Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

THE c-THEOREM, THE DILATON AND THE EFFECTIVE ACTION IN STRING THEORY

S.P. DE ALWIS 1 Theo<v Group. Department of Physics, University of Texas, Austin, TX 78712, USA

Received 6 October 1988

The relevance of Zamolodchikov's "central charge" functional for string theory is discussed. Arguments supporting Tseytlin's ansatz for the effective action are also given.

Recent work [ 1 ] has shown that the (iterative) so- lution of two-dimensional a-model conformal invar- iance conditions yields string (tree level) S-matrix elements ~. This implies that the former are indeed classical equations o f mot ion for the string, as conjec- tured in ref. [ 3 ]. There are however two issues which need to be clarified further. The first is whether the conditions for conformal invariance are derivable from an action and in particular the relation of this action to the fl-function for the dilaton. The second is the question of finding a system of coordinates (in the space of 2D field theory couplings) which makes this action cubic, thus giving a natural interpretation in terms of the joining and splitting o f strings [ 4 ].

In this work the first issue will be discussed in some detail. In particular we will investigate the relation between Zamolodchikov 's c-function [ 5 ] and the ef- fective action for string theory. We find that even on a flat world sheet the naive expression for the fl-func- tion as a gradient of the c-function [5 ] is not in gen- eral true. Secondly we find that the straightforward generalization to a curved world sheet given in ref. [6] is not valid. Finally we give arguments (which are however valid only modulo some technical ques- t ions) in support of the conjecture [ 6 ] that the effec- tive action is given by the path integral with an inser- tion of the stress tensor.

J Research supported in part by the Robert A, Welch Founda- tion and NSF Grant PHY 8605978,

~] The clearest and most general demonstration of this is con- tained in ref. [21.

In a positive metric 2D quantum field theory with a cutoff ~2 A, Zamolodchikov [ 5 ] defines the follow- ing correlation functions (on a flat world sheet) for the components of the stress tensor O_=, 0=:, 0=:= Og (in a complex basis):

C(x, gi) =z4< O=(z)O=(O) >,

O(x, g,) =z3"2< O==(z)O:=(O) ) ,

E(x, g,) =z2~2< O=:(z)O=(0) > . ( 1 )

Here x=A2z~and we take I z[ > 0. The interaction is o f the form 2,g'V, where the sum may include (in the case of the a-model) an integral over coordinates (or momenta) of the target manifold. By 2D Lorentz invariance (SO (2) invariance in the euclidean case ) translation invariance and dimensional analysis, these correlation functions depend only on x, and g, as in- dicated in (1). From the conservation equation for the stress tensor

8:0=- + 0:0:==0, (2)

one then obtains (using zQ-f(x) = g0_-f(x) = ~A (O/OA)f(x) )

A (O/OA )c= - 1 2 E < 0 , (3)

where c = C - 2 D - 3 E , and the inequality follows from the positivity of E. The correlation functions C, D, E should be independent o f the cutoff A for Izl > > A - t Hence we have

~2 In a renormalizable theory the cutoff is to be replaced by the renormalization scale.

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Volume 217, number 4 PHYSICS LETTERS B 2 February 1989

0 =A dc/dA =A Oc/OA + fliO, c, (4)

where fl ,=Adgi/dA, and O,c =- Oc/Og i. We also have the relation ~3

O = = fl ' v~ , (5)

so that defining the (positive definite symmetric) metric

Gij- [z14(V,(z) ~)(0) ) (6)

we have Zamolodchikov's c-theorem in the form

ffO, c = l 2fl'fl'Gq. (7)

(7) implies that at an extremum of c(0 ,c=0, Vi) the theory (since G,j is positive definite) is at a fixed point (i.e. f l ' =0 ) .

Zamolodchikov [ 5 ] gives a conformal perturba- tion theory argument to establish the converse. Choosing the origin in the space o f g ' at a fixed point g*' expand c(g) = c ( 0 ) +giOsc+ .... The second term is given by z4(O=(z)Ozz(O)fg'Vi )g* where the cor- relation function is evaluated at the fixed point. But the latter satisfies a conformal Ward identity and its value is essentially given by ( V~).~. which is zero since V~ has non-zero scaling dimension. (It should be noted however that this involves a certain smooth- ness assumption at least for a non-compact field space. For instance consider a tachyonic perturba- tion of the a-model on a fiat background

f d2zg'Vi~f dZzf T(p) e x p [ i p x ( z ) ] d d p ,

where x/' is the a-model field. The operator in the in- tegrand has dimension p 2 and so the expectation value of the perturbation in the (trivial) fixed point theory vanishes provided that T(p) is not singular a tp2=0 . ) Thus we find that

0,c = %flJ, (8)

for some non-singular matrix ~,j such that

fl~ ~ f l J = f l ' O , j f l s .

Zamolodchikov [ 5 ] actually conjectured that ~s~-----G o and established it to second order in confor- real perturbation theory but this does not seem to be

the case in general. In fact a relation between ~J and G can only be extablished in some rather restrictive circumstances as we shall presently see.

Introduce ~4 the so-called scaling fields [ 7 ] (coor- dinates) in the neighborhood of the fixed point. They are defined by [#~=#i (g ) ]

f l '=A d#'/dA=-fii =a'# i, # ' (g*)=O

(no sum on the RHS of the first equation). (7) now takes the form

( a ' # ' ) 0~c= (aS# ~) (aJ/lJ)Go(#).

We expand c and G o in powers of #' and we have at the fixed point

O,c ° = 0 ,

provided there are no (strictly) marginal operators (i.e. a ' # 0 ) . Thus we have (8) with

~,j= (a j) - ' [0,Sic°+ (1 /2! ) #k0~0j0kC0+...] .

However, the derivatives o f c at the fixed point can- not in general be determined in terms of G~j. At sec- ond order for instance we have

½ (a i +aQ OiOf ° =aiaJa ° ,

so 8iOjc ° is determined provided that a~+aJ¢ O. In fact if we assume that all partial sums Z ;!= ~a '¢ 0 (which is true for instance if all operators are rele- vant or if all are irrelevant) % is determined in terms of G o and its derivatives at the fixed point, but it is certainly not equal to ~a and is not even symmetric in general. The closest we can get is when all scaling dimensions are equal (i.e. a~ is independent of i) in which case

1 . . .# k,,, % = E ~ G(ij,kl...Icn) #kl

and even then it is not equal to G~j though it is symmetric.

Up to now we have not concerned ourselves with the question of defining the stress-tensor in a 2D QFT. An unambiguous way of doing so (which gives sym- metric conserved stress-tensor which is a finite oper- ator when the theory is renormalizable) is to formu- late the theory on a curved world sheet, in a generally

~3 The sum over i in (5) would be infinite for a non-renormaliz- able theory.

~¢4[ a m indebted to Joe Polchinski and Brian Warr for this suggestion.

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covariant manner, take the variat ional derivative with respect to background gravity g , , and then ( i f one is interested in the flat world sheet theory) take the l imit g~ls-~b~n. To be specific let us consider the a-model with just the renormal izable couplings, the tachyon, gravi ton ( G , , ) and an t i symmetr ic two-form field ( B , , ) . It is well known [8] that (on a curved world sheet) the model has an addi t ive renormal iza t ion re- sulting in a "d i l a ton" coupling to world sheet curva- ture ( I n ( 1 / 4 n ) f d2z( \~?R ) (Z> O( x( z, g) ) ).

This addi t ional term implies that even in theflat- space limit #s, the stress-tensor has di laton dependent pieces 0=~0~0, 0 : ~ -0-0~0. The correlat ion func- tions C, D, E will now depend on ~) and there will be a corresponding/?-funct ion, rio. The renormal iza t ion group equat ion (4) is now replaced by ~

dc 0c /~0c ~o0c 0 = / 2 ~ = / x ~ + 0g' + g 0-~' (9)

where the sum over i is taken over all couplings ex- cept the dilaton. Fur thermore , the effect o f the dila- ton term in O= discussed above is to replace [ 10]

(by using the equat ion o f mo t ion ) f l~ f fwhere f id i f - fers from fl by a a -model target mani fo ld diffeo- morphism, in the RHS of ( 5 ).

In addi t ion o f course on a curved world sheet there would be a fl°R (2) term, but obviously it d isappears in the fiat-space l imit . Thus (as observed al ready by Tseytl in [6] ) the c- theorem is modi f ied (even in the flat-space limit) and takes the form

Oc Oc 13 7g~ +13+ ~ = 12f i '?a, j . (lO)

First we observe that at a fixed point ( f l "= f i °=O) the posi t ive defini tness o f G #7 implies that fi~= 0 - ie. scale invar iance implies conformal invar- iance. Hence it follows f rom the conformal pertur-

ba t ion theory argument that f l ' = 0 , / ~ = 0

~5 One can see this even without going off the flat-space limit simply by observing that there is an ambiguity in the stress- tensor corresponding to the possibility of adding a term go6-aCCzt,do,,Oa~ for any scalar field [9 ].

~o Thus the previous considerations are valid only with special choice of the stress-tensor such that ~ is a solution of fl~=0 or Ooc= O.

~7 If the a-model target space has minkowskian signature then one should work in the light-cone gauge. The temporal and longitudinal a-fields are taken to be free and our background fields only have transverse components.

3ic= 3oc= 0 #8. In order words the Zamolodch ikov c- function is still an act ion which is ext remized when all fl-functions ( including the d i la ton) condi t ion are satisfied.

Now clearly the converse is obviously true only to the extent that 3,c=0, 3~c=O~fii=O. Tseytlin has ar- gued that the c o n d i t i o n / f o = 0 is also implied. Let us review his argument. First we write the measure in the functional integral expression for C (or D or E)

f [ d x ] e x p ( - I ) as f d y x / - G ( y ) e x p [ - 2 0 0 , ) ] f i d e ] Xexp[ - l ( y + ~ ) + 2 0 ( y ) ] where y is constant and integrat ion is over all modes orthogonal to the con- stants #~. Now it is assumed that one may unite

c = f d y ~ e x p [ - 2 0 ( y ) ] d ( V O .... ) , (11)

where 6 depends as 0 only through its derivatives. One may just i fy this as follows. The 0 dependence of l ( y+ ~) is ( 1 / 4 n ) f x/g R (2) ¢)o 0'+ ~) where 0o is the unrenormalized dilaton field. But in per turba t ion theory all di la ton counter terms are propor t ional to the derivatives of 0, so the difference l (y+ ~) - 2 ~ ( y ) would seem to be independent of O(Y). The opera- tors O:_-, (9_-: are also independent of O(Y) so that we may conclude that ~ is independent of ~ (y ) .

Now by a general izat ion o f an argument due to Fr iedan [ 11,12 ] we know that at the poin t of confor- mal invar iance / f i= 0, we have a Virasoro algebra with fi~ being the central charge, i.e. ~=fi¢'. Of course this makes sense only i f rio is a c-number at this point. The lat ter can in fact be shown to be true #~o. Armed

#8 Modulo the caveat (for non-compact theories) discussed just before eq. (8).

~9

Note that l (y)= - (1/4n) f ~gR (2~ ~o(Y)=20o(y) since on the sphere Z= - 2.

#~o This was first shown in perturbation theory by Curci and Pafutti [ 13 ]. A simple proof of this statement, due to Pol- chinski [ 14 ] which is independent of perturbation theory goes as follows. Let Zoo,~)... be the path integral with operator in- sertions O(m) .... Then iffl '=0

8Zo - - = (O=(z) O(~)...)=(fl~R~2~(z))O(~)...), ap(z)

where 6p is a Weyl variation of the metric. Now take the sec- ond derivative w.r.t, p and take the fiat-space limit

82Z° k 8p(z)6p(z') ,o=(flO(z)73:62(z-z' )'''7 "

From the symmetry of the LHS under z~z we have (fl~(z)-fl°(z' ))[26Z(z-z ' ) =0 and hence 3.fl~= O:fl ~ =0, i.e. flo is a c-number.

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with these facts let us consider (10) after replacing fl,, flo on the LHS by fl~, fl~. This is possible because of the diffeomorphism invariance (under target space diffeomorphisms) of c and the fact that the barred fl functions differ from the unbarred ones by diffeo- morphisms [ 6 ]. Thus (10) becomes

fl'0,c+/~0oc=/~#G,j. (12)

Consider (12) at the point fl '=0. Writing out the in- tegration explicitly and using the fact that flo is inde- pendent o fy ~ at this point we have

f 8c =rio f dDy 8C =0 (13)

The equation implies that either fi* or f dDy 8c/Sfb(y) are zero. But a theory with confor- mal invariance may certainly have non-zero central charge, so we should consider the latter situation. From ( 11 ) we find

8c --2~exp[-2~(v)] 8~(y)

( +V x / G e x p ( - 2 g ~ ) a V - ~ ,

so that

f 8c - - 2 c . dDy 8O(y)

Thus the vanishing of the LHS implies that c = 0 and hence that fl~ = 0, and so we are forced to the conclu- sion that at the "fixed point" fl '=0, the central charge flo=0. This conclusion is however obviously too strong, for instance the free theory (&,, = q,,,, Bv~ = 0) clearly is consistent with non-zero central charge. So what goes wrong? As far as we can see it must be the case that (10) and (12) are incorrect on a world sheet such as a sphere which cannot be made fiat every- where (since in order to use Zamolodchikov's argu- ment we had to take the fiat-space limit). Indeed on a torus we have no problem. For in that case since the Euler character is zero,

c= f d~'y ~/g(y) e(vo...)

and

f Sc/80(y)=fV(x/gOc/OVO)=O

and fl* = 0 is not implied. In spite of the above, Tseytlin's ansatz for the ef-

fective action may still be correct. Write the renor- realized partition function for the a-model as

ZR[G, B, 4; ~]

= f ddy Gx/-Gx/-Gx/-Gx/-Gx/G~ exp [ - 2O(y) ]

× f d{ exp[ -I(y+{)+2~(y) ]

=Iday X e x p { - 2 0 ( y ) - W[G, H, V0] (y)} ,

where as before y is the constant mode ofx. Making the field redefinition

0(Y)- '0 ' (Y) =0(Y) + 1 W (y ) ,

ZR[G, B, q~] = f day GxflG ~ e x p [ - 2 0 ( y ) ] ,

where we have removed the prime on 4. Then Tseytlin's ansatz is

S= ~ ZR= f d22x/g (Oze)

= - 2 f exp ( -2O) f f ~, (14)

where fro = fl~_ ~ G"" fl,,~. Tseytlin justifies (1) using his extension of

Zamolodchinkov's c-theorem to the spherical world sheet. But for the reasons given earlier this does not seem to be valid. Below we demonstrate directly (on a spherical world sheet with constant curvature met- ric) that the equations of motion obtained from ( 1 ) imply the vanishing of the fl function (fl~-(; =fl~-R = r i°=0) . From the RG equation dZR/dt=O we have (since Off~Or=O)

aS- ( f l{~50 '+f l?~o)Sot

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Volume 2 l 7, number 4 PHYSICS LETTERS B 2 February 1989

In the second equality above we used the fact that

in dimensional regularization

0 l= f lZ-~- dZzO:,= limn_2 ( n - 2 ) I ,

and that

~ O (¢t~-~ : e ) = ( n - 2 ) (@_-) ~ 0

as n-+2 since 8 : - is a finite operator. Using transla- t ion invariance on the sphere and positivity ~1 we

then have the result 0 , S = 0 = O~S~@e=O~ff=O=

The converse result (i.e. that every conformally in- variant a-model is derivable from the action ( 1 ) ). is more difficult to establish. Hopefully the following remarks will make it plausible. Using the RG equa-

t ion for @-: we have

S~- f d2z ( Oz~) =fli ~78ZR + flo 8ZR60 (16)

The derivatives O,ZR, O~,ZR are finite objects being derivatives of the renormalized part i t ion function with respect to the renormalized couplings. They are in fact vertex operators of the form 60~/60 ' ( d ( y - x(z))Pi(Ox,~x) ). The operator in the angular bracket does not have a definite conformal weight. However, it can be expressed as an integral over op- erators with definite weights. For instance around the trivial conformally invariant background

( d(y-x(z) )P~ )

= I d26k exp(iky) (exp[ -ikx(z) ]P, ) .

The operator in the integrand has non-zero weight except on a set of measure zero and hence the integral should vanish modulo the technical assumption made earlier. Hence although we cannot give a rigorous ar- gument the above makes it plausible ~12 that in any conformally invariant background 0iZ= 0oZ=0, so that (from the variat ion of (16) ) fl~=flo=O~O~S= O~S=O.

~t~ If the a-model target space is minkowskian, we may do this only in the light-cone gauge; see footnote 6.

~2 As noted earlier Zamolodchikov's argument for flat a-models also depend on such an assumption.

I wish to thank Joe Polchinski for several valuable discussions, for permission to reproduce his unpub-

lished argument in footnote 7, and for useful com- ments on the manuscript. Thanks are also due to Tom Banks and Brian Warr for useful discussions. Finally I wish to acknowledge the hospitality of the Aspen Center for Physics where this work was completed.

Note added. While writing this up for publication my at tention was drawn to a paper by Osborn [ 15 ] which gives alternative arguments for an action sim-

ilar to (14).

References

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[ 2 ] H. Hughes, J. Liu and J. Polchinski, Texas preprint UTTG- 13-88.

[ 3 ] C. Lovelace, Phys. Lett. B 135 ( 1984) 75; C.G. Callan, D. Friedan, E.T. Martinec and M.T. Perry, Nucl. Phys. B 262 (1985) 593; A. Sen, Phys. Rev. D 32 (1985) 2102; Phys. Rev. Lett. 55 (1985) 1846.

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[6] A.A. Tseytlin, Phys. Lett. B 194 (1987) 63. [7 ] S.-K. Ma, Modem theory of critical phenomena (Benjamin/

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[12] S.P. de Alwis, Phys. Rev. D 34 (1986) 3760. [ 13l G. Curci and G. Paffuti, Nucl. Phys. B 286 (1987) 399;

see also A.A. Tseytlin, Lebedev Institute preprint N342 (1986); H. Osborn, Nucl. Phys. B 294 (1987) 595.

[ 14 ] J. Polchinski, private communication. [ 15 ] H. Osborn, String theory effective actions from bosonic a

models, Cambridge preprint DAMTP/88-16.

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