Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

Spacetime R-duality in discretized string theories

Enr ique Alvarez a nd J .L.F. Barb6n Departamento de Fisica Te6rica, Universidad Aut6noma, E-28049 Madrid, Spain

Received 10 January 1991

A new discretization of the toroidally compactified string theory is proposed, in which spacetime R-duality is valid order by order in the genus expansion. Our model is equivalent in some important particular cases to the Gross-Klebanov truncated model for strings compactificd on a circle.

1. Introduction

When closed strings are compactified in a carte- sian torus of radius R, both the spectrum and the scattering amplitudes are invariant under the sym- metry R - - , R * = o ~ ' / R , which we shall call, following Schwarz in ref. [1 ], "spacctime R-duality". This transformation can be generalized when background fields are present [2 ], and it can also be extended to all orders in string perturbation theory in the follow- ing sense [3]: if we dcfine the formal sum of the per- turbative series as

F(tc, R ) = ~ R ' 2 ( ' g - l ) / ' g ( R ) , (1.1) g = l

where jcz is the string coupling constant, given essen- tially by the vacuum expectation value o f the dilaton field, ( 0 ) = log ~c2; then

FOc, R ) = F ( x*, R* ) , (1.2)

with tc*=~co~'"/2/Rn for a cartesian, n-dimensional torus. It would seem that this is not really a symme- try, but rather a mapping between different theories with different coupling constants; it is easy, however, to realize that the transformation (1.2) is a symme- try of the effective four-dimensional (low energy) theory,, x 2 = ~¢]2, and as such, carries important phe- nomenological implications [4]. As with any other perturbative symmetry, a timely question is what is its nonperturbative fate; that is, whether it is broken or not, and in the case it is, in which way.

It is notoriously difficult to obtain nonperturba-

rive information in string theory. Several attempts have appeared over the years, like string field theory [5] , the grassmanian approach [6 ], etc. but none of them has been particularly fruitful in obtaining new results. A novel approach has been put forward re- cently [ 7 ] in which one performs a clever cont inuum limit ("double scaling") ofdiscretized strings. Some analytic results have actually been obtained, through an equivalent representation as matrix models. The net result is a Borel resummation of a non-Borel summable series (that is, a definite prescription for the contour around the poles in the Borel integral).

In the context ofcompact i f ied strings in a circle S r, this procedure has bccn applied in ref. [ 8 ]. The world sheet action is represented as a Villain model, which, however, fails in general to reproduce thc cont inuum results in perturbation theory. They then truncate the model to a "zero field strength" sector, getting in this way the desired asymptotic expansion. From this point of view, however, this truncation seems rather unnatural.

The purpose of our work is to present another dis- cretization of the string path integral, much in the spirit of the Regge calculus (as extended by Lee and coworkers, cf. ref. [9] ). We shall find that the most natural definition from our point o f view, automati- cally reproduces the known results o f continuum per- turbation thcory, and besides, in some particular cases, it exactly coincides with the Gross-Klcbanov truncated model! While our discussion of the duality propertics will be quite independent o f the particular rcgularization in the gravitational sector, we will re-

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Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

strict ourselves to the case of non-critical strings, where the only dynamical gauge symmetry is Diffin- variance, properly accounted for in Regge's frame- work. A discrete implementation of Weyl symmetry is considerably subtler. However, in ref. [ 10] a dis- cretization of the critical string by means of a regular (non-fluctuating) lattice is proposed; of course, this formulation borrows the result of decoupling of ge- ometry from the continuum framework.

To simpli~, things, we shall consider in this paper the zero leg amplitude (that is, the partition func- tion) only. To include external vertex operators, al- though possible in principlc, would very" much com- plicate the analysis.

2. The definition of the partition function

Our aim is to provide a definition of

,~"(K,/z, R ) = ~ K 2 (g - I ) g=0

× f ~h~b Diff(Zx~ cxp ( -/ZAg ) Zg

× j c.zXexp[-Sg(X)] , (2.1)

where Yg is a genus g Riemann surface, h,b its rie- mannian metric,/z is the two-dimensional "cosmo- logical constant"; Ag- f z , d (Vol) is the area of Zg; and the embeddings of the Riemann surface in the exter- nal spacetime V d are represented by X~':Zg~ Vd. The action is

Sx(X ) = ½ ~ d(Vol)X'~AX~G,p(X), (2.2) IEg

where A is the two-dimensional laplacian and G,p(X) is the metric in Vd (G,/~=O,~ in our case, Va= T, ,×Pa_,) . Our main concern is the case d = n = 1; just to make contact with recent works [8]. Ob- viously, in this case there is no possible embedding (which is by definition a differentiable, 1-1 map- ping), and one has to specify the properties of the functions X'L This is important mainly in defining our discrete model; in the continuum case it is well known (cf. ref. [ 11 ] ) that the functional measure has no support on the continuous functions at least in the gaussian case.

Our point of vicw here is that the mappings X" in our discrete model have to be continuous at least. Otherwise the string would appear as broken from the spaccfime (target space) point of view, contradicting thc basic postulate of the Polyakov approach that all interactions arc geometrical (intrinsic). This point is not mercly an academic one. There have been claims in the litcrature that vortex configurations are impor- tant in the world sheet dynamics and, in particular, that they could induce an infinite order (Kosterlitz- Thouless) phase transition. These configurations are locally of the type

X(z) =I ra log z - z = ° , (2.3) Z -- Z o

and they are obviously excluded by continuity: if a 2zr rotation around Zo is performed, X(z) winds once in the target S 1, which is forbidden if the circle around Zo is contractible in the base space Zg.

The model we are proposing- essentially a restric- tion of (2.1) to piecewise linear Riemann surfaces - is then defined by

,~'(K,/Z,R)= ~ K 2 ( g - l ) g=O

× Z ~ du( / ) exp(-/ZAx)FAg(R ) , (2.4) Ag A~l)(~. ¢ ~)

where A.~ represents a closed abstract complex with no sites, n~ links and n2 plaquettcs, with Euler char- acteristic given by Z - 2 - 2g= n o - nt + n2; and the measure d/z(/) has support on all possible 1-chains A ~) (~ ~ ) obeying some constraints. One can take an independent measure for each link, and add the tri- angular inequalities for each face as a constraint function. Besides, in order to properly take into ac- count the Diff symmetry, we must divide by two overcountings: some variations ofthc link lengths can simply globally rotate the piecewise linear surface, and any vertex fitting in a flat region (null deficit angle) can be moved around changing some link lengths but not the intrinsic geometry.

The area Ag is defined as the sum over 2-simplices (ijk) of the area or each elementary' triangle, l~,k,

A=- Z /,jk, (2.5) (uk)

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Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

161~k = ( l,j + ljk + lk, ) ( l U + ljk - lk, )

× (lo--lik +lk , ) ( --l , j+ljk + l k , ) . (2.6)

The quantity FAg (R) represents the "entropy" of the embeddings in the circle

FA~(R)= J @ X e x p [ - S g ( X ) ] . (2.7)

The support of .~X will be defined, following Itzykson and Drouffe in ref. [ 12 ], by discretizing the embedding in such a way that the surface Zx is "piecewisc linear" as viewed from the target space Vd= T,, × ~d-,,- What this means in practice is that we first embed each site of the complex, A~. °) ~Vd, and we subsequently extend this embedding linearly on the plaquettes.

To be specific, we can always visualize our intrin- sic geometrical data on the complex (that is, the combinatorics and the set of lengths of the edges), as being induced by the pullback of the standard metric in [-qD (with D as large as necessary, for our purposes). That is, there is always an auxiliary embedding (in the mathematical sense, 1-1 and differentiable) of A s into pn such that we can choose barycentric coor- dinates or, otj, Otk>~ O, a i + a j + Ogk= 1 in each triangle A~jk, and an arbitrary point P is represented in ~D as

Y ' ~ t ( P ) = a ~ Y ~ ( i ) + a ~ Y M ( j ) + a k Y ' W ( k ) (2.8)

(M= 1, ..., D). It should be plain that this embedding yM plays only a heuristic role. The physical embed- ding, which wc shall call X u, is the one on the space- time as target S~. We a priori define it in the sites of the complex, X~, and then extend linearly, taking into account all the possible wrappings of the plaquette over S~. This can be implemented just by going to ~, the covering spaccof S~ =~/2nRT_ and look for mul- tivalued embeddings X: Ag---dq that jump by 2nRn~ 2nRT~ along any non-trivial circle in Ax.

Yet another possibility is to consider the cut rep- resentation Ag, that is, the fundamental polygon for the fuchsian group F which defines the Riemann sur- face through A~=H/F (this corresponds to pairwise identification of the sides of the polygon: a + - a T , b f - -b7 ( i= l ..... g) in standard notation; cf. ref. [13]~ Now we consider functions (liftings of At) , X: A~.-,~ with winding boundary conditions

X + = X Z +2nRn, , Xt, + = X K + 2 n R m , . (2.9)

"Piecewise linearity '° of X is defined as pie cewise linearity of the lifted graph X(Ax)=Y~×A 8. We embed each vertex freely to X ~ , and interpolate,

Xl~ok = otiX ~ + oljX~ + OL k Xk . (2.10)

3. The entropy of the embeddings

We compute now the quantity FA,, (R). First we re- call Bander and Itzykson's results; cf. ref. [ 12]. Us- ing the embedding yM introduced before, the two- dimensional metric of the discretized Riemann sur- face is gao=OaYMObY "~t which in barycentric coordi- nates on the face ( i jk) reduces to

M M M M"

g a b = M M M m \ rSk V,k Y~* Yjk ,] ( 3.1 )

(where Yi~ t - Yi ~ - y)~t). This implies that the dis- cretization of the action in (2.5) is

X s,j, ( i j k )

= Z (cot ok +cot +cot o, (3.2)

where the angles are defined through lok-- ~/jfl~ksin Ok and X~ - X~ - X~. We can now construct the "dual" length c% by considering the circumscribed circum- ferences to the two adjacent triangles sharing a com- mon link (/j); say the ( i jk ) and the ( i j k ' ) . If the ra- dius of the circumferences are p and p' , we define

a~j -=p cos Ok + p ' cos Ok,

= ½/~j(cot 0~ +co t Ok ) , (3.3)

so that the action reduces to the Bander-ltzykson form

s=½ Z (3.4) ( 0 lij "" u

Another useful quantity is

Z -,f,J, (3.5) y ( i )

where the sum is extended to all points j linked to i. This coincides with the area of the dual Voronoi cell when the triangles around the point i fit in a plane.

We then define the laplacians

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Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

( - ax), L Z ~ i J ( i ) ~ j x i j '

(3.6)

and Ao= (a~+%)A o. In order to eliminate the zero mode (rigid translation in the target), we fix an ar- bitrary point in the complex (which we shall label "0" ) instead of the usual center of mass prescription. This means that our final form for the integral in (2.5) is

dX,

t ~ A g ( 0 )

g

× Z 1-I x / ~ f i ( X ~ , + - X ~ : +2z~Rn,) na,rni at = 1

× g

[I b~ = 1

VF2n ~( Xb,+ - X~, i- + 2teRm, )

× e x p ( - ~ -..~ / X ~ ) , 1~ ,~z ~ 1 }

where 2 n R

! dXo

(3.7)

is the integral over the zero mode. Now we solve the Dirac delta functions and introduce one-chains a on the dual lattice, as the "closest parallel" to each boundary" in Ag, and the associated "intersect ion" one-chains e~' = + 1, - 1, 0 according to whether the link l crosses the cycle a (in a positive or negative sense) or not. Then the preceding expression can be concisely written as

! dx, t ~ A~ O)

X y. exp ( - -~ ~ ~(X ,÷-X ,_+2rrR , , . n )2 ) , n ~ Z la A(, I ) l

(3.8)

an expression formally identical to the truncated Gross -Klebanov model of ref. [ 8 ].

4. Dual i~ of the genus expansion

It is actually very easy to show that expression (3.8)

is dual to any order in the genus expansion. First of all, by introducing the convenient variables

• O- 0 2g q',----2nR ~ 7 ~ a . n - - Z z r R ~ q,an~, (4.1)

J O ) tO a = I

and completing the square, we get

FA, (R) = R x / ~ ( d e t - A ' ) - ' / 2

× ~. exp(~q~.(-,~')- 'q. n e 7 2 g k,

-2~z2R 2 ~ (~-n) 2 . (4.2)

Here zl' denotes the matrix zi with the zeroth row and column removed. The argument of the exponential, in turn, can easily be shown to be equal to • 'g n £2 "l' (R)nb where l~Z~Z,,t,~l a Ag

-(2- "t' IR~ =2iztR2[d"~'~-tl~( - A ' ) - ' qh ] (4.3) - A g \ !

with ~ = Zw) t ,,aa/l,j. This implies, in turn, that all winding states can bc packed in a big theta function O2g(011-QA¢ (R) ), and the proof of duality given in ref. [ 3 ] for critical strings in the cont inuum perturbation theory' remains essentially valid in our discrctized setting.

Actually, what is involved here is rather the defi- nition of the piecewise linear Riemann surface as the result of an "ironing process" of a smooth Riemann surface. That is, we define a succession Zg (") such that A~(/) = l i m , ,~,Z C"),g uniformly, in, for example, the ~D auxiliary metric. In fact, we can read from (4.3) the discrete period matrix obtained as a limit r ~ ) In order to have a good gravitational en- g ~ T A .

semblc, the set of ra should bc dense in Tcichmiiller space. A requirement which is far from obvious in the case of matrix models, where the link lengths are frozen.

The net outcome is fixed genus duality

Fh, ( R ) = (Rx/~)2-2gFA~(2~) . (4.4)

In order to make contact with the work of other au- thors, we should first remark that our procedure does not work for arbitrary tesselations (that is, dual to a generic Feynman diagram) because we cannot then linearly interpolate in general, without introducing constraints in the measure ~X.

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Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

We can always, however, map an arbitral ; tessela- tion into one of our models, by refining the covering such as to get a triangulation without additional ver- tices (this is always possible). We can then freely embed each of the vertices in such a way that the measure is, again, translationally invariant.

In the work using "matrix models" one takes la= a, which is the cutoff (usually taken as a = 1). This means that in our general formulas we should substi- tute aa/l,~ by 1/x/~.

When the tesselation is obtained from regular squares, the refinement as above is such that the new links do not contribute to the action. This means that our model (3.8) is identical to the truncated Gross- Klebanov model because oa/l,,= 1 for links belonging to the original tesselation by squares, and rr,Jl#=O otherwise.

5. Conclusions

We have proposed a discrete definition of the string path integral in the spirit of Regge calculus which re- produces automatically the symmetries of (contin- uum) perturbation theory such as the spacetime R- duality we have concentrated upon in this letter.

It is remarkable that for regular square tesselations with constant length for links, our model is identical to thc truncated model of ref. [8]. The point of view is completely different, and we forbid vortices to be- gin with. Note that, essentially, we have not made any assumptions on the matter action, besides those im- posed by the gravitational discretization (just a re- striction on the ensemble). We conclude that the in- clusion of vortices through, for example, Villain-type actions (cf. refs. [ 8,10 ] ) is not particularly favoured by the lattice regularization. So the "vortex di- lemma" is, after all, a physical one.

The authors of ref. [8 ] have shown that there is a Kramers-Vannier duality for these "frozen link" models (cf. also ref. [ 14] ) in the sense that

FA~ ( R ) = ( Rx/2~z ) 2- 2~F,,~(2g~) (5.1)

(where A~ is the dual complex). The "ironing pro- cess" we have proposed guarantees that we have as well

/ / \

Both symmetries have the same continuum limit, but they are conceptually different. From our point of view, IA* is also a piecewise linear surface only when

- - * " " r c - the complex is selfdual, Ag-A~, otherwise our fincmcnt process" introduccs additional factors o,j/4j.

We are unfortunately unable to make contact with matrix models exccpt in the case already considered in ref. [8]. We have then no information on the non- perturbative predictions of our models in the general case. Note, however, that the self-dual scries ob- tained in rcf. [8 ] for thc truncated model with ma- trix modcl propagators, does not agrce neither with continuum perturbation theory, nor with our results, since their self-dual functionsf~(R ) are not the usual thcta functions. This can be duc to the singlct trun- cation (perhaps it kills morc than vortex configura- tions), or more likely to the well-known universality problems in D= 1 matrix models (cf. ref. [ 15] ).

Another important physical problem is the inclu- sion of external legs in the amplitudes (either through the use of vertex operators or punctured Riemann surfaces). This should imply considerations of com- plexes with one plaquette removed. The works of duality are then subtler. Work is in progress on these - and related - matters.

Acknowledgement

We are grateful for many discussions with L. Alvarez-Gaumr, C. Bachas, D. Espriu, C. G6mez, A. Gonzalez-Arroyo, L. Ibfifiez, Y. Kogan and Y. Makeenko. This work has been partially supported by CICYT (Spain).

Note added

Bershadsky and Klebanov [16] recently com- puted the integral over the moduli in the genus one partition function for strings compactified in a torus of radius R, and they actually found complete agree- ment with the Gross-Klebanov first selfdual func- tionf~ (R) = 2 R + 1/2R.

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