A simple geometric description of T-duality is given by identifying the
cotangent bundles of the original and the dual manifold. Strings propagate
naturally in the cotangent bundle and the original and the dual string phase
spaces are obtained by different projections. Buscher's transformation follows
readily and it is literally projective. As an application of the formalism, we
prove that the duality is a symplectomorphism of the string phase spaces.Comment: 10 pages, LaTeX (1 reference added

Klimcik, C.

Severa, P.

English

arXiv

A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces

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CERN-TH.7490/94Strings in Spacetime Cotangent Bundleand T-dualityC. Klimck1Theory Division CERN, CH-1211 Geneva 23, SwitzerlandandP.SeveraDept. of Theoretical Physics, Charles University,V Holesovickach 2, CZ-18000 Prague, Czech RepublicAbstractA simple geometric description of T-duality is given by identifyingthe cotangent bundles of the original and the dual manifold. Stringspropagate naturally in the cotangent bundle and the original and thedual string phase spaces are obtained by dierent projections. Buscher'stransformation follows readily and it is literally projective. As an appli-cation of the formalism, we prove that the duality is a symplectomor-phism of the string phase spaces.CERN-TH.7490/94October 19941On leave from Charles University, Prague1 IntroductionTarget space duality keeps attracting the attention of string theorists (see e.g.[1, 2, 3, 4]) mainly because it deepens our understanding of the geometryof spacetime from the string point of view and it is an important tool fordisentangling the full symmetry structure of the string theory.Duality is usually derived in the -model context in appropriate coordi-nates, respecting the isometry of the action. The symmetry is gauged and thegauge elds are constrained to be trivial by introducing a Lagrange multiplier.The latter is understood as a new (dual) coordinate and the gauge elds areintegrated out to end up with the dual -model. The dual metric~G and theskew-symmetric tensor~B are then given by Buscher's formula.This approach is not very transparent from the geometrical point of view,however. An important conceptual simplication was undertaken in [5] wherethe duality was described in a global geometric setting and, following the sug-gestion in [6], it was interpreted as just a canonical transformation of the phasespace of the theory [7]. Attempting to further clarify the concept, we give avery simple global geometric description of duality for a large class of systemsincluding the -models. We show that the dual target, its cotangent bundle,and the elements of the original and the dual phase spaces can be naturallyembedded into the cotangent bundle of the original target. The relation be-tween the original and the dual quantities is now simply given by dierentprojections of invariant objects living in the cotangent bundle. Buscher's for-mula follows extremely straightforwardly and it is literally projective. As anapplication of the developed formalism we present a simple proof that dualityacts as a symplectomorphism of the original and the dual phase spaces, under-stood as appropriate submanifolds of the cotangent bundles of the loop spacesof the targets. The whole picture seems quite natural. We may even guessthat the cotangent bundle will play an important role in developing a naturalframework for the description of string symmetries.The plan of the paper is as follows. In section 2 we identify the cotangentbundles of the original and the dual manifolds and compare their canonicalsymplectic structures. In the next part we lift the dynamical characteristicsof strings into the cotangent bundle and by dierent projections we obtainBuscher's formula. In section 4 we prove that the duality is a symplectomor-phism and in section 5 we consider -models only, recovering the standardresults in a very compact way.12 Dual symplectic structures onthe cotangent bundleConsider a manifoldM with a global vector eld v and a global closed 1-form !such that !(v) = 1.2Let TM be the cotangent bundle ofM . The elds v and! are naturally extended to the whole TM . On TM there is the canonicalsymplectic form . We dene the dual elds ~v and ~! on TM in this way:~! = (:; v) and ! = (~v; :) (1)(so that d~! = 0 and ~!(~v) = 1).There exists another symplectic form~ on TM such that~ + ~! ^ ! = + ! ^ ~!: (2)Then obviously! =~(:; ~v) and ~! =~(v; :) (3)These relations are manifestly dual to (1). This suggests that TM can beinterpreted as the cotangent bundle of some dual manifold~M whose canonicalsymplectic form is~. We obtain~M by the action of the vector eld ~v ona hypersurface M0M such that !jM0= 0. Then clearly M0= M \~M and~!jM0= 0. The dual projection ~ maps (P 2M; 2 TPM) 2 TM into (P02M0; (v)!) 2~M , where P0lies on the same integral curve of v as P does.Every point (P;)2 TM can be understood as a 1-form on~M at the point~(P;) = (P0; (v)!). It associates , to any vector t0annihilated by ~!, thenumber (t0), and to ~v the numberRPP0!. If the integral curves of v are closedthen the function f(P ) =RPP0! is multivalued. In this case we have to identifyTM with T~M factorized by an appropriate discrete group.There remains to show that Eq. (2) holds. Because~ = d~ (and  = d)where  and~ are canonically dened, it is sucient to compare~ and .Every vector t2 TX(TM ) can be uniquely written as t = t0+ !(t)v + ~!(t)~vwhere !(t0) = ~!(t0) = 0. Then by denition (t0) =~(t0), (~v) = 0,~(v) = 0,(v) =RX~! and~(~v) =RX!. Hence~(t) = (t)  !(t)ZX~! + ~!(t)ZX!and Eq. (2) follows. As the formulae suggest, starting with the dual brationand repeating the procedure will bring us back to the original one.We shall argue that the framework described above is very well suited forthe description of the (Abelian) target space duality in string theory.2The form ! is locally the dierential of a coordinate along v.23 Strings embedded into TM and Buscher'sformulaSupposeM is a spacetime in which string propagation is governed by an actionS, invariant with respect to the vector eld v. There is no need to assume thatS is a -model action; we only suppose that S is local, reparametrization-invariant and depending on the rst derivatives of the embedding of the stringworldsheet into M .Because v is a symmetry, by Noether theorem there is a closed 1-form (the density of the v-component of the momentumof the string) on any surfaceextremizing S. Due to this fact, any on-shell string can be naturally lifted toTM along ~v so that the form ~! restricted to the lifted surface gives precisely.3Then we project the lifted surface to~M . We shall show that there existsa ~v-invariant action~S whose extremal surfaces are just the projections to~Mof the lifted surfaces. The action~S will obey the duality property, namely thelift of the dual extremal surface along v, such that the dual Noether form ~ isrestricted !, coincides with the original lift.All that picture should be rened, however, in the case of closed strings.Then the form  need not be exact and the lift of the non-contractible loop onthe worldsheet may give an open curve in TM . Therefore, we have to identifythe points of TM along ~v in such a way thatnIorbit of ~v~! =Iloop = p;where p is the v-component of the total momentum of the string and n is aninteger. As a consequence, the momentum of the string is an integer multipleof some minimal momentum and the dual string winds n times around theorbit of ~v. The picture holds in the dual version, of course.In order to nd the dual action, we have to identify a geometric object onM which encodes the original action, can be naturally lifted to TM , and thenprojected to the dual manifold~M . Because the action is reparametrization-invariant and depends at most on the rst derivatives, the Lagrangian L is afunction of decomposable bivectors b at any point P of M such thatL(b) = L(b);  > 0: (4)This object is not convenient for lifting to TM . Instead, we dene a bilinearmapping ppb(t; u) ddL(b+ t ^ u)j=0; (5)3This lift is dened uniquely up to a uniform shift by ~v. We shall discuss the ambiguityin the next section.3where t; u 2 TPM and t^ b = 0 (the last condition means that the vectort lies in the plane of b and in fact ensures that the argument of L in (5) isdecomposable). Note that pb(t; u) = pb(t; u) or in other words the mappingp depends only on the plane in which b lies. Physically speaking, pb(t; u) isthe density of the u-component of the momentum.4We see that the mapping pb(t; u) contains the essence of dynamical prop-erties of string. Obviously, pb(b) = L(b) and pb(t; v) = (t) where  is thementioned Noether 1-form.A bivector b = v1^v2at a point P 2M can be naturally lifted to a decom-posable bivector b22TP(TM )'2(TPM + TPM ):b= (v1+ pb(v1; :))^(v2+ pb(v2; :)): (6)We observe the simple formula2L(b) = (b): (7)The dual Lagrangian~L is dened in such a way that lifting~b = ~(b) by~Lgives b. Then obviously52~L(~b) =~(b): (8)From (2), (7) and (8) Buscher's duality transformation follows:~L(~b) = L(b)  (~! ^ !)(b) = L(b)  ( ^ !)(b): (9)By construction, the dual Lagrangian is ~v-invariant. In a way, it may be in-terpreted as the Routh function, because ! is the dierential of the coordinatealong the symmetry eld and  is the corresponding momentum. Later weshall write the formula in the familiar -model context. The formula can bederived also relaxing the condition of reparametrization invariance. However,its derivation is slightly less straightforward.We should demonstrate that we obtain, by minimizing the dual action,the surfaces in~M projected from the lifted extremal surfaces of the originalaction. First notice that for lifting a surface F M we only need d = 0. Upongauging the symmetry v the variation of the action S(F + v) S(F ) is equaltoRFd ^ . It means that d = 0 i F is extremal with respect to arbitrary(non-uniform) variations in the direction of v. By construction, the same thing4We may say that pb(t; u) denes for any embedding of string into M a 1-form on theworldsheet with values in 1-form on the target. In arbitrary coordinates on the worldsheetand Xon the target it can be written as p= @(L)=@(@X), where S =RLd0d1.5Speaking more exactly, the value of  is the same for any bivector on TM obtained byacting by the vector elds v; ~v on b, because the symplectic form  is v; ~v-invariant. Thismeans that we can transport binto a point of the dual manifold~M embedded in TM andwrite the formula (8) there.4is valid for dual objects, i.e. the liftable surfaces are exactly those extremizingS and~S with respect to the v- and ~v-direction respectively. Therefore we shallrestrict our attention to these surfaces only. Now from (9) one easily observes~S(~F ) = S(F ) ZF~! ^ !; (10)where Fis the common lift of FM and~F~M . The dierence between theactions is an integral of a closed form so it does not feel any variation.4 Duality as a symplectomorphism ofstring phase spaceBy the phase space of a closed string theory we understand the space of allclassical solutions having the topology of a cylinder. On this space there isa natural symplectic form Phcoming from the action. However, we had toquantize the momentum for the duality to make sense. If we x the momentumthen we obtain a hypersurface in the phase space. This is not a symplecticspace because Phis not invertible on it. To obtain a symplectic structureone has to factorize this hypersurface: one identies each string F with all thestrings obtained by tranlating F by the vector eld v. This is (the simplestcase of) the Marsden-Weinstein reduction [8]: if there is a function p on asymplectic space generating a vector eld w then one sets p = const: to obtaina hypersurface and then factorizes by w; the result is a symplectic space. Thisfactorisation is perfectly suited for the duality because the dual string~F isonly dened up to a shift by ~v and does not depend on shifting F by v. So,duality is a one-to-one mapping of the reductions. There is a pretty physicalreason for the factorization: if a string has an exact value of the momentumthen its state does not change if we shift it by v.Now we prove the following proposition: the duality is a symplectomor-phism of the (reduced) phase spaces.Proof: Let LM be the loop space of the target M . As usual, we obtain thephase space from the cotangent bundle TLM , on which there is the canonicalsymplectic form Ph= dPh. Namely, we identify some submanifold in TLMand then factorize it appropriately.6The construction goes as follows: if wehave a string worldsheet F and a loop l on it, then we dene a correspondingelement lF2 TlLM . To describe how lFacts on a vector u 2 TlLM , rstrealize that u can be thought of as a family of vectors u(X) 2 TXM where X6We proceed conceptually as in the case of a relativistic particle in a background; in the-model case the submanifold is dened by the Virasoro constraints.5runs along l. ThenlF(u) Ilpb(:; u(X)): (11)If we take all lF's for all possible F 's we obtain the mentioned submanifold ofTLM . Now we identify all lF's coming from the same extremal F and obtainthe phase space.In this framework we can easily prove the proposition. Let H be a surfacein the original phase space, i.e. a 2-parametric family of on-shell strings, andlet on each F 2 H be a loop l(F ). Then by (11)ZHPh=I@HPh=ISF2@Hl(F )pb(F )(:; :)The last integral is over a closed surface in M. We will prove that if~H is acorresponding family in~M (dened up to an independent shift of each~F by~v) thenISF2@Hl(F )p =IS~F2@~Hl(~F )~p:We will compare the two expressions using the common lifted family H. Oneimmediately checks that if tand uare vectors at a point of a lifted surfaceF, ttangent to Fand uarbitrary, then~p(~t; ~u)  p(t; u) = (! ^ ~!)(t^ u)so thatIS~F2@~Hl(~F )~p ISF2@Hl(F )p =ISF2@Hl(F)! ^ ~! = 0because ! ^ ~! is closed and the closed surface over which we integrate is aboundary.5 -model and projective transformationsIn what follows we shall study the duality in the familiar context of the non-linear -model. In this case there is a metricG and a 2-formB on the manifold.The action S is minus the area of the surface plus the integral of B over thesurface. We assume the surface to be time-like everywhere, i.e. there are tworeal light-like tangent vectors k; l at any point of the surface; we always chooseboth of them lying on the same light cone (future or past). Then the area ofk ^ l is simply  k  l, i.e.L(k ^ l) = G(k; l) +B(k; l)  E(k; l): (12)6It means thatpk^l(k; :) = E(k; :) and pk^l(l; :) =  E(:; l): (13)The duality transformation follows directly from Buscher's formula (9), butthere is a simpler geometric way of deriving it in the -model context. UsingEq. (13) and the decomposition TP(TM) ' TPM + TPM ' TP~M + TP~M weinterpret k= k + pk^l(k; :) = k + E(k; :) as~k +~E(~k; :), and accordingly forl, thus obtaining the dual bilinear form~E =~G +~B. That is, we interpretthe graph of E, E = ft+E(t; :)jt 2 TPMg, from the dual point of view as thegraph of~E.The projective formula for~E is self-evident now. One simply exchangesTPM and TP~M by the linear transformationR of TP(TM), v$ ~v and w 7! w,if !(w) = ~!(w) = 0. Then~t+~E(~t; :) = R(t+ E(t; :)).Our formalism can be easily extended to the case of d commuting symme-tries. Then R is an element of a group O(d; d;Z) preserving the natural metricon TP(TM), (t+ )2= (t).References[1] T.H. Buscher, Phys. Lett. B194 (1987) 51, B201 (1988) 466.[2] M.J. Du, Nucl. Phys. B335 (1990) 610.[3] M. Rocek and E. Verlinde, Nucl. Phys. B373 (1992) 630; A. Giveon andM. Rocek, Nucl. Phys. B380 (1992) 128; A. Giveon and E. Kiritsis, Nucl.Phys. B411 (1994) 487.[4] E. Kiritsis, Nucl. Phys. B405 (1993) 109.[5] E.Alvarez, L.Alvarez-Gaume, J. Barbon and Y. Lozano, Nucl. Phys.B415 (1994) 71.[6] A. Giveon, E. Rabinovici and G. Veneziano, Nucl. Phys. B322 (1989) 167;K.A. Meissner and G. Veneziano, Phys. Lett. B267 (1991) 33.[7] E.Alvarez, L.Alvarez-Gaume and Y. Lozano, A Canonical Approach toDuality Transformation, preprint CERN-TH-7337/94.[8] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag 1978.7

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