Space-time duality and discrete strings

~UCLE,L~ P~YSCS E

Nuclear Physics B (Proc. Suppl.) 25A (1992) 31-37

PROCEEDINGS SUPPLEMENTS

North-Holland

SPACE-TIME DUALITY AND DISCRETE STRINGS J.L.F. Barb6n Dep. FiRica Tedrlea. Universidad AItdnoma de Madrid, ~8049 Madrid, Spain

1. Introduction Spacetime duality is ,~ discrete symmetry of strings propagating in background manifolds with non contractible loops (see !1] for a basic review). The simplest of them presenting this property is the cartesian torus of radius R, in which both the spectrum and the scattering amplitudes are invariant under the transformation R . . 4 ~" e~I = R*. To ilustrate the point, consider a bosonie string in 1"1.25 x S t and do standard canonical analysis. The bosonic coordinate in the circle dimension ca~, be expanded as X = Xo + a ' P r + La + oscillators

(1)

In the right hand side the first term is the string center of mass, the second is the string moment u m ( r is the wcr!d-sheel ,qme). The third describes the winding of the string on S 1 (a is the world-sheet space) and the oscilator terms are the s t a n d a r d string vibrations. For radius R, P and L are quantized as P = n / R , L = mR; n, m = integers. The spectrum is ( mR V where NR and NL are the right and left oseilator numbers. Since it is invariant under n ~ m, R ~ a~/R symmetry, we see that R-physics is equivalent to 1 / R physics provided we interchange the roles of winding states (m) and stan-

dard Kaluza-Klein m o m e n t , . , n ~es ( - I In calculating the partition fm]ction, since we compute a trace over the entire spectrum of states, we find an exact R-duality symmetry. From the physical point of view, this duality between large and small scale physics points certainly to the existence of a fimdamental length (aprox. the Planek length) in stch,g t L ~ t y . Similar conclusions arc drawn from the study of high ererg~ string eolisions [2]. This symmetry is also present in more complicated backgrounds like orbifolds and Calabi-Yau compactifications. Moreover, it can be generalized to all orders in string interactions, t h a t is, to all string topologies. We'll see that R-duality is a classical effect on the matter sector, and depends on modular properties. So the works of duality are eualitatively independent of the central charge and our remarks apply to critical as well a-s to non critical strings. 2. R-duality to all orders Consider the bosonic string partition function and expand it in the topological series 0o Z ( ~ , R ) - S'~ , F ° - ~ Z g ( R )

(3)

9=1

g is the world-sheet genus and ~" is the string coupling constant, given essentially by the vac-

0920-5¢t32/92/$05.00 ~) 1992- Elsevier Science Publishers B.V. All rights reserved

J.L.F. BKb~n / Space-time dualityand discretestrln~s uum exnectation value of the dilaton field, (~b) = log ~2; then Z(~, R) = Z(~*, R*)

(4)

with R* = l / 2 r R , ~* = ~/v/~R (from now on we set a" = 1 / 2 1 r ) . Note that an R-dependent shift of the dilaton is needed for g ~ 1. Let's briefly ilustrate the proof following ref. [3]. Given the fixed genus partition function as a sum over random surfaces ~l: Z, = ~ e-~A"oU%(R) {%l

(5)

In Polyakov's language, the matter contribution is a kind dspaee-time entropy for eadl Riemman surface ~z: F%(R) ~A t being the laplacian in ~ I ' The functional integral is computed over all embeddings of ~e into the target space T. In order to have winding modes, we assume for simplicity only one non contractible loop in the target; ~I(T) = Z, and continuity of X, a suitable coordinate around this loop. These mappings can be clasified ~. ording to their homotopy type: since we have 29 non contractible loops in ~l (for example, the usual a~and bi homology cycles in the standard canonical parametrization [4]), then we have 2g integer winding numbers of these cycles in T. Furtilermore, a natural splitting can be made: X=X0+X'

(7)

where X~ is harmonic and X ' is homotopicalty trivial. Then the harmonic (so called solito.'.ic) contribution factorizes from the primed one, which is in~nsi~iw to the compactness of the target. Because of the gaussian character of the action, the solitons pack in a theta function Fz,(R) = 02s [Ollf~t(R)] F~0(R = co)

(8)

where t~#(R) is a 2g x 29 matrix:

n,(R) = 2,iR, ( ~',r2'~, -rsr~ - - ~ l r+1", l ri i 1 )

(9)

and 7" = ~1-I-i~'2 is the period matrix of the Rierre man surface ~e" In these general setting the previous duality transformation between the windlug modes and the Kaluza-Klein modes is represented by the modular transformation between a-cycles and b-cycles:

,.

.)o(Oo

So that, after Poisson resummation we have r~,(R) = (Rq'~)2-2"r~,

~

which obviously implies the relation (4). This is, however, a formal result for the critical case, because of the tachyon divergence. It is worth remarking that a similar proof exists for heterotic strings, while type I! superstrings doesn't exhibit duality; R-duality is a rather tricky symmetry in the fermionic sector. From the point of view of string theory we have here an amusing symmetry. First of all, as wc mentioned above, it shows in ~iwple terms the appearance of a dynamical cutoff in string theory. Second, if we compactify the euclidean time, we formulate strings at finite temperature, and spacetime duality becomes high-low temperature duality. In these case the theHnodynamic behaviour is rather counter intuitive unless we assume a non pcrturbative breaking, because duality drives the partition function to zero at high temperatures (see [5]). On the other hand, this is an extremely useful symmetry from the phenomenological point of view. It is a true symmeLly ~f the effective fo~lr dimensional field theory, since it leaves inva:i~ ant the effective gauge coupling ~ ~ ~-n~2(n

J.L.F. Barbdn / Space-tlme duality and discrete strinp is the number of compact dimensions). It considerably reduces the moduli space of independent low energy models and it can be traded in a Higgs-like mechanism [I]. In addition, some recent works [6] have exploited a non perturbative duality symmetry which, upon spontaneous symmetry breaking, constrains some traditional free p,~rameters, like the compactification radius. The nice thing is that this fixing comes along with the supersymmetry breaking via gaugino condensation. Given that duality survives to all orders in the genus expansion, the most interesting problem is of course its non perturbative fate, specially because the whole perturbative series (3) is divergent [7] and not even Betel summable. This probably means that the results obtained in string perturbation theory are not a good indication of the f hyping! p.~iction~ of the theory. UrJortunately, it is notoriously difficult to obtair. non pertmbative information in string theory. Several attemps have appeared over the years, like string field theory, the grassmanian approach, etc, but none of them has been particularly fruitfull in obtaining new results. A novel approach has been put forward recently [8] in which one performs a clever continuum limit (double sealing) of discretized strings. Some analytic results have actually been obtained through -n equivalent representation as matrix models. There are suitable matrix models for the study of t'::o dimensional gravity coupled to c < I confors~,~l matter, and even c -- I conformal matter (strings in ~ne dimensional target 3pace). The net r ~ :Ig is a Betel resummation of a non Borel summai,!e series (that is, a definite pro~ription for the coutour around the poses in the Borel integral). In the context of compactified strings in a circle S l this procedure has been applied in [9]. According to this authors, non perturbative R-

33

duality is safe provided the theory is truncated to the zero vorticity sector. In fact, vortex configurations destroy duality order by order in perturbation theory.

3. Discrete strings and vortices A natural set up for discrete string theory would be the following: Consider a gennc 9 world-sheet discretized in the form of a pieeewise fiat two dimensional surface. We take a simplicial complex AI with no sites, nl links and n2 plaquettes with Euler characteristic X = 2 - 29 = no - n t + n2. The u~tric propert'cs are encoded in the abstract complex (that is, the combinatories of the triangulation) and the set of lengths of the edges, ll/. In particular, in the work using '~matrix models" one t a k ~ ll/ = a = const., which is the cutoff, ;.h;.le in the classical Regge calculus one fixes the triangulation and considers the link lengchs a~ dynamical variables of gravity. In the matter part, which is the relevant one for duality, ",:e can attach a copy of the target space to eac~ vertex of the lattice, and design a local discrete action whose continuum limit reproduces Polyakov action. For a compact target S I, a sensible construction is given in [I0] by regarding the circle as tile real line divided by discrete translations: S~ -- R/2a'f/Z. So we can consider the mattcr as a gauge theory of the di.~crcte group g (addilive), identifying site variab1~ ~ Xi Xi q- 2~rRni, ni E Z. Al~er constructing the corresponding covariant derivative ( D X ~ 0 = X i X j + 2~rRmij, m l j E g we may gauge the free action:

(ij)

3.L.F. Barbdn/ Space-timeduality and discretestrings

34

where m 0 plays the roie of Wilson's link variable. The resulting entropy of random surfaces is:

I t 2"a

F(Av')(R) = ~

~/s •

\

dXi)

Z

e-S(m)

(131

mijEZ

exactly the Villain model, very well known in statistical mechanics as a gaussian version of the XY model. For a fixed lattice (gravity decouplcd), this system exhibits a Kesterlitz-Thouless phase transition between the spin-wave phase (R > RIfT ~ Planck length), where the theory scales in the continuum to the gaussian model, and the Coulomb gas phase (R < RIFT) where X disorders and becomes massive. In fact one cannot scale a conformal field theory in this regime. The field configurations which drive the phase transition are the vortices. In the gauge analog ,hove. eorre~p,~nd . . . . . to. ~-~n . . tr;.:';.a! . . ~ a ~_trc;:g'.hs, that is, we can define the vortex number for each plsquette as vorticity = ~ #o

mij

(14 I

2O

mij=m~-mj+Zt~.ina,

n. e Z

(16)

a--I The closed up.e-chains mij are gi:'en by the exact ones (pure gauge) plus the homologieally non trivial ones. They are generat¢d by the onechains (~ a.ssoeiated to ?g non trivial homology cycles, arbitrary chc~en in the dual lattice: t~ = + 1 , - 1 , 0 according to whether the link I crosses tile cycle a (in a positive or negative sense I or not. Upon substitution in the Villain model we find [9]:

F^,(Rl = ~ (i~ dXi) n~Z2 e-S°x(n' 1

..

SGK(n) "---~ ~...~(Xi - Xj -F2~rR~ijn)2 (ij)

(17)

(18 I

'l-.his expression for the space-time entropy has the remarkable property of being KramersVannier self-dual: changing variables to the dual lattice A.~ and performing a Fourier expansion one has r^,(P~) = (nvq;~)7-2~rA;(2--~R I

(19)

Vortex configurations are localy of the type

X(z) ~ Im log ~ z0

(151

If a 27 rotation around z0 is performed, X(z I winds once in the target. Thus they're lik~ local winding states. The~e vortices interact by means of an effecfive Coulomb pote;,tial and they're bound in dipoles fvr P > ,~-/c~'. However , for R < RIfT these dipoles dissociate mtt, z ~ortex-antivortex plasma, and a mass gap is generated for X. ~Ve can eliminate vortex configurations just by truncating the theory to the zero field s~rength sector: vorticity = 0. This is an homology problem in the group H i ( h i , Z ) ~ Z2g easily solved by:

llence, the vortices are exactly responsible for the perturbative violation of R-dualtity. The authors of ref.[9] present a non perturbative analysis by means of the matrix quantum mechanics. The full partition function (matter + gravity) is regarded as the free energy of a quantum mechanical system of N x N hermitian matrices, interacting by means of a suitable quartie potential (thus generating square plaquettes). Formulated in this way, the mat'~er action is not exactly gaussian, but it is believed to be in the ~ame universality class. The vortex configurations are ,.onjectured to correspond (see however Kazakov's leetu~c~ in these proceedings for a proof) to non singlet staten under the groUl~ U(N !. Incidentally, the singlet trunc zi;on gives_a

J.L.F. Barbdn / Space.time d~alily and discrele strings R-dual result even non perturbatively; the correspnndilJg asymptotic expansioP has been checked for the genus one case in ref. [11] in the framework of Liouville theory, the iutegral of the soiltonic theta funcion over the moduli space gives the matrix model result. The Kosterlitz-Thouless scenario has been checked also in presence of fluctuating lattices [9]. llence, at this point we have two different scenarios for high temperature strings, depending on whether we neglect vortices or not. In tide first ,ase we get the usual continuum picture, including R-duallty, in the ~;ccond case the cornpact dimension disappears beyond RKT. From the point of view of lattice strings, as pr~cnted here, it seems rather unnatural to truncat,' '~,. theory, but we'll see in the following section that this is not quite .so.

ter action: just the restriction of the continuum one to piecewise flat manifolds. This procedure is valid even in the case of pure gravity where, as shown in [12], the celebrated Regge action is exactly tile Einstein-llilbert action evaluated in piecewise flat manifolds. This has been done in the ganssian case (real line as target space) by Bander and ltzykson in [13]. One simply chooses baricentric coordinates in each triangular face (ijk): oi,r~j,a~ >. O, a i + oj 4- ok = 1 and computes the Polyakov action ia these particular coordinate system. The result is the usual one:

5,= I Z t-~-(.Y~ ~# - xj f (0)

where lij is the link length and ~ij is a kind of "dual" link length, defined by 1

It has been argued in the last section that vortices are like local winding states, inserted at some point of the surface. In fact, whenever the vorticity is non zero at some plaquette (or dual lattice site), the embedding function is wrapping that plaquette an integer number of times onto the target circle. This means that the Villain action measures embeddings of grids, and not complete surfaces with full plaquettes, because the boundary of a plaquette is contractible in itself. Hence, at some point the Villain model is i n t r ~ ducing discontinous surfaces (remember that tide basic gauge symmetry is highly discontinous). In that way we break the world-sheet when embedding it, thus introducing non geometrical interactions. So we see that the principle of contil~nity of the wurld-sheet,quite natural m string theory, forbids vortices to begin with. Actually, there is a very natural prescription for the mat-

(~o)

"

#ij = .~l,) (cotO~ + eotOk' ) 4. "Ab initio" vortex truncation

35

(21)

in terms of the angles lying opposite to the link (ij) ill lhe adjacent triang'.es (ijk) and (ijk'). Tile quotient aij/lq is a volume element, which reduces to a trivial constant for lattices of the matrix-model type. It i,~ easy to generalize this setting to the compact targct case with arbitrary world-sheet topology. We just compute the space-time entropy as a sum over all continous embeddings of piccewise flat manifold% preserving the piecewise fiat structure as seen from the target space, and weighted by the standard Polyakov action evaluated in the lattice manifolds. Flatness in the target is equivalent to linearity of the embeddiug function. Furthermore, since we maintain continuity, the homotopi: dasification sketched in section 2 holds here; we simply map the full lattice surface onto the circle, embedding the sites of the complex, and then extending linearly, taking into account all the possible wrappings of tide plaquette over S~. This can be imple-

36

J.L.F. Barbdn / Space-time

duality and discrete strings

merited just by going to R, the covering space of S~ = R/2:rRZ and look for nultivalued embeddings X : As - . I t that jump by 2xRn ~ 2~rRZ along any non trivial cycle in Aa. Yet another posibility is to consider the cut representation As, that is, the fundamental polygon for the fuchsian group F which defines the Riemman surface through As = H / r (this corresponds to pairwise identification of the sides of the polygon: a + = aT, b+ -'- b7 (i = 1 ..... 9) in standard notation [4]). Now we consider functions (liftings of.~s) , X :/~s ~ I t with boundary conditions

qin -----27tR~'~ ~ e , j n -2~R~'-~r/~'na (25) J(0 ': a=l (the first sum is over the closest neighbours of the site "i'). Completing the square, we get

XL = X7. + 2 - R n , , XT. = X~ + 2 , R m ,

F^g(R) = R v ~ ' ~ ( d e t - A ' ) - l / z

(22)

"Piecewise linearity" of X is defined as piecewise linearity of the lifted graph X(As ) C I t x As. We embed each vertex freely to Xi E It, and interpolate, Xl~,, , = niX, + a j X j + akXk

(23)

Now we write the entropy of random surfaces

dual cycles (a = I, ..., 2g) are chosen as tile "closest parellel" to each boundary of the cut surface, so that the c~'j one-chains exactly implement the solitonic jump in (22). It is actually very easy to show that this expression is self-dun] in the same sense ~m in the continuum case. Fi~t of all, we introduce the convenient variables 2g

nEZ~t

llere/X' denotes the lattice laplacian/k with the zeroth row and column removed. The argument of the exponential in turn. can easily be shown to be equal to iTr~'~lb=l nafl,~6(R)nb where n~b(R) -- 2i~rR2 [6°b~a --

iEAl,

n o , m , a,ml S

(' •

IEA~

)

where this time we treated carefully the translational zero mode, by fixing for example the 0'th site. Now it is easy to realize that, after solving the Dirac delta, ff,-.ctions, we get exactly the Gross-Klebanov truncated model in (17), (18) (with general volume elements ¢/[). Here the

,f~(--/k')-1,/b] (27)

with ~o - - ~(ij).LaO'ij/lij. This implies in turn, that all winding states can be packed in a big theta function Ozs [0]]t~^(R)], and the proof of duality given in ref.[3] for critical strings in the continuum perturbation theory remains essentially valid in our discretized setting. Actually, what is involved here is rather the definition , f the piecewise flat surface as the result of an "ironing process" of a smooth surface. That is, we define a succession E ('*) such that As(I) = lim~-oo E(s~), uniformly. In fact, we can read from (9) and (27) the discrete period matrix obtained as a limit r ('0 ~ rA. Smooth properties like the modular transformation in (10) survive the limit and we can prove the strong result:

3.L.F. Barbdn / Space-time duality and discrete strings 1

rA,(R) = (R,/~)~-2gr^,(2-~R)

(28)

Note that it is not necessary to go to the dual lattice! So this is different from the KramersVannier duality shown in (19), although both havc the stone continuum limit. 5. Conclusions

We have proposed a natural lattice construction for the compactified bosonic string, close in the spirit to Bander and ltzykson works [13], which reproduces the usual results of continuum perturbation theory, including R-duality of the genna expansion. We conclude that the inclusion of vortices, through for example Villain actions (13), is not particularly favoured by the lattice regularization. So tl:e "vortex dilemma" is, after all, a physical one. It is quite non trivial that the lattice model enjoys exact R-duality (28) and not only KramersVannier duality (19). Both symmetries have the same continuum limit, but they are conceptually diffe,~nt. In particular, in the proof of (19) it is :.~-.tial to keep the winding states, by means of the intersection one-chains ¢O. This explains why the continuum proofs of R-duality based on the continuum analog of Kramers-Vannier (cg~,X ~ OoX* = ¢ c o 0 ~ X ) fail in reproducing the shift of the dilaton ( ' ~ R ) 2-2g) [14]. At the world-sheet level, space-time duality involves not only local hut also global transformations (10).

Acknowledgements This work is based on ref. [15], in colaboration with E. Alvarez, to whom 1 express my thanks. I also would like to thank the orsa~izers of this workshop for giving m e ti~e opportunity to present this material. This work has

37

been partially supported by CICYT and MECFPI (Spain). References [I] J.H. Schwan, "Spacetime duality in string theory", Cahech prep¢int CALT-68-1581 (1989). [2] G. Veneziano, "Physics with a hmdamental length", Cem preprint CERN-TH 5581[89. [3] E. Alvarez and M.A.R. Osodo, Phys. Rev. D 40

09s9) 167. [4] D. Mtmlford, Tara lectures on theta, vol.I ~BirkhSuser, Basel, 1983). [5] J.J. Atick and E. Witten, Nucl. Phys. B 210 (1988) 291. E. Alvarez and M.A.R. Osorlo, Physica A 158 (19891 449. [6] A. Fruit, L E. lb,ifiez, D. Lilst and F. ~ucv~]o, Phys. Lctt. B 245 (1990) 401; S. Ferrara, N. Magnoli, T.R. Taylor and G. Veneziaalo, Phys- L~tt. ~ "~.~ (1990) 409. [~ D.J. Gross and V. Pcriwal, Phys. Rev. Lett. 60

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ao (198v) 41s. [11] M. Bershadsky and I. Klebanov, Phys. P.ev. Lett. [12] R. Frledberg ~ld T.D. Lee, Nucl. Phys. B 2¢2 (1984) 145; G. Feinherg, [L Friedberg, T.D. Lee and [|.C. Ren, Nucl. Phys. B 245 (1984) 3t3. [13] C. ltzykson and J.M. Drouffe, Statistical field theory (Cambridge U.P., Cambridge, 1989). [141 J. Molera and B. Ovrut, Phys. Rev. D 40 (198'3) 1146. [15] E. Alvarc~zand J.L.F. Barb6n, Phys. Lett. B 258

(1991) 7s,