Letters in Mathematical Physics 18: 67-76, 1989. �9 1989 Kluwer Academic Publishers. Printed in the Netherlands.

67

Conformal Gauge Fixing in Minkowski Space

R A I N E R DICK Institut fiir Theoretische Physik, Universitiit Karlsruhe, D 7500 Karlsruhe, West Germany

(Received: 9 January 1989)

Abstract. Conformal gauge fixing on simply connected parts of world sheets in Minkowski space is examined in detail. It is proved that conformal gauge fixing can be imposed, including the usual boundary conditions.

AMS subject classifications (1980). 30F99, 53A30, 53C50.

1. Introduction

The local reparametrization symmetries of the actions of Nambu (see, e.g., [ 1]) and Polyakov [2] make it necessary to eliminate gauge degrees of freedom by imposing gauge conditions which become part of a then well-defined Cauchy problem.

The popular conformal gauge G~a ,,, r/~a for the metric G~a on the world sheet seems to have been introduced in physics by Goddard, Goldstone, Rebbi, and Thorn in their pioneering paper on strings [3].

Meanwhile, the investigation of the various mathematical aspects of string theory has reached a high standard, as a look into introductory textbooks proves [4]. However, to many physicists, the possibility of conformal gauge fixing seems to be guaranteed by geometric imagination or by a simple counting of variables without any deeper examination. In spite of this, Polyakov [5] and others examined the role of Killing vectors and of the zero modes of the operator adjoint to the Lie derivative with respect to the scalar product on the space of metrics, which was introduced in [2]. The Killing vectors and the zero modes of the adjoint operator, respectively, destroy the uniqueness or existence of the conformal gauge. This investigation, however, is limited to the tangent plane of the space of metrics generated by Diff0(M), thus excluding the action of the modular group, and to the Euclidean case, because the scalar products involved should be definite for the arguments to go through.

Segal, on the other hand, considered the problem globally in Minkowski space [6], but made explicit use of the equations of motion, thus providing on-shell results.

A correct local on-shell argument in Minkowski space has already been provided by Goddard, Goldstone, Rebbi, and Thorn [3].

In Section 2, I will give a detailed prescription to introduce the conformal parameters on world sheets off-shell in Minkowski space by using elementary methods. Section 3 concludes with some remarks.

68 RAINER DICK

2. Conformal Gauge Fixing

Before I can formulate the fundamental theorem of conformal gauge fixing in Minkowski space, some basic notions of string theory must be reviewed.

A world sheet is a two-dimensional differentiable manifold with or without boundaries, embedded in Minkowski space, such that all boundaries are either timelike or lightlike, and every internal point p has some neighborhood Up serving as a chart for local coordinates

~: Up-oR2: z=a~176 a=at=dpl(p'),

such that the induced metric

G~,tj = (O~,XU)(OtjXOnuv

is nondegenerate, differentiable, and indefinite in Up. U, will be referred to as open or closed if q~(Up) is open or closed with respect to

the usual topology in R 2. The requirement of indefiniteness is equivalent to the statement that we may

choose z and a with G0o < 0, G~1 > 0 (1), which is, henceforth, assumed. The variational principle of string theory then affirms that whenever a simply

connected piece M of a world sheet is selected by intersecting the world sheet with an arbitrary pair of disjoint spacelike hypersurfaces (Figure 1), the volume S of M is extremal.

In an obvious manner, M may serve as a coordinate chart in an atlas of the world sheet, and one can choose coordinates ~ and a such that Goo < 0, GI~ > 0 with the possible exception of lightlike boundaries, and the intersections correspond to , = 0 or z = T, respectively, while the boundaries correspond to a = 0 or a = Z,

respectively. The classical equations of motion resulting from

s = f,,, d a4:-6 are then

= , / = - o . {,i," - O ' ( O ~ X , ) ( ~ X O } . O'~ ~. O~X. = 0

and

in the case with boundaries, or

x,(r ~ + x) = x,(r ~)

in the case without boundaries.

(2a)

(2b)

CONFORMAL GAUGE FIXING IN MINKOWSKI SPACE

/ f

I / Fig. 1.

69

Equation (1), which immediately generalizes to an arbitrary dimension of the world sheet, asserts that the transverse components of the target space vector

V ~' = G ~ t~ t~pX ~

vanish, leaving the longitudinal components as gauge degrees of freedom. Confor- real gauge fixing then amounts to transforming the operator G ~ a~ t3p to hyperbolic normal form on M and corresponds to the requirement of vanishing longitudinal components of the generalized acceleration V ~.

The fundamental theorem of conformal gauge fixing can now be formulated:

In any simply connected chart o f a world sheet, one can choose parameters such that

Before entering the proof, I want to stress, that the world sheet is not assumed to be on-shell and G~ can be any nondegenerate, differentiable, and indefinite metric without the requirement of being the induced metric.

A fairly simple proof works in the following fashion: A reparametrization on M

~, ~ --, ~ +(~, G), ~ -(~, ~r)

transforms the inverse metric according to

n + + = G~176162 +)2 + 2Gm(do~ +)(~ ~ +) + G~I(~ r +)2, (3a)

H - - = G~176 -)2 + 2GO,(do~ -)(~, ~ - ) + G"(t3, ~ -)2, (3b)

H + - = G~176 +)(dog - ) + G~ +)(dl ~ - ) + (d, ~ +) x

x (~o~-)) + G"(~, ~ +)(~ ~-). (3c)

To construct conformal parameters on M, I employ a standard trick in partial- differential equations and require H + + = 0, H - - = 0.

A-priori, Equations (3a-c) only make sense for G 5 0 . The case of points on lightlike boundaries, where Goo = 0 and G0~ = 0 hold, is omitted for the moment and

70 RAINER DICK

will be dealt with later. Then one finds, as necessary and sufficient conditions for a nondegenerate solution of H § § = 0, H - - = 0:

~o~ + - A+ �9 ~91~ + = 0, (4a)

C~o~ - - A_ �9 ~9, ~ - = 0, (4b)

G O, I(G~ G li 1

A+ = - a - - ~ + q t aoo ) - ~ = -6-77,, (ao, + , I ~ 6 ) , (Sa)

with

Go, /(GOly G" 1 A_ = -GO---- 6 - x[\GOO ] - ~ = ~ (Go, - x//--~). (5b)

These coefficients are bounded:

0 ~< A+(r, a) ~< M+, - M _ ~<A_(z, a) ~<0

and different,able on P : 0 ~< z ~< T, 0 ~< a ~< Z. Furthermore, they can only vanish on lightlike boundaries.

As a consequence of different, ability, these functions also satisfy Lipschitz conditions in P:

IA+(z, a') - A• ~)1-< N• I~ ' - ~1. Because any function ~(z, a) is a solution of (4a) or (4b), respectively, if and only if it is to first order in dz invariant under transformations z--+z + d'c, a-->a - A • a) dr, the solution of Equations (4a,b) is equivalent to the determi- nation of all the corresponding characteristics

d4,+ dz - - A + [ z , r ao = r

= r ~o, *o) ~ ~+(T, a) = ~ +(~o, *o),

de_ dz A [ z , r (z)]; ao = *_(Zo)

(6a)

a = * _ ( z ; Zo, ao) ~ ~ -(% a) = ~ -(Zo, ao) (6b)

in P.

Solving the initial-value problems (6a) and (6b) for suitable sets ao, Zo such that P is covered with characteristics, yields solutions of (4a) and (4b) if one prescribes smooth functions ~ •177 along curves a =f:~(z) which intersect every charac- teristic a = O• Zo, ao) exactly once.

Because the characteristics (6a) are decreasing, while the characteristics (6b) are increasing, suitable sets of initial values are provided by ao =f(zo) for (6a) and a o = Z - f ( z o ) for (6b), with 0 ~<Zo ~< T, i f f ( z ) is a uniformly increasing function satisfying f (0 ) = 0, f ( T ) = Z.

CONFORMAL GAUGE FIXING IN MINKOWSKI SPACE 71

The proof of the existence of solutions of Equations (4a,b) thus reduces to a proof

of the global existence of solutions of the initial-value problems (6a,b) in P. This, in turn, is achieved by an application of the Picard-Lindel/Sf theorem.

To apply this theorem globally in P, one has to extend the domain P, where A+ and A are defined, to the domains

D+:O<~z<~T, - M + ' T < ~ a < ~ E + M + ' T

for A+ and

D :O<~'r<~T, - M "T<~a<<.E+M__.T

for A .

The functions L• a) defined on this extended domains are required to be integrable, to have the same extremal values like A_+, to satisfy the same Lipschitz conditions like A• and to be equal to A• in P.

First, one observes that such extensions exist in a canonical fashion for the closed string due to periodicity in a.

For the open string, the following construction is sufficient:

[A • (z, E),

L• a) = ~A+_(r, tr), (A, (~, o),

0 ~ o ~ E ,

- M •

The existence of unique solutions (1)• to, o0) of the initial-value problems (6a,b) for 0 ~< z ~< T, (z o, ao) ~P now follows exactly like in the proof of the Picard-Lindel6f theorem (see, e.g., [7]) from the uniform convergence of Picard's iterative solution

�9 ,~ (z ; to, Cro) = ao - dOL • [0, ~,~_ i(O ; "to, ao)], O

�9 g (r : r,,, Oo) = oo

for 0 ~< r ~< T due to the boundedness and Lipschitz continuity of L+ in D• Thus,

q~+(z ; to, ao) = ao - dOLe[O, 0• to, ao)] 0

= lim ~,~(z;ro, ao)

exists uniquely for

O~<r~<T, O~<ro~< T, O~<a0~E.

Moreover, due to the semidefiniteness of L+ and L , (1)+(z) and (I) (z) are nonincreasing or nondecreasing functions, respectively. Hence, the parts of the characteristics satisfying 0 ~< 0• ~< E also satisfy

q~_+(r ; To, ~o) = a o - dOA•177 ao)]. 0

72 RAINER DICK

Suppose now that qJ+(z; Zo, ao) is another solution for another extension, L'+ of A_+ in D+. I will prove that, in P, the resulting characteristics agree:

~ = @ +(Z ; Zo, ao) = q '+(Z ; Zo, ao), if O~<tr ~<X.

The equation holds, at least for z = r o. If the equation would not hold everywhere in the interior of P, there would exist, e.g., some value t+ in (0, T) such that

0 ~< qJ+(t+ ; to, ao) = @+(t+; Zo, ~ro) ~< E,

but this equation would be invalid either for all z in [ t + - 6, t+) or for all z in (t+, t+ + 6] for some 6 > O, while it holds between ro and t+. Then there would also exist some e in [ - 3 , O) or (0, 6] such that ]qJ+(z) - @+(r) I attains its maximum in t+ + a for the interval under consideration.

Hence, by Lipschitz continuity,

Iq'+(t+ + e ; to, ao) - @+(t+ + e ; to, ao)l

= '++ dO{A+(O;q)+(O)) -A+(O;U?+(O))} +

N+' �9 I r § 30, - v + U + + to, o)1

~< N+ - 6 �9 Iw+(t+ + e ; ro, tro) - * + ( t + + e; to, tro) I.

This immediately yields 1 ~< N+ �9 6 in contradiction to the fact that 3 may be chosen as arbitrarily small. Thus, by continuity, W_+ and @_+ describe the same characteris- tics in P including the boundaries.

Up to now, nondegeneracy of the metric has been assumed to hold throughout on the world sheet, including the boundary. For internal points and the induced metric, G < 0 certainly holds due to the definition of the world sheet. Other metrics under consideration in string theory are also required to have this property. On lightlike boundaries, however, values of r in (0, T) may exist such that Gol(r, 0) = 0 or Got(r, E) = 0 holds. A well-known example for this behaviour is a rigid rotator of length L with induced metric and spacelike intersections for X~ and X~ T (see, e.g., [8], where another, partial gauge fixing is demonstrated).

A 'naive' and a conformal parametrization is provided by

X ~ = r = u ,

x

The points (r, O) and (r, Z) in P where G - 0 holds, and the corresponding points on the boundaries of the world sheet will be called degenerate points.

CONFORMAL GAUGE FIXING IN MINKOWSKI SPACE 73

The proof given before for the unique existence of characteristics should then be applied to reduced world sheets originating from cutting out all degenerate points together with open neighbourhoods. A unique definition of characteristics on the original world sheet everywhere satisfying equations (6a,b), is then provided by using those characteristics on the original world sheet which emerge from the characteristics on reduced world sheets by a continuous reduction of the open neighbourhoods which have been cut out.

In the example above, the lightlike boundaries themselves are characteristics, from which other characteristics branch off everywhere, to the interior of the world sheet. The construction just mentioned clearly amounts to discarding the use of the boundaries as characteristics in the construction of the conformal light-cone parameters ~ + and ~- described below. Cases, where lightlike characteristics are only approached asymptotically, provide no difficulties at all for the construction of conformal light-cone parameters.

The global unique construction of the characteristics in P can now be employed to construct the conformal light-cone parameters in the already indicated fashion by prescribing their value on every corresponding characteristic in the form

-+ (r, f_+ (3)). (See the discussion following Equations (6a,b).) Conformal parameters u, v are now provided via

I + u=,~(r + 4 ), v = � 8 9 1 6 2 ), (7)

G"" = -G""=~HI + , G u t ' = O . (8)

This concludes the proof of the existence of conformal parameters on M.

For matters of convenience, I henceforth assume that

2 H + - = (~,~+)(~,~-) < 0,

Gl~

i.e. v(z, a) is spacelike and u(z, a) is timelike. This can always be achieved by a simple reflection ~- ~ -

There is still the possibility of choosing the values of ~+-(z,f• which correspond to the conformal Killing vectors within the infinitesimal framework. First of all, one has to choose these functions uniformly increasing or decreasing on o =f• to have an isomorphism z, a ~ ~+, r

Another requirement implicitly used in conformal gauge fixing, reduces our freedom to choose ~ • (r, f_+ (r)) still further: For the explicit solution of the boundary equations (2a) or (2b), it is necessary to map the boundaries a = 0, a = E to the boundaries v(r, 0 ) = 9, v(~, E ) = E', which implies the following boundary conditions for ~- . +"

r + (z, O) = ~ - (r, 0), (9a)

r E) = G-(r, E) + 2E'. (9b)

74

This yields simple boundary equations in the conformal gauge:

OX~'(u, v)

o r

R A I N E R DICK

X~'(u, v + Z') = X~(u, v).

In addition to (9a,b), one may also preserve the simple notation XU(0, a) for the initial state by demanding u(0, a) = 0, i.e.

r a) = - ~ - ( 0 , a). (10)

One can satisfy all the requirements (4a,b), (9a,b) and (10) by choosing a smooth uniformly increasing function ~+(0, o ) = - r a) obeying ~ + ( 0 , 0 ) = 0 , ~+(0, E) =E ' and

O~+(0' a) ~ = o O a = c~ +(0' ~ o=~ = 0 " c ~ a

This function must then be transported from the left boundary of P along the characteristics o = (r ; 0, Oo) according to

r +(r, O_+(r; 0, ao)) = r ao)

and, according to (9a,b), get reflected at the upper and lower boundaries. The conditions on the derivatives of r +(0, a) ensure smoothness, i.e. continuous

differentiability, of the resulting functions ~-+(r, o) aross the characteristics, where they are pieced together. To ensure continuous higher-order derivatives requires the vanishing of the higher-order derivatives of r +(0, o) for a = 0 and a = Y:.

Schematically, the reparametrization then looks like the example shown in Figure 2. Usually, it will be impossible to map P again into a rectangle. This, however, is not necessary to receive the well-known simple form of the equations of motion of the string in a conformal gauge.

o v ~ §

~t

T ~"

Fig. 2.

\

CONFORMAL GAUGE FIXING IN MINKOWSKI SPACE 75

3. Remarks

For a Euclidean embedding of the world sheet, the parameters A• in (5a,b) are complex conjugates of each other and the method of characteristics is not applicable. Accordingly, the Euclidean problem has been solved, not by the method of characteristics but by use of the Green's function of the Laplacian in two dimensions, which, in turn, is not available in the hyperbolic case. For treatment of the Euclidean problem, see [9-11].

It is an immediate consequence of the construction in Section 2 that any surface with a finite cover embedded in Minkowski space such that, at every point, a timelike direction exists, admits a global conformal gauge. Moreover, this global conformal gauge induces a complex structure with spacelike reflections a ~ - a as conjugation.

In a basis

~o=0+ + 3 , c~1 = ~ + - 8 _ ,

7:+ ----T-4-O ", T ----T--O',

Equations (4a,b) read

? r 1 6 2 =0, ( l la)

? + ~ - - U + ?-~ =0, ( l lb)

with

A + - I A +1 j [ ~ + - - - - - - - - _ _

A + + I " P+ A - 1 '

which clarifies the relation of the A• parameters to the Minkowski space analogue of the Beltrami parameters.

In the conformal gauge, the Beitrami parameters vanish and Equations (1 la,b) imply 'holomorphic' transition functions among conformal light-cone parameters ~+__,~+,.

Excluding lightlike boundaries, the conformal light-cone parameters satisfy the differential equations

( ' ) (~o- A_+ ~1)r = 0 ~--~ ~l - ~+-_+ 0o ~ - - + = 0 ,

with

A + = G ~ 1 7 6 -+ i l _GOl,/ + l - G ~ 1 7 6 ' ~ = G 1'

Hence, the equivalent differential equations may be written as

( , / ~ . a~176 ~o + , / - ~ . G o' a, _+ai)~ • =0 ,

( - , / - a a '1 a, - ~ - G . G '~ ao _ 0o)~ -+ = 0,

which also holds on lightlike boundaries.

76 RA1NER DICK

Therefore, Equations (4a,b) imply

�9 c -+) = 0,

i.e. the conformal parameters satisfy the bulk equations of motion (1) of the target space variables X ~'.

This is another proof of the fact that the conformal gauge can be strengthened by an additional condition

n+~ + + n ~ =a~,X"

with constant coefficients n+, at,, provided a,X" is kept on-shell.

Acknowledgement

The author wishes to acknowledge the support of Studienstiftung des deutschen Volkes.

References

1. Scherk, J., Rev. Mod. Phys. 47, 123 (1975). 2. Polyakov, A. M., Phys. Left. 103B, 207 (1981). 3. Goddard, P, Goldstone, J., Rebbl, C., and Thorn, C. B., Nucl. Phys. B56, 109 (1973). 4. Green, M. B., Schwarz, J. H., and Winen, E, Superstrmg Theoo', Volumes l and II, Cambridge

Umversity Press, Cambridge, 1987. 5. Polyakov, A. M., Gauge Fields" and Strings, Harwood Academic Pubhshers, Chur, 1987. 6. Segal, G., Commun. Math Phys. 80, 301 (1981). 7. Stepanow, W. W., Lehrhuch der DiHerentialgleichungen, 4. Auflage, VEB Deutscher Verlag der

Wlssenschaften, Berhn, 1976. 8. Schild, A , Phys. Ret. DI6, 1722 (1977). 9. Chern, S. S., Proc Amer. Math. Soc. 6, 771 (1955).

10. Courant, R. and Hilbert, D., Methods o/Mathematwal Physics, Volume II, Interscience Publishers, New York, 1962

11. Lehto, O., in W. J. Harvey (ed), Dt.wrete Groups and Automorphic Functions, Academic Press, London, 1977, p. 121.