-abelian charge. In the case of an ISO(2,l) gauge theory, the Casimir invariants are pap, and p”jo, which may be naturally identified with the mass and spin of the source (m2 and ma to be precise), and we expect to recover the dreibein and spin connection appropriate to a source with mass m and spin (+ (via the gauge field Ar> when we couple such a non-abelian charge.

The required form of e”, and w”@ for a massive spinning point particle may be read off directly from the metric (51, bearing in mind that there is a gauge freedom in their definition. In a gauge naturally defined by the co-ordinates of (5), the gauge field is given by [191

e”=dt+4Gad0 ‘)

e’=(l-4Gm) -‘cosedr-rsin8d0,

e2 = (1 - 4Gm) -‘sinedr+rcos8dB,

o’=J=o. (15)

This gauge field of course has a curvature and torsion singularity at the origin which may now be evaluated explicitly as

To = 8rGd2( r) dx A dy ,

R”=8rGma2(r)dxAdy. (16)

Given (15) and (16), we may calculate the holonomy about the origin, which is gauge invariant up to conjugation by a group element of ISO(2,l). This holonomy

198 M. E. 0rti.z / Gmuitarional anyons

will turn out, as one would expect, to play an important role in evaluating the Aharonov-Bohm effect below. It has been evaluated by de Sousa Gerbert and Jackiw [19], who obtained the perhaps predictable result

The holonomy can be calculated explicitly if we gauge transform (15) to the gauge equivalent configuration

e0=4God0, w” = 4Gm de (17)

(with all other components vanishing) as in this configuration we can forget about the path ordering since the gauge field components commute. However, in this gauge the dreibein is no longer invertible, and we have thus lost the notion of a metric.

Consider now a non-abelian source with

pa = m8” 0, j” = US:. (18)

In a non-abelian gauge theory, the current j” is Lie algebra valued (via the charge), and couples to the gauge field through an interaction term

Ii,,, = /(A,, j’) d3X. (19)

Here

j”( y, t) = /I( 7)tpiS3( yp -X*(T)) dr

and tp = dx“‘/dr is the tangent to the worldline x~(T) of the particle. The gauge field equations are then

and for the source above, the curvatures agree exactly with those of eq. (161, confirming that a charge (18) corresponds to a massive spinning point particle.

More generally, a source pa, j” will give rise to a gauge field with holonomy

w = e8~G(p”J,,+j’Po)

up to conjugation. Under a gauge transformation the conjugation changes the value of W so that it takes the same form but with pa’ and y’, which are related to p” and j” by an ISO(2,1> gauge transformation. We thus see a form of non-abelian

M. E. Ortiz / Graoitational anyons 199

Stokes’s theorem, where the charge determines the curvature, which in turn determines the holonomy, although only up to conjugation.

In order to get a complete picture of particle scattering in 2 + 1 dimensions, it is necessary to include terms in the lagrangian determining the trajectories of the point particles in space-time and internal space as well as their interaction with the gravitational (gauge) field. In the standard theory of non-abelian charges as developed by Wong, one expects both a kinetic term for I and a kinetic term for xp. It is the equation of motion for X” which determines the space-time trajectory of the particle, whilst there is a second set of equations

or

dp” - + $,c~brpc = D,p” = 0, dr

dj”

(fixed by the Bianchi identities obeyed by Fob) determining the motion of the charge in the internal space. These equations leave the Casimir invariants pap, =

m2 and p”j, = mu unchanged as required. In the ISO(2,l) theory of gravity, it does not seen possible to write down a

kinetic term for XP since this requires the definition of a metric on space-time. As mentioned above, no such metric can be constructed from the gauge field. The scattering problem must therefore be approached in a different way to that suggested by the Wong formalism, since all the dynamics takes place in the internal space inhabited by the charges of the point particles.

Let us now describe how we may employ point particle scattering in ISO(2,l) gravity to shed light on the gravitational Aharonov-Bohm effect. Our treatment follows those of de Sousa Gerbert [19] and Koehler et al. [20].

Firstly, for a point particle with no intrinsic spin, Witten [43] wrote down the action

I, = \dr vobpaD,qb + h( p2 - m2) (20)

200 ME. Ortiz / Gravitational anyons

where the integral is taken along the trajectory of the particle in space-time. Here A is a lagrange multiplier, D,q” = dq’/dT + eQ7 + EPbCWbrqC and the charge qa is related to p” and j” via ja = &‘b& p b ’ from which it is obvious that p”j, = 0 and thus u = 0. This action may be seen to split up into the usual free action for a point particle plus an interaction term of the type (19). The action is invariant under gauge transformations where the gauge field transforms as in (12) and the charges q” and pa transform as

6q” = pa - EabcKbqC )

6p” = -+,cKb$.

For more than one source we must include an action I, for each source, and we then label the charges of each particle (cu) as

The phase space of an n particle system has been discussed at length in the various papers on this topic. On a topologically trivial space-time manifold, taking into account the constraint equations derived from the total action I, + Icsw, the phase has dimension 4n - 6 where the 4n degrees of freedom may be identified with the freedom in qPCa) and p’(,,, and there are 6 gauge degrees of freedom to remove. The easiest way to understand this phase space is in terms of Wilson loops. Given a fixed point of reference, the phase space is determined by the Wilson loops about this point. The constraints arising from the action I, + Icsw determine the value of these Wilson loops to be non-zero only when a loop encircles a point particle. If we use variables p” and j” to now label the Wilson loops, then the constraints further specify the representation of ISO(2,l) labelled by the p’s and j’s. A comprehensive treatment may be found in refs. [18,20,44].

In the case where point particles carry non-zero intrinsic spin, a new action must be written down in terms of j’ and pa, subject to the constraints that pap, = m2 and p’j, = mu. Such an action has been given by de Sousa Gerbert [19], and is a straightforward generalisation of (201, and of the free action for a spinning point particle given in ref. [42]. We refer the reader to the original reference for details of this action. For our purposes, it is sufficient to note that the action is again the sum of a free kinetic piece and an interaction term of the form (19). De Sousa Gerbert has also discussed the phase space for the case of non-zero intrinsic spin,

ME. Ortiz / Gravitational anyone 201

and this is identical to that for spinless particles with the obvious difference that it is labelled by co-ordinates paCu) and jaCa) with p°CajjoCa) = m,,,~~,,. It is again possible to use p’s and q’s rather than p’s and j’s.

The step from the classical to quantum theory is straightforward, and has been outlined in the various papers on this subject 118-201. We shall consider the most simple situation of a test particle moving in the background gauge field of a second particle, and we shall follow the similar analysis for spinless point particles carried out by Koehler et al. [20]. We are only interested in computing the topological phase resulting from, at this stage, a rotation of 27r of the test particle about the source particle (the rotation of 2rr is in physical space rather than in internal space). We may therefore consider only the interaction part of the hamiltonian arising from the action for the test particle, which is of course of the form (19).

The amplitude to go from an initial state I i> to a final state If) is given by

(f] c Texp( -i/dTH(r))li). paths

where the sum over paths refers to the different possible paths t@ in space-time and may be most easily understood as a sum over unobserved degrees of freedom. As mentioned above, in order to isolate the topological phase, we need only retain the interaction part of the hamiltonian, i.e.

c Texp (-i/d~{e’,p, +&TjO})]i). (21) paths

where now p” and i” are quantum operators generating a representation of ISO(2,l). The operator in expression (21) may be thought of as the monodromy matrix [45] in the approximation where one particle is taken as a source particle.

For a trajectory defined by tp in space-time which is closed (which will pick out the Aharonov-Bohm phase), the exponential is just a holonomy, and so the sum over paths is seen to be a sum over homotopically distinct particle trajectories. Expression (21) then becomes [19]

c exp (-8aGi{Paj,+PpU})]i). paths

Here P” and J” define the holonomy due to the source particle, and are determined by the curvature and torsion of the field due to the source up to an ISO(2,l) transformation. We begin to see at this stage the nature role of the ISO(2,l) holonomy of the source particle in giving rise to a non-trivial phase for homotopically different paths. The final state is given by a sum over paths, each

202 ME. Orth / Gravitational nnyons

contributing a different phase,

If>= f e~~Gi(Wo+~pd 1 i> .

n= -02

if we take the holonomy to be given by P” = M and Jo = S. In a general non-abelian gauge theory, an operator such as the one above is not

a phase but the monodromy matrix (in this case infinite-dimensional), so that the “Aharonov-Bohm phases” are defined as the eigenvalues of this matrix. Thus, in general, there are a series of statistical phases and none of these has a unique right to be thought of as the Aharonov-Bohm phase. However, in the case we are considering, a particular Ii) is picked out by requiring that in the frame (or gauge) picked out by the source particle, the test particle also be in its rest frame. This is physically reasonable since we wish to isolate a statistics phase from any dynamical phase. We therefore look at the effect of the monodromy matrix on the state

PO19 = mli>, joli) = ali).

Each revolution about the source flux-tube contributes an additional phase exp(8rGi(Ma + Sm}). It is simple to see that for the state vector and source holonomy we have chosen, the non-abelian Aharonov-Bohm effect splits approxi- mately into two abelian Aharonov-Bohm effects.

If the spin and mass of the test particle are the same as the parameters defining the holonomy of the source particle, then the two particles of course belong to the same irreducible representation of ISO(2,l) (both particles have the same mass and intrinsic spin). We may think of such particles as being identical. In this case the Aharonov-Bohm phase is given by

e-S~Gi(mtr+om) = ,-16rGmai (22)

We have chosen to use a canonical treatment above to illustrate the gravitational Aharonov-Bohm effect. As such, our derivation uses the approach of Koehler et al. We cold equally have reproduced the same result by following the approach of de Sousa Gerbert. Indeed, in his calculations, de Sousa Gerbert alludes to the presence of an Aharonov-Bohm effect, although its nature, and the role of the holonomy of the source particle, are not so evident. A third approach would be to study the problem in a path integral formalism. It is clear from the above analysis that we expect topologically distinct trajectories to contribute different phases.

ME. Ortiz / Gravitational anyons 203

However, a detailed analysis of this type has not yet been carried out. Such an analysis is perhaps the most appropriate framework in which to view 2-particle scattering, and as such would be of considerable interest.

4. Two types of gravitational anyon

The calculation of the non-abelian Aharonov-Bohm phase above shows clearly that both the curvature and torsion of the source field act as “flw points”. The off-diagonal inner product (13) has the effect of coupling flux in the form of curvature to charge in the form of spin, and flux in the form of torsion to charge in the form of mass. If we think of the Aharonov-Bohm effect therefore as coming from two abelian Aharonov-Bohm effects, then that with torsion as the flux corresponds to the gravitational Aharonov-Bohm effect of Dowker and others [g-12]. It is also equivalent to discussions on spinning strings by Mazur [14] (although in Mazur’s paper there appears to be a misunderstanding about the consequences of a flux point, or flux tube for a string in 3 + 1 dimensions [46,47]). The possibility of an Aharonov-Bohm effect around a spinning cone was also demonstrated by Jackiw and de Sousa Gerbert [28].

It is easy in the ISO(2,l) formulation to understand why there is no Aharonov-Bohm effect for identical massive, spinless particles, since in that case there are no torsion “flux points”; a similar result applies for spinning, massless particles. However, the treatment above also indicates that localised curvature can act as a flux point for particles with intrinsic spin. This result is in agreement with a discussion by Dowker on fermions in a conical space-time 191.

The non-abelian Aharonov-Bohm phase (22) should at first sight be equal to the statistics phase corresponding to the interchange of two particles (there is a double effect since two particles move in each other’s fields, but each particle moves through an angle rr rather than 27r), although without further analysis of the two-body problem there is no rigorous proof. We should like however, to at least be able to demonstrate an equivalence between the ndive phase one would expect from our generalised gravitational Aharonov-Bohm effect and the results on gravitational anyons obtained by Deser and McCarthy. Comparing eq. (22) with their result (lo), and recalling that in expression (10) the spin u is equal to m/p, we see that two answers differ by a factor of 4 which must be explained if we are to identify the two effects.

A factor of 2 should be accounted for by the phase resulting from the change in the field configuration as each particle moves around the other by analogy with the abelian case [4]. In this non-abelian case, however, it is not straightforward to verify this explicitly. The second factor of 2 arises because we are considering gravitational anyons in a pure Einstein theory, whereas Deser and McCarthy looked at anyons in topologically massive gravity..In Einstein theory, each particle

204 ME. Ortiz / Grauitational anyom

has both mass and spin, and gives rise to both curvature and torsion. Thus there is a double effect from the off-diagonal inner product. In topologically massive gravity, Deser and McCarthy considered particles with mass but no spin. The effect of the gravitational Chem-Simons term is to induce a torsion “flux point” around the massive particle. Thus there is a single effect corresponding to a massive particle encircling a torsion “flux point” only.

5. Conclusions

Having recalled the relationship between Deser’s gravitational anyon model and the gravitational Aharonov-Bohm effect, we were led by various features of the theory to seek a covariant interpretation of the Aharonov-Bohm effect in the gauge theory formulation of (2 + D-dimensional gravity developed by Witten. In this context, we were able to identify the gravitational analogues of magnetic flux, these being not only torsion, as suggested by the linearised gravitational Aharonov-Bohm effect, but also Riemann curvature. We have also seen that the off-diagonal nature of the inner product used on the Lie algebra of ISO(2,l) explains the absence of an Aharonov-Bohm effect for a massive particle moving around a conical singularity.

By casting gravitational anyons in a gauge theoretic framework, the similarity between this model and the standard U(1) Chem-Simons model for anyons may be seen. This is a result not anticipated by Deser. In particular it shows explicitly how the Einstein term in the gravitational action behaves precisely as a Chem-Simons term. The need for a gravitational Chem-Simons term in Deser’s model stems in ISO(2,l) language from the off-diagonal inner product; a particle with only one type of charge (mass), gives rise to the wrong type of flux point for exotic statistics in ordinary Einstein gravity. The effect of the gravitational Chem-Simons term is to ensure that it gives rise to both types of flux point (curvature and torsion), making the particles gravitational anyons.

There is no gauge theory equivalent of the gravitational Chem-Simons term (it is certainly not equivalent to adding an ISO(2,l) F2 term which would be possible in a gauge theory), and this is reflected in the fact that it is not possible to write topologically massive gravity as a gauge theory; the gravitational Chem-Simons term breaks the ISO(2,l) symmetry. However, since the Einstein term dominates the behaviour of the gravitational field at large distances from a source, we may still think of the source-test particle problem in topologically massive gravity in terms of our ISO(2,l) treatment. For a test particle, the gravitational field of the source defines, at large distance from the source, a gravitational field which may be thought of as a flat ISO(2,l) connection, allowing us to make use of the ISO(2,l) formalism. In this sense we might say that the ISO(2,l) symmetry broken by the gravitational Chem-Simons term is restored at large distance.

M.E. Odz / Gravitational anyom 205

Finally, we mention that our discussion has been in terms of homotopically distinct trajectories in the space-time manifold on which ISO(2,l) charges are defined. It has been established by those authors who have studied the scattering of point particles in the CSW theory [18-201 that results obtained for scattering in the internal degrees of freedom agree with those obtained in a standard gravita- tional treatment by ‘t Hooft [21], Deser and Jackiw [27] and Jackiw and de Sousa Gerbert [281. In this last paper, scattering on a cone is considered, and as mentioned above, solutions with Aharonov-Bohm type behaviour are found. If these solutions are also present in the theory of scattering in internal space for CSW gravity, then it is natural to expect some relationship between them and the space-time Aharonov-Bohm phases we have discussed. A path integral approach to the problem of scattering might help to shed light on whether any duality exists between topologically non-trivial trajectories in internal space and space-time.

I would like to thank Gary Gibbons, Steven Carlip, Stanley Deser and Roman Jackiw for helpful discussions, and the SERC for financial support.

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