On the obstruction to light-cone gauge fixing at higher genus in string theories

Volume 256, number 3,4

PHYSICS LETTERS B

14 March 1991

On the obstruction to light-cone gauge fixing at higher genus in string theories A d r i a n R. L u g o Departamento de Particulas Elementales, Universidad de Santiago, E- 15706 Santiago de Compostela, Spain Received 2 December 1990

We analyze the possibility of fixing a light-cone gauge in the operator formulation of string theories on arbitrary Riemann surfaces, and show that the standard prescription leads to ambiguities and is incompatible with Lorentz covariance.

I. Introduction

In spite of string theory being first formulated in an operator context [ 1 ], the path integral formulation became more popular after Polyakov's work [2 ]. The fundamental reason lies in the fact that it allows to define a perturbative series for the scattering amplitude among the (infinite in n u m b e r ) excitations of the string as a sum over genus g Riemann surfaces, g denoting the loop order in the expansion [ 3 ]. The operator formulation, though more "physical" in the sense that it clearly exposes the spectrum of the theory as states on a Hilbert space, was essentially restricted to the genus zero and one case, the last one reached from the free theory ( g = 0) by "gluing" procedures and unitarity arguments [ 1 ]. A mixed treatment to define the g-loop contribution to a scattering amplitude was developed by Mandelstam [4], but it showed to be intractable for explicit computations at higher genus, although it furnished a link to demonstrate the unitarity of the Polyakov approach [5 ]. However, in recent work [ 6-10 ] a global covariant Fock space formulation of perturbative string theory (and quantum conformal field theories in general) was developed by using bases for m e r o m o r p h i c tensors, holomorphic outside two distinguished points P+ and P_, introduced by Krichever and Novikov [ 11,12 ], which closely resembles the genus zero formulation, and was showed to be equivalent to the path integral formulation. As it is well known, there are two ways of operato-

rially quantizing free string theory; a manifestly Lorentz covariant approach ~ la G u p t a and Bleuler (or an equivalent BRST formulation), and a lightcone gauge fixing procedure. The last one has the advantage of completely eliminating the spurious degrees of freedom, leaving a physical theory without ghost; however Lorentz covariance is obscured and the main problem to be solved is to show that the Lorentz algebra is realized on the Hilbert space, that it happens in the critical dimension D = 2 6 (or 10 for the superstring), and for certain values of the ground state energy [ 1 ]. In this letter we analyse the possibility of fixing a light-cone gauge in the interacting theory within the operator context of ref. [ 7 ], restricting ourselves to the bosonic closed string theory.

2. Some background and conventions

Here we briefly summarize the formalism developed in refs. [ 7 - 9 ] (we refer the reader to them for details, in particular for the definitions and properties of the bases A,,, oJ n, e~, £2n, for meromorphic tensors of weight 0, 1, - 1, 2, respectively). The spacetime coordinates of the string X%/t = 0, ..., D - 1, obey OSXU(Q)=O, QCP+, P , and so are expanded according to

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

387

Volume 256, number 3,4

PHYSICS LETTERSB

xit( Q ) = x i t - ipitz( Q ) +~

i

~ [hi

> g/2

[auO~ (Q) + a u ~ . ( Q ) ] ,

(2.1)

where z (Q) = Re fee)g/2 is the light-cone time [4,1 1 ] whose level lines define contours C~ going to small circles around P+ when r goes to ~ oo, and the set of ~., 0. completes the basis of harmonic functions (together with 1 and z), regular outside P+ ( P _ ) for n < - g / 2 ( > g / 2 ) [7]. The operators in (2.1) satisfy the commutation relations ~ [Ol.l~; Og~] : ~ n m q i t " ,

[Xit;pU]

=it/u"

(2.2)

,

where t / i t " = d i a g ( - 1 , 1..... 1) and 7.m= (1/2~i)>( ¢c. dA.Am. Let us introduce the following product (,d=-i(O-O)): (01(0) = ~

(0,drp-~0,d0),

(2.3)

Cr

where 0, ~0 are harmonic outside P± (by Green's theorem (2.3) is independent of z). Then the generators of spacetime translations are given by Pit = 2i (Xit I 1 ) = p i t ,

(2.4)

Jit"=_ 2 ( Xul X ") =lit" + Eit" + E ~" ,

(2.5a)

lit"= x i t p " - x"pit ,

(2.5b)

~

n< --g/2,m>g/2

It v a,,am = .. a ,It amv : +t/it'9.,. •

(2.8)

We introduced in the last equations the a~ for n e I - [ - g / 2 , . . . , g / 2 - 1] given by (see ref. [7] for the constant matrices A, B) a~=

(A. m a mIt + B n m a- ,#. ) ,

~ Iml

n~l ,

(2.9)

>g/2

and by definition a~/2 =62~/2 - -pU/x//2.

3. The standard light-cone gauge on Riemann surfaces Let us assume that it is possible to fix the standard light-cone gauge (X +_= (X o + X D- l ) / x / ~ ' X = ( X 1, ...,

X D-2 ) )

X+ ( Q ) = - i p + r ( Q )

.

(3.1)

In this gauge we have equivalently from eqs. (2.6)

(3.2a) 1 x / ~ p + (L~X-Lr)= ~m lmg/2a7~'

(3.2b)

(O,,lOm)(a~a~-a~aUm) . (2.5c)

The holomorphic component of the energy-momentum tensor is given by T(Q)-

Here C~,m are the structure constants and •nm is the cocycle [ 11,10 ]. The normal ordering is defined by (for its relation to the Fock vacuum and the constants f,,, see refs. [ 7-9 ] )

i 0X-(Q) = -~Te,~(Q)[:OX(Q)'OX(Q): + T ( Q ) ] ,

and the Lorentz generators by

Eit"=i

14 March 1991

-:OX(Q).OX(Q):

= ~ L,g2"(Q),

(2.6a)

where e~= (oJg/2) -~ and " T " stands for the transverse oscillator contribution. We can formally invert (3.2b) to get 1 O/m ---- _

l,s

n

L.=½ 21~ s:ak'a~:,

l,-~

k,s

e, ofo) ~,

(2.6b)

¢

~r S m r ( T j ) ( L T r - L r ) '

(3.3a)

Cz

where the generalized Virasoro operators L~ ( K N operators) satisfy the algebra [ 11 ] D [L,; Lm] = Y. C S , , L , + - i ~ X n m . s

~1 " A n t i h o l o m o r p h i c " ted in what follows.

388

--

(barred)

(2.7)

p a r t s will b e c o n s i s t e n t l y o m i t -

A~('c) = ~ i

e~Am°f°)" Cr

Smr(r) = ~ i

e'~A"f2r'

(3.3b)

Cr

and analogous expressions for the e2m. We have made the time dependence o f A ~ , Smr explicit in the above

Volume 256, number 3,4

PHYSICS LETTERS B

equations. The reason is very easy to see; the merom o r p h i c vector field e~ has 2g poles outside P+ corresponding to the critical points (zeros) of~og/2 where interactions take place, a n d there these coefficients pick up contributions. In fact this vector field was sued in ref. [ 7 ] to define the (time d e p e n d e n t ) worldsheet h a m i l t o n i a n and m o m e n t u m o f the theory

( l l " ( r ) - A ~ / 2 ( r ) ): :a,.a,:

= 2 Sg/sr(r)Lr~- Z Sg/2r(~')I~,

P(r)=~

(3.4a)

r

Y~ lL'(z) : a l ' a , : Ls

--½ ~ P ( r ) I.s

:a,.as:

(3.4b)

r

If we want to impose the classical constraints Lr=O (up to normal ordering at the q u a n t u m level, see below) for any r and consider eqs. (3.3) as giving a y in terms o f the " p h y s i c a l " transverse oscillators, then we find ambiguities due to the time d e p e n d e n c e m e n t i o n e d above. A n d what is more, (3.3a) would be i n c o m p a t i b l e with (2.9) for m e I . We can still go ahead taking into account that in this range lrm g / 2 = 0 if Irl>~go-3g/2 and S m r ( ¢ o o ) ~ S m r ( ~ - ) - ~ - - O if 2 r >~_go and m >
~

l,s

=

Y~

Sx/2"(+_)L~-2a+_

r~g o

= ~ / v , ( _ + ) :0~t'o~: -2a+_ /,s

=

~

S~./f( + ) E T - - 2 a -+

(3.6)

S,,r( _+ )L'lr" ,

m >g2/--g/2 <

(the second line comes from the /5+go c o n d i t i o n ) . These constraints are no other than the analogues o f the " m a s s shell c o n d i t i o n s " o f g = 0 . In terms o f H and P they read

H ( ~ oo) - 2 S g / 2 +go + )a + = 0 ,

= 2 Sg/2r(75)Lr- Z ~'~g/2r(T)L" r

p + p - = ~ lZ*( + - ) :al'o~,: -2a+_

r~go

H ( ' r ) = ½ 2 F s ( r ) :o#.o~,: +½ ~ P ( r ) I.s l.s r

14 March 1991

"

(3.5)

C%o We are still left with m = g / 2 and here we find an unsolvable ambiguity; we must choose one o f the two possible conditions L +_go = a -+ for some constants a +, leading to ~3

P(-y-oo)=0.

The genus zero constraint is known to be H = 2 ( a n d because e~ has no poles outside P+ ( -= z = 0, ov ) they are univocally defined! ). I f we take it as an ansatz for the mass shell conditions near the " i n " and " o u t " vacua [ 12 ], then we obtain the values a + -~-" ~±go + ) -= [Sg/2 +_go( _+ ) ] - 1, which in terms o f the K N operators corresponds to L +_go- E~go) = 0 ~4. These are exactly the conditions that satisfy the " i n " and " o u t . . . . prim a r y " fields defined in ref. [ 12 ] in the framework of constructing the representations o f the algebra (2.7). However let us stress that we must choose one possibility in order to define eqs. (3.6), i.e., near P+ or near P_. A n d what is crucial in the present treatment is that the generators (2.5a) close in the Lorentz algebra after expressing t h e m in terms o f the transverse oscillators. It is well known that at g = 0 this r e q u i r e m e n t fixes a = 1 and D = 2 6 . The a n o m a l o u s c o m m u t a t o r to be considered is that involving J ' - . After tedious but straightforward c o m p u t a t i o n s we get 1

[ji-;jj-]_ ~2 Not implementing conditions on Lr for Ir I go, see refs. [ 11,6]. ~3 There are no normal ordering indefinitions for Ir l > go in (2.6), but they appear for r = -+go instead, see refs. [ 12,10].

(3.7)

CO=

~ n<

(3.8a)

p+2 (C°+C°),

' J A,,,(a,O~m

j

, ),

- - OL n O ( m

(3.8b)

--g/2,m>g/2

where (A~(-T-ov)=-A~( +__)) ~4 The coefficients that appear here correspond to the following expansions [11]: ~+) z~+.-go+ ~[1 +O(z±)la~., e, =E,-

z+ (P~) =0. 389

Volume 256, number 3,4

PHYSICS LETTERS B

A,,,=2[l'm( + ~ ) - a + - ( ¢ , [ O , , ) ] , D-2 2

r< --g/2,s>g/2

A, (+)mr

14 March 1991

X+(Q)=V+(Q) ,

(4.2)

with V ~' a harmonic arbitrary Lorentz vector (worldsheet scalar). In order that this gauge condition be compatible with Lorentz invariance, any Lorentz observer should get it by making a diffeomorphism by means of ( 4.1 ), i.e., it should exist a vector field E (Q) such that

(0nI¢,)(0~I¢,,)

(--)(~'pf~ltk-~lf~tp)

l,p~ft

r< --g/2,t,p>g/2

5AX+(Q)+E(Q)dX+(Q)=A+~V"(Q)

× Z A~( + )A'['q( + )TlrTsq I,s,q

+

~.

p,r< --g/2,t>g/2

e(Q);3v + (Q) +g'(Q)g)v + (Q) = A + , [ V"(Q) - X ~ ( Q ) ] .

q,l,5

Z

(4.3)

for an arbitrary Lorentz transformation A. Put in other form

(Opl¢,)(~mlC/)~)

× ~ A~,q( - )A~( - )Tz,gqs +

,

(0~ 10z) (¢,10,)

(4.4)

These commutators (and therefore Anm) should be zero if covariance holds. At g = 0 the awful anomaly (3.9) has the particularity of reducing to

This last equation clearly displays the obstruction; the LHS is a sum of h o l o m o r p h i c + a n t i h o l o m o r p h i c pieces, while the RHS is harmonic. At g = 0 we can choose V~= - i p " r : in this way the harmonic part o f the expansion (the " t i m e " ) drops, leaving a sum compatible with (4.4). It is easy to see in this case that the vector field e is given by

A,m- 6,+mo(2~m~ (m3_m)+2m(l_a))

2 + e ( z ) = i ~ A ,Xrigh,(z)z.

p,r< --g/2,l,t> g/2

× 2 A~q( - )A',"'( + )TorTls. q,s

(3.9)

whose cancellation leads to the values mentioned. But if g > 0 it is not possible to get A,m=O for any choice in (3.6) and any values of the parameters D, a, and then the light-cone gauge seems to be incompatible with the closure of the Lorentz algebra.

4. The holomorphic obstruction The question that naturally arises is, why does it happen? The light-cone gauge should be possible to fix because of the residual symmetry under diffeomorphisms of the punctured surface which leave the conformal gauge g,a=p6,p invariant; these are generated by the vector fields of the form

E(Q) = e ( Q ) + ~ ( Q ) ,

(4.1)

where e(Q) ( ~ ( Q ) ) is holomorphic (antiholomorphic) on the punctured surface, expressed as linear combinations of e, (On), I nl >~go (see footnote 2). This fact is well known on the sphere. In order to analyse the case on general surfaces, let us consider a general gauge condition of the form 390

(4.5)

But a g > 0 it is not possible to do this, all we can do is to choose the difference ( V " - X ~) as a linear combination of the An, A, ~5. Let us remark that the symmetry under diffeomorphisms generated by (4.1) does exist, but we could use it only to fix a gauge of the type mentioned above i f we would like to preserve Lorentz covariance. The usefulness of these kind of gauges would require further study.

5. Concluding remarks We have showed here that in the operator formalism the light-cone gauge fixing is in contradiction with Lorentz covariance on surfaces o f genus greater than zero. We do not know how this fact is manifested in the path integral formulation (and we do not know if some attempt to analyse the problem in this frame~s The origin of the problem may be traced to the fact that at g=0, Cn=-(1/n)A_,=-z-~/n and the bases (r, An, ,~,) and ( 1, r, ¢,, ~,) are essentially the same, this does not happen for g> 0.

Volume 256, number 3,4

PHYSICS LETTERS B

work was m a d e ) , b u t we guess that there m i g h t arise c o m p l i c a t i o n s in the d e f i n i t i o n o f the c o r r e s p o n d i n g F a d e e v - P o p o v d e t e r m i n a n t . Also the c o n n e c t i o n ( i f a n y ) with Liouville t h e o r y a n d 2 D " i n d u c e d " q u a n t u m gravity [ 13 ] s h o u l d be i n v e s t i g a t e d in the present framework. We w o u l d like to r e m a r k that M a n d e l s t a m ' s a p p r o a c h [4] in the light-cone has n o t h i n g to do with the subject treated here; o f course by gluing " p h y s i c a l " cylinders (the case m o r e extensively s t u d i e d in the literature is the t o r u s ) it is possible to reach higher genus surfaces; the p r o b l e m is in the global d e f i n i t i o n o f the gauge. We finally m e n t i o n that we have f o u n d o b s t r u c t i o n s [ 14] o f the s a m e origin in the a t t e m p t o f carrying out the F r e n k e l - K a c [ 15 ] a n d D D F c o n s t r u c t i o n s [ 1 ] at higher genus.

Acknowledgement We w o u l d like to t h a n k fruitful c o n v e r s a t i o n s with L. B o n o r a a n d J. Russo in the early stages o f this work, a n d to the Santiago's Particle Physics G r o u p for w a r m hospitality. T h i s work is s u p p o r t e d by a post-doctoral fellow o f the M i n i s t e r i o de E d u c a c i o n y C i e n c i a o f Spain.

14 March 1991

References [1 ] M.B. Green, J.H. Schwartz and E. Witten, Superstring theory, Vols. I, I1 (Cambridge U.P., Cambridge, 1987), and references therein. [2] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [3] See for example D. Friedan, in: Les Houches 1982, Recent advances in field theory and statistical mechanics, eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984); O. Alvarez, Nucl. Phys. B 216 (1982) 125. [4] S. Mandelstam, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986 ). [5] E. D'Hoker and S. Giddings, Nucl. Phys. B 291 (1987) 90. [6] L. Bonora, A. Lugo, M. Matone and J. Russo, Commun. Math. Phys. 123 (1989) 329. [7] A. Lugo and J. Russo, Nucl. Phys. B 322 (1989) 210. [8] J. Russo, Phys. Lett. B 220 (1989) 104. [9] A. Lugo, Intern. J. Mod. Phys. A 12 (1990) 2391. [ 10 ] A. Lugo, The expectation value of the energy-momentum tensor and the Krichever-Novikov algebra in CFT on Riemann surfaces, Len. Math. Phys., to appear. [ 11 ] I.M. Krichever and S.P. Novikov, Funk. Anal. Pril. 21, No. 2 (1987) 46. [ 12] I.M. Krichever and S.P. Novikov, Funk. Anal. Pril. 21, No. 4 (1987) 47. [ 13] A.M. Polyakov, Mod. Phys. Len. A 2 (1987) 893. [ 14 ] A. Lugo, Ph.D. thesis, unpublished. [15] P. Goddard and D. Olive, in: Vertex operators in mathematics and physics (Springer, Berlin, 1984).

391