Infinite-Dimensional Symmetries of Two-Dimensional Coset Models
aa r X i v : . [ h e p - t h ] F e b Infinite-Dimensional Symmetries of Two-Dimensional CosetModels
H. L¨u, Malcolm J. Perry and C.N. Pope , George P. & Cynthia W. Mitchell Institute for Fundamental Physics,Texas A&M University, College Station, TX 77843-4242, USA. DAMTP, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, England.
DAMTP-2007-107 MIFP-07-28
ArXiv:0711.0400
Abstract
It has long been appreciated that the toroidal reduction of any gravity or supergrav-ity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensionalglobal symmetry. This symmetry is an extension of the finite-dimensional symmetry G in three dimensions, after performing a further circle reduction. There has not beenuniversal agreement as to exactly what the extended symmetry algebra is, with differentarguments seemingly concluding either that it is ˆ G , the affine Kac-Moody extension of G ,or else a subalgebra thereof. We take the very explicit approach of Schwarz as our start-ing point for studying the simpler situation of two-dimensional flat-space sigma models,which nonetheless capture all the essential details. We arrive at the conclusion that thefull symmetry is described by the Kac-Moody algebra ˆ G , whilst the subalgebra obtainedby Schwarz arises as a gauge-fixed truncation. We then consider the explicit exampleof the SL (2 , R ) /O (2) coset, and relate Schwarz’s approach to an earlier discussion thatgoes back to the work of Geroch. ontents G symmetry . . . . . . . . . . . 62.3 Virasoro-like symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 SL (2 , R ) /O (2) Coset Model 19 δ and some example δ transformations . . . . . . . . . 26 ˆ G H Versus Kac-Moody Algebra ˆ G
29B The Virasoro-type Symmetry and the Schwarz Approach 33
The study of supergravity theories, and their symmetries, have played a very important rˆolein uncovering the underlying structures of string theory. Especially significant are the U-duality symmetries of the string, which have their origin in the classical global symmetriesexhibited by eleven-dimensional supergravity and type IIA and IIB supergravities aftertoroidal dimensional reduction. For example, if one reduces eleven-dimensional supergravityon an n -torus, for n ≤
8, the resulting D = (11 − n )-dimensional theory exhibits a global1 n symmetry [1–3]. In the cases n ≥ E symmetry that one finds after reduction to three dimensions,it is natural to push further and investigate the symmetries after further reduction to twodimensions, and even beyond. It turns out that the analysis of the global symmetry for areduction to two dimensions is considerably more complicated than the higher-dimensionalones. There are two striking new features that lead to this complexity. The first is that,unlike a reduction to D ≥ L ∼ √− gR , with no scalar conformal factor.) The inability toreach the Einstein conformal frame in two dimensions is intimately connected to the factthat √− gR is a conformal invariant in two dimensions. It has the consequence that themetric in two dimensions is not invariant under the global symmetries.The second striking new feature is that an axionic scalar field ( i.e. a scalar appearingeverywhere covered by a derivative) can be dualised to give another axionic scalar field inthe special case of two dimensions. This has the remarkable consequence that the globalsymmetry group actually becomes infinite in dimension. This was seen long ago by Geroch,in the context of four-dimensional gravity reduced to two. There are degrees of freedom intwo dimensions that are described by the sigma model SL (2 , R ) /O (2), and under dualisationthis yields another SL (2 , R ) /O (2) sigma model. Geroch showed that the two associatedglobal SL (2 , R ) symmetries do not commute, and that if one takes repeated commutatorsof the two sets of transformations, an infinite-dimensional algebra results [4]. The precisenature of this symmetry, now known as the Geroch Group , was not uncovered in [4].The feature of having an infinite-dimensional symmetry in two dimensions is not re-stricted to situations where gravity is involved, and in fact the same essential mechanismoperates in a similar fashion if one considers a sigma model in a flat two-dimensional space-time. Thus, a natural preliminary to investigating the symmetries of two-dimensional re-ductions in supergravity is to study the symmetry of a flat two-dimensional sigma model G / H . Considerable simplifications arise if one restricts attention to symmetric-space sigmamodels, and since these in any case always arise in supergravity dimensional reductions, thespecialisation to this class of models is a very natural one. We shall use the acronym SSMto denote a symmetric-space sigma model.There is quite a considerable literature on the subject of the infinite dimensional sym-metries of two-dimensional symmetric-space sigma models, both in the flat and the curvedspacetime cases (see, for example, [5–15], some of which considers also principal chiralmodels). A very clear and explicit presentation of the global symmetry algebras of two-2imensional SSMs has been provided by Schwarz, whose papers formulate the problem ina very transparent way. He first considers the problem of two-dimensional theories in flatspacetime in [16], and then generalises to the case of a curved two-dimensional spacetimein [17]. He also gives an extended history of the earlier literature, and rather than attempt-ing to repeat that here, we refer the reader to his papers for further details.Our work in the present paper is concerned entirely with the case of symmetric-spacesigma models in flat two-dimensional spacetime, and we follow very closely the approachtaken by Schwarz in [16]. The results in [16] differ somewhat from those in much of theliterature, where the infinite-dimensional global symmetry algebra of the SSM G / H is foundto be ˆ G , the affine Kac-Moody extension of the underlying algebra G of the “manifest” globalsymmetry group G . The generators of ˆ G may be represented by J in , satisfying[ J im , J jn ] = f ijk J km + n , (1.1)where f ijk are the structure constants for the Lie algebra G , whose generators T i satisfy[ T i , T j ] = f ijk T k .By contrast, Schwarz obtained a certain subalgebra ˆ G H of ˆ G as the global symmetryalgebra, essentially generated by J ′ in = J in ± J i − n , where the + sign is chosen if i lies in thedenominator algebra H , and the − sign if i lies in the coset K = G / H .We find that by extending the techniques developed by Schwarz, we can construct ex-plicit global symmetries for the entire ˆ G Kac-Moody algebra, expressed purely in termsof local field transformations. As far as we are aware, it is only through the use of theconstruction that Schwarz developed that it has become possible to obtain explicit localtransformations for the entire Kac-Moody algebra. We find also that the subalgebra ob-tained by Schwarz can be viewed as a gauge-fixed version of this full Kac-Moody symmetry.In order to understand this, we recall that the possibility of dualising axions to give newaxions in two dimensions means that the original theory can be reformulated in terms ofnew fields that are non-locally related to the original ones (since the process of dualisationrequires differentiation and Hodge dualisation, followed by integration, to obtain the newvariables). A convenient way to handle this is to enlarge the system by introducing auxiliaryfields, so that the manifest global symmetries of the original and the dualised sigma modelscan be exhibited simultaneously, in purely local terms. In fact to do this, one has tointroduce an infinite number of auxiliary fields. The full set of Kac-Moody symmetries,generated by J in with −∞ ≤ n ≤ ∞ , acts on the complete set of original plus auxiliaryfields. However, the “negative half” of the Kac-Moody algebra ˆ G , generated by J in with n <
0, acts exclusively on the auxiliary fields, whilst leaving the original sigma-model fieldsinert. In fact these symmetries are essentially constant shift transformations of the auxiliary3elds, reflecting the arbitrariness of the choice of constants of integration that arose whenthe non-local dualisation was recast into a local form in terms of the auxiliary fields.The subalgebra ˆ G H of symmetries found by Schwarz can be viewed as a gauge fixing inwhich the values of the original and the auxiliary fields are all set to prescribed values atsome chosen point in the two-dimensional spacetime. Effectively, the level-0 transformations J i that lie in K are used up in gauge fixing the original fields to their prescribed values,and the entirety of the J in transformations with n < SL (2 , R ) /O (2).We show how one needs to introduce an infinity of auxiliary fields in order to describesimultaneously the original SL (2 , R ) symmetry and the SL (2 , R ) symmetry of the dualisedversion (we denote this by SL (2 , R )). We also show how each generator of each copyof SL (2 , R ) can be precisely matched with a corresponding generator in the Kac-Moodyalgebra, and this allows us to show explicitly that the Geroch algebra generated by takingmultiple commutators of SL (2 , R ) and SL (2 , R ) transformations is exactly the same as thefull Kac-Moody algebra \ SL (2 , R ).We also examine a further symmetry of two-dimensional symmetric-space sigma models G / H , again basing our analysis on the work of Schwarz [16]. This is again an infinite-dimensional symmetry, but this time a singlet under the original G symmetry. It turns outto be related to the centreless Virasoro algebra. We shall begin by considering an arbitrary symmetric-space sigma model (SSM) in a flattwo-dimensional spacetime background, with coset given by K = G / H , where G is a Liegroup with subgroup H . The commutation relations for the corresponding generators of4he algebra take the form[ H , H ] = H , [ H , K ] = K , [ K , K ] = H . (2.1)The condition that K is a symmetric space is reflected in the absence of K generators onthe right-hand side of the last commutation relation. The symmetric-space algebra impliesthat there is an involution ♯ under which K ♯ = K , H ♯ = −H . (2.2)In many cases, such as when G = SL ( n, R ) /O ( n ), the involution map is given by Hermiteanconjugation, K † = K , H † = −H , (2.3)and later, we shall typically write formulae under this assumption. In some cases, such as G = E (8 , , H = O (16), the involution ♯ is more involved.Let V be a coset representative in K . We may then define M = V ♯ V , A = M − dM . (2.4)Under transformations V −→ h V g , (2.5)where g is a global element in the group G and h is a local element in the denominatorsubgroup H , we have shall have M −→ g ♯ M g , A −→ g − Ag , (2.6)since it follows from H ♯ = −H that h ♯ = h − .The Cartan-Maurer equation d ( M − dM ) = − ( M − dM ) ∧ ( M − dM ) implies that thefield strength for A vanishes: F ≡ dA + A ∧ A = 0 . (2.7)The Lagrangian for the coset model may be written as L = − tr( ∗ A ∧ A ) (or, using indices, L = − η µν tr( A µ A ν )) and hence the equation of motion is d ∗ A = 0 . (2.8)The Lagrangian is clearly invariant under the global G transformations, and the equations(2.7) and (2.8) transform covariantly under G .As discussed in [16], the equations (2.7) and (2.8) can both be derived from the integra-bility condition for the Lax Pair of linear equations (cid:16) ∂ + + tt − A + (cid:17) X = 0 , (cid:16) ∂ − + tt + 1 A − (cid:17) X = 0 , (2.9)5o admit a solution X ( x ; t ), where t is an arbitrary constant spectral parameter . Theseequations are written in light-cone coordinates on the two-dimensional flat spacetime, inwhich the metric is ds = 2 dx + dx − . We prefer to use the language of differential forms,for which A = A + dx + + A − dx − . On 1-forms we have ∗ = +1, where ∗ is the Hodge dualoperator, and ∗ dx ± = ± dx ± , and so ∗ A = A + dx + − A − dx − . (2.10)It is useful also to record the following properties for 1-forms u and v : ∗ u ∧ v = ∗ v ∧ u , ∗ u ∧ ∗ v = − u ∧ v , (2.11)and for Lie-algebra valued 1-forms A and B : ∗ A ∧ B = − A ∧ ∗ B , ∗ A ∧ ∗ A = − A ∧ A . (2.12)In terms of differential forms, the Lax pair (2.9) becomes simply the single equation t ( d + A ) X = ∗ dX . (2.13)We shall call this the Lax Equation . By taking the appropriate linear combination of thisand its dual, we obtain dXX − = t − t ∗ A + t − t A . (2.14)Thus the integrability condition for the existence of a solution X ( x ; t ) to the Lax equation,which follows from the Cartan-Maurer equation d ( dXX − ) = ( dXX − ) ∧ ( dXX − ), gives d ∗ A + t ( dA + A ∧ A ) = 0 . (2.15)Since this must hold for all t we indeed derive (2.7) and (2.8). Note that (2.14) is anequivalent formulation of the Lax equation; an appropriate linear combination of (2.14) andits dual gives back (2.13). Thus we may use the term “Lax equation” interchangeably for(2.13) and (2.14). We have already noted that the global G transformations (2.6) are a symmetry of the zero-curvature condition (2.7) and the equations of motion (2.8) of the two-dimensional cosetmodel. In fact, these symmetries are merely the tip of an infinite-dimensional “iceberg”of global symmetries. These extended symmetries are a special feature that arises because6he coset model lives in a two-dimensional world volume, and they may be understood in avariety of ways. An intuitive understanding, which we shall turn into a concrete discussionin section 4 for the example of the coset SL (2 , R ) /O (2), is that the axionic scalars canbe dualised into new, non-locally related sets of axions in two dimensions, and that themanifest global symmetries in the different duality pictures do not commute, but insteadtheir commutators close only on an infinite-dimensional extension of the finite-dimensionalsymmetries that are manifest in each individual duality choice. In the present section, we shall begin by following a construction given in [16], whichshows how the formalism of the Lax equation may be used to derive the infinite-dimensionalalgebra. Our description will be formulated in the language of differential forms rather thanlight-cone coordinates. The details of our calculation differ somewhat from those in [16], andour conclusions differ also. Specifically, we find that the full symmetry of the symmetric-space sigma model is precisely the affine Kac-Moody extension ˆ G of the manifest G globalsymmetry, and not merely the subalgebra of ˆ G that was found in [16]. (We shall commentfurther about this later in this subsection, and in appendix A.)At the infinitesimal level, the transformation (2.5) becomes δ V = V ǫ + δh V , (2.16)where ǫ is an infinitesimal global element of the Lie algebra G and δh is a local elementof H . In order to exhibit the infinite-dimensional extension of this symmetry algebra, onemay consider more general transformations of the form [16] δ V = V η + δh V , where η = X ( t ) ǫX ( t ) − . (2.17)The meaning of this equation is as follows. As before, V is a coset representative for G / H ,and thus it depends on the scalar fields parameterising the coset, which themselves dependon the two spacetime coordinates x , but it does not depend on the spectral parameter t .The function X ( t ) is the solution of the Lax equation (2.13) and thus it depends on thespacetime coordinates x (we are now suppressing the explicit indication of this dependence)and on the spectral parameter t . The quantity δh , in the denominator algebra H , is afunction of the spacetime fields and it may now depend upon t . On the left-hand side of(2.17) there is t -dependence only in the variational symbol δ itself, and it is to be interpretedas δ = δ ( ǫ, t ) = X n ≥ t n δ ( n ) ( ǫ ) . (2.18)Thus by equating powers of t on the two sides of (2.17) we obtain a hierarchy of transfor-mations δ ( n ) that act upon the scalar fields in the coset representative V . The lowest set of This idea dates back to a paper on four-dimensional gravity reduced to two dimensions, by Geroch [4],although at that time the precise nature of the infinite-dimensional algebra was not addressed. i.e. for n = 0, just correspond to the original infinitesimal G transforma-tions that were manifest in the coset model from the outset. By contrast the transformations δ ( n ) with n >
0, which all involve t -dependent terms in X ( t ), are non-local expressions interms of the original fields of the scalar coset. To check that (2.17) does indeed give symmetries of the theory, one must check thatthe corresponding variation of the equation of motion (2.8) vanishes. First, one sees from M = V † V and A = M − dM that (2.17) implies δM = M η + η † M , δA = Dη + D ( M − η † M ) , (2.19)where the G -covariant exterior derivative is defined on any G -valued function f by Df = df + [ A, f ] . (2.20)It can also be seen from the definition of η in (2.17), after making use of the Lax equation(2.13), that Dη = 1 t ∗ dη , D ( M − η † M ) = t ∗ d ( M − η † M ) . (2.21)Thus we conclude that under (2.17), δA = ∗ d (cid:16) t η + t M − η † M (cid:17) , (2.22)which indeed verifies that d ∗ δA = 0.In order to read off the symmetry algebra one needs to calculate commutators of theform [ δ ( m ) , δ ( n ) ]. Since, as noted above, the variations δ ( n ) involve X ( t ), which itself dependsnon-locally on the fields of the scalar coset, one first needs to calculate the variations of X ( t ) with respect to the hierarchy of transformations δ ( n ) . This was obtained in [16], andwith a small but important modification that we shall discuss later, it is given by δ X = t t − t ( η X − X ǫ ) + t t − t t M − η † M X . (2.23)Here δ (with no parentheses around the 1) denotes δ ( ǫ , t ) = P n ≥ t n δ ( n ) ( ǫ ), whilst X denotes X ( t ) for a different and independent choice of spectral parameter t . By equating Note, however, that all the transformations become local if one introduces an infinite set of auxiliaryfields, as we shall do later. From this point onwards, we shall assume for simplicity, and to make the expressions look more palatable,that the involution of the symmetric space algebra is implemented by Hermitean conjugation, as in (2.3).In a case where the more general ♯ involution operator is required, all † symbols in what follows should bereplaced by ♯ . t m t n on both sides of (2.23), one can read off the variation under δ ( m ) ofthe t n term in the series expansion of X ( t ).In order to derive (2.23), we follow the method used in [16], which amounts to varyingthe Lax equation (2.13) under (2.17), with δA given in (2.19) and δX given by (2.23), andverifying that the varied equation is also satisfied. Thus, one must substitute (2.23) into[ t ( d + A ) − ∗ d ]( δ X ) + t ( δ A ) X = 0 , (2.24)or in other words, after using (2.19), into[ t ( d + A ) − ∗ d ]( δ X ) + t [ Dη + D ( M − η † M )] X = 0 . (2.25)After some algebra, again involving the use of the Lax equation, the desired result follows.Using (2.23) one can calculate the commutator of transformations on M = V † V , finding(in a similar manner to [16]) that[ δ , δ ] M = t δ ( ǫ , t ) − t δ ( ǫ , t ) t − t M , (2.26)where ǫ = [ ǫ , ǫ ]. It is also straightforward to show, after some lengthy algebra, that[ δ , δ ] X = t δ ( ǫ , t ) − t δ ( ǫ , t ) t − t X , (2.27)If the transformations δ given in (2.17) and (2.23) were the only ones extending G thenwe would have essentially “half” of the affine Kac-Moody extension ˆ G . However, there areadditional transformations, which we shall denote by ˜ δ , that also extend G . These leave M invariant but they do act non-trivially on X . They are given by˜ δ M = 0 , ˜ δ X = t t − t t X ǫ . (2.28)Again, the notation here is that ˜ δ = ˜ δ ( ǫ , t ) = P n ≥ t n ˜ δ ( n ) , and X = X ( t ). (Note thatthere is no n = 0 term here in the expansion of ˜ δ , as can be seen from the absence of a t term on the right-hand side of (2.28).) It is easy to verify that (2.28) describes symmetriesof the Lax equation. The easiest way to do this is to note that (2.28) implies ˜ δ ( dXX − ) = 0,and so since ˜ δA = 0, it is evident that the Lax equation (2.14) is indeed invariant under ˜ δ .The commutators of the ˜ δ transformations give[˜ δ , ˜ δ ] X = t t − t ˜ δ ( ǫ , t ) X − t t − t ˜ δ ( ǫ , t ) X , (2.29)9here again, ǫ = [ ǫ , ǫ ]. (This commutation relation is vacuous, of course, when actingon M .) Finally, we may calculate the commutators of δ and ˜ δ transformations, finding[ δ , ˜ δ ] X = t t − t t δ ( ǫ , t ) X + 11 − t t ˜ δ ( ǫ , t ) X . (2.30)(The commutator on M is the same, except that there is no ˜ δ term on the right-hand sidesince ˜ δM = 0.)In summary, we therefore have in total the commutation relations[ δ , δ ] = t t − t δ ( ǫ , t ) − t t − t δ ( ǫ , t ) , (2.31)[ δ , ˜ δ ] = t t − t t δ ( ǫ , t ) + 11 − t t ˜ δ ( ǫ , t ) , (2.32)[˜ δ , ˜ δ ] = t t − t ˜ δ ( ǫ , t ) − t t − t ˜ δ ( ǫ , t ) . (2.33)From these, one can read off the towers of modes in the t -expansions, using δ ( ǫ, t ) = P n t n δ n ( ǫ ), etc. For example, (2.31) gives X m ≥ X n ≥ t m t n [ δ ( m ) ( ǫ ) , δ ( n ) ( ǫ )] = 1 t − t X p ≥ ( t p +11 − t p +12 ) δ ( p ) ( ǫ ) , = X p ≥ p X q =0 t q t p − q δ ( p ) ( ǫ ) , = X m ≥ X n ≥ t m t n δ ( m + n ) ( ǫ ) , (2.34)whence we obtain [ δ ( m ) ( ǫ ) , δ ( n ) ( ǫ )] = δ ( m + n ) ( ǫ ) , m ≥ , n ≥ . (2.35)The analogous calculations for (2.32) and (2.33) give[ δ ( m ) ( ǫ ) , ˜ δ ( n ) ( ǫ )] = δ ( m − n ) ( ǫ ) + ˜ δ ( n − m ) ( ǫ ) , m ≥ , n ≥ , (2.36)[˜ δ ( m ) ( ǫ ) , ˜ δ ( n ) ( ǫ )] = ˜ δ ( m + n ) ( ǫ ) , m ≥ , n ≥ , (2.37)where in (2.36) it is to be understood that δ ( n ) = 0 for n ≤ − δ ( n ) = 0 for n ≤ ( n ) of variations, defined for all n with −∞ ≤ n ≤ ∞ , according to∆ ( n ) = δ ( n ) , n ≥ , ∆ ( − n ) = ˜ δ ( n ) , n ≥ . (2.38)10t is then easily seen that (2.35), (2.36) and (2.37) become[∆ ( m ) ( ǫ ) , ∆ ( n ) ( ǫ )] = ∆ ( m + n ) ( ǫ ) , m, n ∈ Z , (2.39)with ǫ = [ ǫ , ǫ ]. This defines the affine Kac-Moody algebra ˆ G . In terms of currents J i ( σ )defined on a circle, with J i ( σ ) = ∞ X n = −∞ e inσ J in , (2.40)the commutation relations (2.39) are equivalent to[ J im , J jn ] = f ijk J km + n , (2.41)where f ijk are the structure constants of the Lie algebra G . Specifically, we have theassociation ∆ ( n ) ( ǫ i ) ↔ J in , (2.42)where ǫ = ǫ i T i , and T i are the generators of the Lie algebra G .Since we have arrived at a somewhat different conclusion from Schwarz, who finds onlya subalgebra of the Kac-Moody algebra ˆ G as a symmetry of the SSM [16], we shall discussin appendix A exactly why the difference has arisen. In essence, the key distinction is thatwe include the transformations ˜ δ defined in (2.28) as independent symmetries. They arenon-trivial symmetries of the Lax equation, even though they act trivially on the scalarfields in the coset representative V itself. In section 4, we shall study the explicit exampleof the SL (2 , R ) /O (2) coset model, in order to illustrate this point in greater detail. Weshall show that a natural formulation of the model involves introducing an infinite numberof additional scalar fields, in terms of which X appearing in the Lax equation (2.13) can beexpressed as a local quantity. The ˜ δ transformations act on this infinite tower of additionalfields. We shall also show how this infinity of extra scalars can be interpreted as fieldsthat one introduces in order to exhibit in a local fashion the symmetries arising from theclosure of the two non-commuting SL (2 , R ) symmetries of the original theory and a dualisedversion.A further remark about the Kac-Moody transformations δ and ˜ δ is also in order. The ˜ δ transformation defined in (2.28) are of the general form ˜ δX ∼ Xǫ . It can be seen that thesecond of the three terms on the right-hand side of the δX transformation given in (2.23)is also of this general form. This means that as far as obtaining symmetries of the Laxequation is concerned, one could have omitted the second term in (2.23) altogether, sinceit is itself a distinct symmetry in its own right. However, it actually serves an importantpurpose in (2.23), namely to subtract out what would otherwise be a pole at t = t ifone had only t η X / ( t − t ) rather than t ( η X − X ǫ ) / ( t − t ). (The third term is in112.23) is necessary in addition, in order to get a symmetry, but there is no pole associatedwith this term, since we expand t and t around zero.) Now, the derivations of the δX and ˜ δX transformations as symmetries involved considering the variation of ( dXX − ) inthe Lax equation (2.14). In the case of the ˜ δ transformation we have ˜ δA = 0, and onemay view ˜ δX as the solution of the homogeneous equation ˜ δ ( dXX − ) = 0, whilst δX is the solution of the inhomogeneous equation δ ( dXX − ) = (non-zero source). Thus theinclusion of a ˜ δX contribution as the second term in (2.23) can be viewed as the necessaryaddition of a solution of the homogeneous solution that is needed in order to ensure thatthe inhomogeneous solution satisfies the necessary boundary condition ( i.e. that δ X beregular at t = t ).This discussion also emphasises the point that it is really the δ transformations found bySchwarz, appearing in our slightly modified form in (2.23), that lie at the heart of the Kac-Moody symmetries of the symmetric-space sigma models. The ˜ δ transformations, althoughthey are of course equally necessary in order to obtain the complete Kac-Moody symmetry,are somewhat secondary in nature since they are already present within the construction ofthe δ transformations.It is also worth remarking that we have obtained the full Kac-Moody algebra as asymmetry of the SSM by means of a purely perturbative analysis, which involved a small- t expansion of X ( t ) around t = 0. One may also consider instead a large- t expansion of X ( t ), around t = ∞ . The result is in fact equivalent. This can be seen by letting t = ˜ t − ,whereupon the Lax equation (2.14) becomes dXX − = − ˜ t − ˜ t ∗ A − − ˜ t A . (2.43)If we let X = M − ( e X − ) † , we arrive at a Lax equation that is identical in form to theoriginal expression (2.14), namely d e X e X − = ˜ t − ˜ t ∗ A + ˜ t − ˜ t A , (2.44)showing that the large- t expansion is equivalent to the small-˜ t expansion. One would there-fore reach identical conclusions had one performed a large- t expansion instead of a small- t expansion. It would be interesting to study the regime where the small- t and large- t expan-sions overlap. Although the Lax equation is regular in both regions, it becomes singular at t = ±
1. Even if such a non-perturbative analysis could be performed, we would not neces-sarily expect to find a larger symmetry algebra than the full Kac-Moody algebra, which isalready found in our perturbative approach.12 .3 Virasoro-like symmetry
The symmetry discussed in section 2.2 is an infinite-dimensional extension of the manifest G symmetry of the G / H symmetric-space sigma model. As such, the transformation pa-rameters ǫ in (2.17) are themselves G valued. There is an additional infinite-dimensionalsymmetry of the SSM, with transformation parameters that are singlets under G , whichturns out to be a subalgebra of the Virasoro algebra. Our discussion here again beginsby using an approach that is very close to that of Schwarz [16], although with certainmodifications and elaborations.The transformations in question act on the coset representative V as follows [16]: δ V ( t ) V = V ξ , where ξ = − t ˙ X ( t ) X ( t ) − . (2.45)By equating the coefficients of each power of t in (2.45), one obtains an infinite set oftransformations δ V ( n ) of the scalar fields in the SSM, with δ V ( t ) = X n ≥ t n δ V ( n ) . (2.46)Note that it is because of the explicit t factor in the definition of ξ in (2.45) that the sumin (2.46) does not include n = 0.To see that (2.45) indeed describes symmetries of the theory, one must show that theequation of motion d ∗ A = 0 is preserved. It follows from (2.45) that δ V A = Dξ + M − dξ † M = Dξ + D ( M − ξ † M ) , (2.47)where as usual Dξ = dξ + [ A, ξ ]. Differentiating the Lax equation (2.13) with respect to t , and subtracting the Lax equation premultiplied by ( ˙ XX − ) and postmultiplied by X − ,one finds that D ( ˙ XX − ) = 1 t h ∗ d ( ˙ XX − ) − − t A − t − t ∗ A i , (2.48) We should really include an infinitesimal parameter as a prefactor in the definition of ξ in equation(2.45). However, since it is a singlet it plays no significant rˆole, and so it may be omitted without any riskof ambiguity. Our transformations (2.45) differ slightly from those given in [16], in which the lowest-order term issubtracted out and the overall t -dependent factor is different. Our choice for the explicit t -dependent factoris made so that the algebra takes the simplest possible form. The subtraction was shown to be necessary inthe context of principal chiral models in [16], and was carried over into the discussion of the SSM case inthat paper. In fact, the subtraction becomes optional in the SSM case, which amounts to saying that theSSM has an additional mode in the symmetry transformation. We shall discuss this in further in appendixB. D ( M − ( ˙ XX − ) † M ) = t ∗ d ( M − ( ˙ XX − ) † M ) + 11 − t ∗ A + t − t A . (2.49)Substituting into (2.47), we find δ V A = ∗ d (cid:18) t ξ + tM − ξ † M (cid:19) + A , (2.50)and from this is follows that d ∗ δ V A = 0, thus proving that δ V is a symmetry of the equationsof motion. The next step is to calculate the commutator of the δ V transformations, in order todetermine their algebra. As a preliminary, we need an expression for δ V X . Guided by thediscussion in [16], we find that it is given by δ V X = Y X , Y = 1 t − t h t ξ + t ( t − − t t ξ i + t t − t t M − ξ † M . (2.51)The verification that (2.51) is correct is achieved by substituting (2.47) and (2.51) into theLax equation (2.14).After lengthy calculations of the commutators [ δ V , δ V ] M and [ δ V , δ V ] X , we find that[ δ V , δ V ] = − t t (cid:20) t − t ) + 1(1 − t t ) (cid:21) δ V + t t (1 − t )( t − t )(1 − t t ) ˙ δ V − [1 ↔ , (2.52)where ˙ δ V denotes the derivative of δ V with respect to its argument t , and the symbol[1 ↔
2] indicates the subtraction of two terms obtained from those that are displayed byexchanging the 1 and 2 subscripts everywhere.To derive the mode algebra, we substitute the mode expansion (2.46) into (2.52), andcollect terms associated with each power of t and t . We then find that the abstract algebraof the δ V transformations is given by[ δ V ( m ) , δ V ( n ) ] = ( m − n ) δ V ( m + n ) − ( m + n ) δ V ( m − n ) , (2.53)where it is understood that δ V ( n ) with negative mode numbers n is defined to be given by δ V ( − n ) ≡ − δ V ( n ) , n ≥ . (2.54) It is because of the cancellation in (2.50) of the contributions proportional to ∗ A coming from the twoterms in (2.47) that there is no need to make the lowest-order subtraction that was found in [16] to benecessary in the PCM case. t = t in (2.52)would have presented difficulties in interpreting the algebra, but in fact one finds thatcancellations imply there are no such poles. One way to make this manifest is to note that(2.52) can be rewritten as[ δ V , δ V ] = − t t ( δ V − δ V )(1 − t t ) − t t ( t − t )1 − t t ∂ ∂t ∂t h t ( t − t − ) δ V − t ( t − t − ) δ V t − t i . (2.55)We may define a current K ( σ ) in which we associate the mode K n with the symmetrytransformation δ V ( n ) : K ( σ ) = ∞ X n = −∞ e inσ K n . (2.56)The reflection condition (2.54) implies that the modes K n satisfy K n = − K − n , and from(2.53), they satisfy the algebra[ K m , K n ] = ( m − n ) K m + n − ( m + n ) K m − n , (2.57)This is clearly not the Virasoro algebra, but it is closely related to it. Specifically, if weintroduce generators L m for a centreless Virasoro algebra,[ L m , L n ] = ( m − n ) L m + n , (2.58)then we find that the modes K m may be represented as K m = L m − L − m , m = 0 (2.59)(Recall that that (2.45) contains no δ V (0) transformation, and so K is not present in thealgebra.) If we define the usual Virasoro current T ( σ ) = ∞ X m = −∞ L m e imσ , (2.60)then it follows from (2.56) and (2.59) that K ( σ ) = 2 i ℑ ( T ( σ )) . (2.61)It is interesting to contrast this result with the analogous one that was obtained in [16]for the case of a principal chiral model, where it was shown that K m = L m +1 − L m − andhence K ( σ ) = − i sin σ T ( σ ). In that case, one could view this relation as a definition ofthe energy-momentum tensor T ( σ ) in terms of K ( σ ), save for the degenerate points σ = 015nd σ = π at the ends of the line segment. By contrast, the relation (2.61) for the SSMcannot be used to define the whole of T ( σ ), but only its imaginary part. Thus the Virasoroalgebra itself is not described by the symmetry transformations (2.45).We may also calculate the commutators of the Virasoro-like transformations δ V with theKac-Moody transformations δ and ˜ δ of section 2.2. These commutators must be evaluatedon X , and not merely on M , in order to capture the resulting terms that correspond to ˜ δ transformations, since M is inert under these.By calculating the commutator [ δ V , ˜ δ ] acting on M and on X , we find that[ δ V , ˜ δ ] = t t (1 − t t ) δ ( t , ǫ ) + t t (cid:20) − t t ) + 1( t − t ) (cid:21) ˜ δ ( t , ǫ ) − t t ( t − t ) ˜ δ ( t , ǫ ) − t t ( t − t − t )(1 − t t ) ˙˜ δ ( t , ǫ ) , (2.62)where ˙˜ δ ( t , ǫ ) denotes the derivative of ˜ δ ( t , ǫ ) with respect to t .Similarly, calculating the commutator [ δ V , δ ] acting on M and on X , we find[ δ V , δ ] = t t (1 − t t ) ˜ δ ( t , ǫ ) + t t (cid:20) − t t ) + 1( t − t ) (cid:21) δ ( t , ǫ ) − t t ( t − t ) δ ( t , ǫ ) − t t ( t − t − t )(1 − t t ) ˙ δ ( t , ǫ ) , (2.63)As in the case of (2.52), although there are ostensibly poles in (2.62) and (2.63) at t = t ,these in fact cancel. Expanding in powers of t and t , and making use of the definition(2.38) for the full set of Kac-Moody transformations ∆ m , we find that[ δ V ( m ) , ∆ ( n ) ] = − n (∆ ( n + m ) − ∆ ( n − m ) ) . (2.64)In terms of the Kac-Moody current-algebra modes J in and Virasoro-like modes K n that weintroduced earlier, we therefore find[ K m , J in ] = − n ( J in + m − J in − m ) . (2.65)One may verify that this is consistent with the Jacobi identity [ K m , [ K n , J ip ]] + · · · = 0, afterusing our result (2.57) for the commutator [ K m , K n ]. A slightly different approach to describing the symmetries of two-dimensional symmetric-space coset models was taken in [18], and it is useful to summarise some salient aspects16ere, since we shall make use of some of the formalism in section 4. It is again an approachwhere the SSM is viewed as an integrable system, and it is essentially equivalent to thedescription in [16] in terms of the Lax equation.Starting from the coset representative V that we introduced previously, one may define d VV − = Q + P , (3.1)where Q is the projection into the denominator algebra H and P is the projection into thecoset algebra K . From the Cartan-Maurer equation d ( d VV − ) = ( d VV − ) ∧ ( d VV − ), onecan then read off the equations dQ − Q ∧ Q − P ∧ P = 0 , (3.2) DP ≡ dP − Q ∧ P − P ∧ Q = 0 . (3.3)Under the transformations (2.5) one has Q −→ hQh − + dhh − , P −→ hP h − , (3.4)which shows that D = d − Q ∧ − ∧ Q can be viewed as an H -covariant connection. P trans-forms covariantly under H and is invariant under the global right-acting G transformations.From (2.4), and making the convenient assumption again that the involution ♯ is imple-mented by Hermitean conjugation, we see that with M = V † V A = M − dM = V − (cid:16) d VV − + ( d VV − ) † (cid:17) V = V − ( Q + P + Q † + P † ) V = 2 V − P V , (3.5)since under the involution we shall have Q † = − Q , P † = P . It follows from (2.14) that V dXX − V − = 2 t − t ∗ P + 2 t − t P , = 2 t − t ∗ P + 1 + t − t P − P , = 2 t − t ∗ P + 1 + t − t P + Q − d VV − , (3.6)and hence d ˆ V ( t ) ˆ V ( t ) − = Q + 2 t − t ∗ P + 1 + t − t P , (3.7)where we define ˆ V ( t ) ≡ V X ( t ) . (3.8)17he Kac-Moody transformations δ and ˜ δ , which we defined in (2.17), (2.23) and (2.28),can now be applied to ˆ V . We find δ ˆ V = t t − t ˆ V X − η X − t t − t ˆ V ǫ + t t − t t ˆ V ( M X ) − η † M X + δh ˆ V , (3.9)˜ δ ˆ V = t t − t t ˆ V ǫ , (3.10)where as usual η = X ǫ X − , δh is an H compensating transformation and ˆ V = V X .The quantity A = M − dM can be thought of as a G -valued conserved current, sinceas we noted in section (2.1), it transforms under global G transformations V → h V g as A → g − Ag , and it satisfies d ∗ A = 0. We see from (3.5) that A = 2 V − P V . One canconstruct a hierarchy of conserved currents ˆ J ( t ), for which ˆ J (0) = A , by definingˆ J ( t ) = 21 − t ˆ V − (cid:18) t − t P + 2 t − t ∗ P (cid:19) ˆ V . (3.11)That ˆ J is conserved can be seen from the following calculation, which also provides [18] asimpler expression for the currents:ˆ J = 21 − t ˆ V − ∗ (cid:18) t − t P + 1 + t − t ∗ P (cid:19) ˆ V , = ˆ V − ∗ ∂∂t (cid:18) t − t P + 2 t − t ∗ P (cid:19) ˆ V , = ˆ V − ∗ ∂∂t (cid:18) Q + 1 + t − t P + 2 t − t ∗ P (cid:19) ˆ V , = ˆ V − ∗ ∂∂t (cid:16) d ˆ V ˆ V − (cid:17) ˆ V , = ∗ d ˆ V − ∂ ˆ V ∂t ! . (3.12)Note that using (3.8), we can also write ˆ J asˆ J = ∗ d ( X − ˙ X ) . (3.13)It is also useful to define the quantity v ( t ) = X − ( t ) ˙ X ( t ) = X n ≥ t n v ( n ) , (3.14)such that J = ∗ dv . 18he quantity v ( t ) has a simple transformation under the ˜ δ Kac-Moody symmetries, with˜ δ v ( t ) = t (1 − t t ) ǫ + t t − t t [ v ( t ) , ǫ ] . (3.15)In terms of the mode expansion in (3.14), this implies˜ δ ( m ) ( ǫ ) v ( n ) = mδ m,n +1 ǫ + [ v ( n − m ) , ǫ ] . (3.16)The generalised currents ˆ J = ∗ dv also transform nicely under the Kac-Moody transfor-mations ˜ δ . From (3.15) we find ˜ δ ˆ J = t t − t t [ ˆ J , ǫ ] , (3.17)where ˆ J ≡ ∗ d ( X − ˙ X ). If we expand ˆ J as a power seriesˆ J ( t ) = X n ≥ t n ˆ J ( n ) , (3.18)then (3.17) implies that ˜ δ ( m ) ( ǫ ) ˆ J ( n ) = [ ˆ J ( n − m ) , ǫ ] , n ≥ m . (3.19)One might be tempted therefore to regard ˆ J as defining a hierarchy of Kac-Moody cur-rents. However, although they transform covariantly under the “lower half” of the Kac-Moody symmetries corresponding to ˜ δ , their transformations in general under the “upperhalf” of the Kac-Moody symmetries, corresponding to δ , are very complicated, and onecannot express δ ( m ) ( ǫ ) ˆ J ( n ) as any linear combination of ˆ J ( p ) currents with field-independentcoefficients. SL (2 , R ) /O (2) Coset Model
The simplest non-trivial example that illustrates the constructions we have described inthis paper is provided by the symmetric-space sigma model SL (2 , R ) /O (2). We begin bydefining the SL (2 , R ) generators H = (cid:18) − (cid:19) , E + = (cid:18) (cid:19) , E − = (cid:18) (cid:19) . (4.1)19he O (2) denominator group is generated by the anti-Hermitean combination E + − E − ,whilst the generators in the coset are the Hermitean matrices H and E + + E − . A convenientway to parametrise the coset representative V is in the Borel gauge, for which V = e φ H e χ E + . (4.2)The fields φ and χ are the standard dilaton and axion of the SL (2 , R ) /O (2) sigma model,with the Lagrangian L = − tr( A µ A µ ) = − ( ∂φ ) − e φ ( ∂χ ) . (4.3)From (3.1) we find Q = ( E + − E − ) e Q , P = HP φ + ( E + + E − ) P χ , (4.4)with e Q = e φ dχ , P φ = dφ , P χ = e φ dχ . (4.5)The standard SL (2 , R ) symmetry of the sigma model is given by δ ( ǫ ) V = δh V + V ǫ , (4.6)with ǫ = ǫ H + ǫ − E + + ǫ + E − , where δh is the appropriate O (2) compensating transformationto restore the Borel gauge choice. Thus we have δφ = 2 ǫ + 2 ǫ + χ , δχ = ǫ − − ǫ χ + ǫ + ( e − φ − χ ) . (4.7)The next step is to define ˆ V , whose relation to X is given in (3.8). Following the generalidea described in [18], we do this by introducing scalar fields ˆ φ , ˆ χ and ˆ ψ , which depend onthe spectral parameter t as well as the spacetime coordinates, and writingˆ V ( t ) = e φ ( t ) H e χ ( t ) E + e ψ ( t ) E − . (4.8)We require that ˆ V smoothly approach V , defined in (4.2), as t goes to zero, and so φ (0) = φ , χ (0) = χ , ψ (0) = 0 . (4.9)In terms of power-series expansions for φ , χ and ψ , we may therefore write φ ( t ) = φ + tφ + t φ + · · · ,χ ( t ) = χ + tχ + t χ + · · · ,ψ ( t ) = tψ + t ψ + · · · . (4.10)20ince Q , P and ∗ P in (3.7) are independent of the spectral parameter t , it follows thatby substituting (4.8) into (3.7) we can read off a hierarchy of equations for the fields φ i , χ i and ψ i . At order t , we simply obtain the expressions for e Q , P φ and P χ already given in(4.5). At order t , we find ∗ P φ = dφ + χ dψ , (4.11) ∗ P χ = e φ ( dχ + φ dχ − χ dψ ) + e − φ dψ , (4.12)0 = e φ ( dχ + φ dχ − χ dψ ) − e − φ dψ , (4.13)where the last equation comes from the absence of t -dependence in the denominator groupterm e Q . It can be used to simplify the ∗ P χ expression, to give ∗ P χ = e − φ dψ . (4.14)By equating the t expressions (4.11) and (4.14) for ∗ P φ and ∗ P χ to the duals of the t expressions for P φ and P χ in (4.5), we obtain, together with (4.13), the t equations ofmotion ∗ dφ = dφ + χ dψ , (4.15) e φ ∗ dχ = dψ , (4.16)0 = dχ + φ dχ − ( χ + e − φ ) dψ . (4.17)At order t we find P φ = dφ + χ dψ + χ dψ , (4.18) P χ = e − φ ( dψ − φ dψ ) , (4.19)0 = dχ + φ dχ + ( φ + φ ) dχ − ( χ + e − φ ) dψ +[ φ e − φ − χ (2 χ + φ χ )] dψ , (4.20)and we therefore obtain in total 3 equations at this order, after equating these expressionsfor P φ and P χ to those in (4.5). One can continue this process to any desired order in t .The SL (2 , R ) symmetry δ ( ǫ ) in (4.7) extends to the higher-level fields via the con-struction (2.23), with t = 0. Thus we have δ ( ǫ ) X = [ X, ǫ ], and so using (3.8) to write X = V − ˆ V , together with (4.8), we find we can write the SL (2 , R ) transformations as δφ = − ǫ − ψ + 2 ǫ φ + 2 ǫ + χe φ − φ ,δχ = ǫ − (1 + 2 χψ ) − ǫ χ + ǫ + e − φ − φ (1 − χ e φ ) , (4.21) δψ = − ǫ − ψ + 2 ǫ ψ + ǫ + (1 − e φ − φ ) . v defined in (3.14): δv = [ v, ǫ ].Expanding out (4.21) in powers of t , using (4.10), we recover (4.7) at order t , and atthe next couple of orders we find δψ = − ǫ + φ + 2 ǫ ψ ,δφ = 2 ǫ + ( χ + χ φ ) − ǫ − ψ ,δχ = − ǫ + (2 χ χ + χ φ + e − φ φ ) − ǫ χ + 2 ǫ − χ ψ ,δψ = − ǫ + ( φ + φ ) + 2 ǫ ψ − ǫ − ψ ,δφ = ǫ + (cid:16) χ + 2 χ φ + χ ( φ + 2 φ ) (cid:17) − ǫ − ψ ,δχ = ǫ + (cid:16) − χ ( χ + χ φ ) − χ ( φ + 2 φ ) − χ + e − φ ( φ − φ ) (cid:17) − ǫ χ + 2 ǫ − ( χ ψ + χ ψ ) . (4.22)The hierarchy of equations of motion for the higher-level fields, for which we presentedthe first two orders in (4.15)–(4.17), and (4.18)–(4.20), are invariant under the SL (2 , R )transformations (4.21). An interpretation of the higher-level fields can be given as follows. The equations of motionfor the original level-0 fields, following from the Lagrangian (4.3), are d ∗ dφ + e φ ∗ dχ ∧ dχ = 0 , d ( e φ ∗ dχ ) = 0 , (4.23)Since we are in two dimensions, the axion χ can be dualised to another axion ¯ χ , such that d ¯ χ = e φ ∗ dχ . (4.24)Substituting this into the φ equation of motion, we can remove a derivative from thisequation too, obtaining ∗ dφ = dσ + χ d ¯ χ , (4.25)for some new field σ . Defining ¯ φ = − φ , the original Lagrangian (4.3) can be written in adualised form, terms of the barred fields, as L = − ( ∂ ¯ φ ) − e φ ( ∂ ¯ χ ) . (4.26)We see, comparing (4.24) and (4.25) with (4.15) and (4.16), that¯ χ = ψ , σ = φ . (4.27)22he dualised Lagrangian (4.26) clearly also has an SL (2 , R ) symmetry, which we shalldenote by SL (2 , R ). Denoting its infinitesimal parameters by ¯ ǫ ± and ¯ ǫ , this symmetryacts on ¯ φ and ¯ χ exactly analogously to the action of the original SL (2 , R ) on φ and χ :¯ δ ¯ φ = 2¯ ǫ + 2¯ ǫ − ¯ χ , ¯ δ ¯ χ = ¯ ǫ + − ǫ ¯ χ + ¯ ǫ − ( e − φ − ¯ χ ) . (4.28)(For notational reasons that will become clear shortly, we switch the + and − indices on¯ ǫ ± , relative to ǫ ± , when passing to this barred version of (4.7).)One may also define an infinite tower of higher-level barred fields for the dualised sigmamodel, precisely analogous to the unbarred ones defined above. For example, in order toobtain the barred version of (4.15)–(4.17), we should make the identifications¯ φ = − φ , ¯ χ = ψ , ¯ φ = − φ − χ ψ . (4.29)The barring operation is an involution, with the bar of a bar being the identity operator,and so there is an analogous version of (4.29) in which all barred and unbarred fields areexchanged. The relations (4.29) can be extended to all levels, as we shall now discuss.What we are seeing here is that although the original ( φ , χ ) fields are non-locallyrelated to the dual fields ( ¯ φ , ¯ χ ) (because of the differential relation (4.24) expressing ¯ χ in terms of χ ), there exists a purely local relation between the full hierarchy of fields( φ i , χ i , ψ i ) and their barred analogues. This relation can be established to any desiredhigher order in level number, by systematically examining the systems of equations thatfollow from (3.7), which we presented at level-1 in (4.15)–(4.17) and level-2 in (4.18)–(4.20).There is, however, a simpler way of presenting the entire hierarchy of relations in a compactform.To do this, we first introduce a barred version of ˆ V , which was defined in equation (4.30):ˆ¯ V ( t ) = e
12 ¯ φ ( t ) ¯ H e ¯ χ ( t ) ¯ E + e ¯ ψ ( t ) ¯ E − . (4.30)Here ¯ H and ¯ E ± are SL (2 , R ) generators that satisfy identical commutation relations to H and E ± , namely[ H, E ± ] = ± E ± , [ E + , E − ] = H ; [ ¯ H, ¯ E ± ] = ± E ± , [ ¯ E + , ¯ E − ] = ¯ H . (4.31)This is already enough to ensure that the barred hierarchy of fields will satisfy identicalequations of motion to the unbarred hierarchy; they are derived from the barred version of(3.7). Next, we note that we may make the following choice for the barred generators interms of the unbarred ones:¯ E + = t E − , ¯ E − = 1 t E + , ¯ H = − H , (4.32)23ince this is consistent with (4.31). Thus we haveˆ¯ V ( t ) = e −
12 ¯ φ ( t ) H e t ¯ χ ( t ) E − e t − ¯ ψ ( t ) E + . (4.33)We now impose the relation ˆ¯ V ( t ) = ˆ V ( t ) (4.34)which therefore establishes a relation between these barred and unbarred fields, which havealready been established to satisfy the same system of equations. This is easy to solveexplicitly, since one has only to exponentiate 2 × t -dependence of all the fields) ψ = t ¯ χ χ ¯ ψ , χ = 1 t ¯ ψ (cid:0) χ ¯ ψ (cid:1) , φ = − ¯ φ − (cid:0) χ ¯ ψ (cid:1) . (4.35)Expanding in powers of t allows us to read off the relation between the entire hierarchiesof barred and unbarred fields. At the leading order, we find precisely the relations (4.29) thatwe obtained previously when we started the level-by-level process of mapping the unbarredequations of motion into barred ones. If one carries out such a sequential calculation, onefinds that the entire hierarchy of relations between barred and unbarred fields uniquelyfollows, once the leading-order relations (4.29) are fed in. Thus, we may conclude that sincethe all-level relations (4.35) match (4.29) at the leading order, they represent the uniquecompletion of this relation to all orders.The barred hierarchy of fields transforms under SL (2 , R ) in precisely the same way asthe unbarred hierarchy transforms under SL (2 , R ). For example, for the first couple oflevels, the barred fields will transform under the dual SL (2 , R ) symmetry according to thebarred version of (4.22) (with the exchange of ¯ ǫ + and ¯ ǫ − , as we discussed previously for ¯ φ and ¯ χ in (4.28)). The SL (2 , R ) transformations of the entire hierarchy of dual fields canbe succinctly expressed as the barred analogue of (4.21), which is therefore given by¯ δ ¯ φ = − ǫ + ¯ ψ + 2¯ ǫ ¯ φ + 2¯ ǫ − ¯ χe ¯ φ − ¯ φ , ¯ δ ¯ χ = ¯ ǫ + (1 + 2 ¯ χ ¯ ψ ) − ǫ ¯ χ + ¯ ǫ − e − ¯ φ − ¯ φ (1 − ¯ χ e φ ) , (4.36)¯ δ ¯ ψ = − ¯ ǫ + ¯ ψ + 2¯ ǫ ¯ ψ + ¯ ǫ − (1 − e ¯ φ − ¯ φ ) . Since we also have the relation (4.35) between the barred and the unbarred fields, itis now a straightforward matter to work out the transformations of the original unbarred24elds under the dual SL (2 , R ) symmetry. From (4.35) and (4.36) we find¯ δφ = − ǫ − ǫ − (cid:20) tχe φ + φ + 1 t ψ (cid:21) , ¯ δχ = 2¯ ǫ χ + ¯ ǫ − (cid:20) tχ e φ + φ + 1 t (1 + 2 χψ − e − φ + φ ) (cid:21) , (4.37)¯ δψ = t ¯ ǫ + − ǫ ψ + ¯ ǫ − (cid:20) te φ + φ − t ψ (cid:21) . Expanded, as usual, in powers of t , these equations give the transformations of the entirehierarchy of original fields ( φ i , χ i , ψ i ) under the dual SL (2 , R ) symmetry. Note that thereare no negative powers of t in the expansions.It is evident from (4.37) that the ¯ ǫ transformation in SL (2 , R ) is the same (modulo asign) as the ǫ transformation with respect to the original SL (2 , R ) (see equation (4.21)).The ¯ ǫ + transformation in (4.37) is also very simple, with¯ δ (¯ ǫ + ) φ = 0 , ¯ δ (¯ ǫ + ) χ = 0 , ¯ δ (¯ ǫ + ) ψ = t ¯ ǫ + . (4.38)In terms of the expansions (4.10), this means that all fields ( φ i , χ i , ψ i ) in the hierarchy areinert except for ψ , which suffers the shift transformation¯ δ (¯ ǫ + ) ψ = ¯ ǫ + . (4.39)It is easy to see that this is precisely the same as the transformation given by ˜ δ X inequation (2.28), at order t and with ǫ taken to be just ǫ + , i.e. ˜ δ (1) (¯ ǫ + ) X = t X ¯ ǫ + . (4.40)This shows that the ¯ ǫ + transformation in SL (2 , R ) is implemented by the Kac-Moodygenerator J + − (see (2.42)).This leaves the ¯ ǫ − transformation in SL (2 , R ) still to be identified. In fact, this isprecisely a δ X transformation as given in (2.23), at order t and with ǫ taken to be just¯ ǫ − . Using (2.23), this is given by δ (1) ( ǫ ) X = 1 t [ X , ǫ ] − ˙ η ( ǫ , X + t M − ǫ † M X , (4.41)with ǫ = ¯ ǫ − , where η ( ǫ , t ) ≡ X ( t ) ǫ X − ( t ). Substituting X = V − ˆ V into this, and using(4.8), one straightforwardly reproduces the ¯ ǫ − transformation in (4.37). This shows thatthe ¯ ǫ − transformation in SL (2 , R ) is implemented by the Kac-Moody generator J − (see(2.42)). 25t this stage, we have arrived at a complete understanding of all six transformationsin the original and dual symmetry groups SL (2 , R ) and SL (2 , R ). The original SL (2 , R )transformations ǫ ± and ǫ of course correspond to the level-0 Kac-Moody generators J ± and J . We have also shown that the dual SL (2 , R ) transformations ¯ ǫ + , ¯ ǫ − and ¯ ǫ correspondto the Kac-Moody generators J + − , J − and J : SL (2 , R ) : ( J +0 , J − , J ) ,SL (2 , R ) : ( J + − , J − , J ) . (4.42)It is indeed clear from the Kac-Moody algebra (2.41) that both these triplets selected fromthe generators J in form SL (2 , R ) subalgebras. It is also clear that the two triplets do notcommute. In fact, from the two triplets one can fill out the entire Kac-Moody algebra, bytaking appropriate sequences of multiple commutators.Thus we have shown in a very explicit and precise way that the affine \ SL (2 , R ) Kac-Moody symmetry of the two-dimensional SL (2 , R ) /O (2) symmetric-space sigma model isgenerated by taking multiple commutators of the two SL (2 , R ) symmetries of the originaland the dualised formulations of the theory.It is interesting to note that the entire “negative half” of the Kac-Moody symmetry( i.e. J in with n < J + − with J in with n ≥
0, emerges from the humble shift symmetry ¯ δψ = ¯ ǫ + that we obtained in (4.39).This emphasises the point, which we remarked on earlier, that the negative half of theKac-Moody algebra arises through symmetries that are realised only on the infinite towerof fields ( φ i , χ i , ψ i ) with i > ˜ δ and some example δ transformations It is not hard to work out the explicit form of all the ˜ δ transformations on the fields( φ i , χ i , ψ i ). From (2.28), (3.8) and (4.8) we find˜ δ ψ ( t ) = t t − t t (cid:0) ǫ + + 2 ǫ ψ ( t ) − ǫ − ψ ( t ) (cid:1) , ˜ δ χ ( t ) = t t − t t (cid:0) − ǫ χ ( t ) + ǫ − (1 + 2 χ ( t ) ψ ( t )) (cid:1) , ˜ δ φ ( t ) = 2 t t − t t ( ǫ − ǫ − ψ ( t )) . (4.43)26ollecting the powers of t and t , we find for n ≥ m ≥ δ ( m ) ( ǫ ) φ n = 2 δ mn ǫ − ǫ − ψ n − m , ˜ δ ( m ) ( ǫ ) χ n = − δ mn ǫ χ n + δ mn ǫ − + 2 ǫ − n − m − X p =0 χ p ψ n − m − p , ˜ δ m ( ǫ ) ψ n = δ mn ǫ + + 2 ǫ ψ n − m − ǫ − n − m − X p =1 ψ p ψ n − m − p , (4.44)where it is understood that on the right-hand side χ n = 0 for n < ψ n = 0 for n < δ ( m ) φ n = 0, ˜ δ ( m ) χ n = 0 and ˜ δ ( m ) ψ n = 0 whenever m < n . Of course we also have˜ δ ( m ) φ = 0, ˜ δ ( m ) χ = 0.The symmetries ˜ δ in (4.44) are essentially just shift transformations of φ n , χ n and ψ n byconstant parameters ǫ , ǫ − and ǫ + (with independent sets of these SL (2 , R ) parameters ateach of the negative Kac-Moody levels), with the extra terms being the necessary “dressings”that ensure that the transformations leave the equations of motion invariant. In accordancewith an observation we made previously, the ˜ δ transformations could therefore be used inorder to “gauge fix” the auxiliary fields ( i.e. ( φ i , χ i , ψ i ) for i ≥ SL (2 , R ) /O (2)example) to any desired set of values at one chosen point in spacetime. Since the auxiliaryfields also transform under the δ symmetries, one could view the ˜ δ transformations, in sucha gauge-fixed situation, as compensating transformations that restored the fields to thesechosen values after having performed δ transformations. This is effectively what happensin the construction of Schwarz’s subalgebra of the full Kac-Moody algebra.As we observed in section 3, the ˜ δ transformations become more elegant if they areapplied to the quantities v ( n ) defined in (3.14), for which we have (3.16). In fact v ( t ) iseasily calculated in terms of φ ( t ), χ ( t ) and ψ ( t ), giving v − = ˙ χ + χ ˙ φ , v = ˙ φ + ψ ˙ χ + χψ ˙ φ , v + = ˙ ψ − (1 + χψ ) ψ ˙ φ − ψ ˙ χ . (4.45)Thus, as can be seen by expanding in powers of t , the v ± ( n ) and v ( n ) are are just certaincombinations of the φ m , χ m and ψ m fields, v − (0) = χ + χ φ , v (0) = φ , v + (0) = ψ , etc. (4.46)The δ symmetries in (2.17) and (2.23) are more non-trivial, but again they are completelylocal transformations of the fields ( φ i , χ i , ψ i ), which can be read off explicitly to any desiredorder of non-negative Kac-Moody level, and to any desired order in the t -expansion of the27elds. For example, we find for the SL (2 , R ) /O (2) example that at Kac-Moody level 1, thetransformations on ( φ , χ , ψ , χ , ψ ) are given by δ (1) ( ǫ ) φ = 2 ǫ + χ + 4 ǫ χ ψ − ǫ − ψ ,δ (1) ( ǫ ) χ = − ǫ + ( φ e − φ + 2 χ χ + χ φ ) + ǫ − ( φ + 2 χ ψ )+2 ǫ ( ψ e − φ − χ − χ φ − χ ψ ) ,δ (1) ( ǫ ) φ = ǫ + (cid:0) χ + χ (2 + 2 φ − φ ) + 2 χ e φ (cid:1) + 2 ǫ − ( ψ + χ e φ )+2 ǫ (1 + 2 χ e φ + 2 χ ψ + 2 χ φ ψ ) ,δ (1) ( ǫ ) χ = ǫ + (cid:0) (1 + φ ) e − φ − χ − χ χ + χ ( φ − φ ) − χ e φ )+ ǫ − ( φ + 2 χ ψ + 2 χ ψ − φ + χ e φ )+ ǫ (cid:0) − χ + χ ( φ − φ − χ ψ ) − χ φ ψ − φ ψ e − φ − χ e φ (cid:1) ,δ (1) ( ǫ ) ψ = − ǫ + ( φ − φ + χ e φ ) + ǫ − ( e φ − ψ )+2 ǫ ( ψ − φ ψ − χ e φ ) . (4.47) In this paper, we have studied the global symmetries of flat two-dimensional symmetric-space sigma models. This can be viewed as a preliminary to studying the somewhat moreintricate problem of curved-space two-dimensional sigma models, which arise in the toroidalcompactification of supergravity theories. Both the curved and the flat cases share thecommon feature that the global symmetries include an infinite-dimensional extension of themanifest G symmetry of the G / H sigma model.There has been some controversy over the precise nature of the infinite-dimensionalextension. Whilst most authors have asserted that the symmetry is the affine Kac-Moodyextension ˆ G of G , Schwarz [16] found instead a certain subalgebra ˆ G H of the Kac-Moodyalgebra. One of our goals in this paper has been to resolve the discrepancies.In our work we made extensive use of Schwarz’s results which have, it seems for the firsttime, provided explicit expressions for the key transformations that underlie the positivehalf of the Kac-Moody symmetry algebra. By synthesising this with earlier work where theidea of introducing an infinity of auxiliary fields in order to provide a local formulation wasdeveloped, we have been able to construct a fully local description of the entire Kac-Moodyalgebra of global symmetry transformations.We have also shown how the subalgebra found by Schwarz can be viewed as a conse-quence of making a gauge choice, in which the values of the complete set of fields are fixed28o prescribed values at a chosen distinguished point in the two-dimensional spacetime.In order to make some of the ideas more concrete, we also studied a simple explicitexample, where the coset of the sigma model is taken to be SL (2 , R ) /O (2). We showedhow our present analysis could be related to much earlier work by Geroch [4], in which theinfinite-dimensional symmetry was obtained by commuting SL (2 , R ) symmetry transfor-mations of the original sigma model and its dual version. In particular, we were able toexhibit the precise correspondence between the two sets of SL (2 , R ) transformations andcertain generators of the Kac-Moody algebra. This provides an explicit demonstration thatthe Geroch algebra formed by taking commutators of the two SL (2 , R ) transformations isthe same as the Kac-Moody algebra \ SL (2 , R ). Acknowledgements
We are very grateful to John Schwarz for discussions, and for drawing our attention toreferences [16, 17]. We thank also Gary Gibbons, Herman Nicolai and Kelly Stelle for dis-cussions. This research has been generously supported by George Mitchell and the MitchellFamily Foundation. The research of H.L. and C.N.P. is also supported in part by DOEgrant DE-FG03-95ER40917.
A Schwarz Algebra ˆ G H Versus Kac-Moody Algebra ˆ G In [16], the Lax equation (2.14) is solved for X as a non-local function of the originalsigma-model fields, by writing X ( x ; t ) = P exp h Z xx (cid:16) t − t ∗ A + t − t A (cid:17)i , (A.1)where P denotes path ordering along the integration path, and x is an arbitrarily-chosenpoint. This is a significantly different approach from the one we have followed, where X isexpressed locally in terms of an infinity of auxiliary fields.Our transformation (2.23) for δ X is not quite the same as the one given in Schwarz’sdiscussion [16]. Let us denote his expression by δ ′ X ; it is given by δ ′ X = t t − t ( η X − X ǫ ) + t t − t t ( M − η † M X − X M − ǫ † M ) , (A.2)29here M = M ( x ), and x is chosen as the lower limit of the integral expression (A.1) for X ( t ). Thus the relation between δ ′ and our expression δ is δ ′ = δ − t t − t t X M − ǫ † M . (A.3)In [16], Schwarz calculates the commutator [ δ ′ , δ ′ ] M , finding[ δ ′ , δ ′ ] M = t δ ′ ( ǫ , t ) − t δ ′ ( ǫ , t ) t − t M − t t − t t (cid:0) δ ′ ( ǫ ′ , t ) − δ ′ ( ǫ ′ , t ) (cid:1) M , (A.4)where ǫ = [ ǫ , ǫ ] , ǫ ′ = [ M − ǫ † M , ǫ ] . (A.5)(In obtaining this result, one must hold M fixed.) The right-hand side of (A.4) involves δ ′ transformations again, and so the algebra appears to be closing. However, Schwarz doesnot calculate [ δ ′ , δ ′ ] X . Let us denote his result in (A.4) as [ δ ′ , δ ′ ] M = δ S M . After somealgebra, we find that[ δ ′ , δ ′ ] X = δ S X + t t − t t X ( M − ǫ † M − ǫ ′ ) + t t − t t X ( M − ǫ † M + ǫ ′ ) . (A.6)This shows that on X , the commutator of δ ′ transformations does not close merely on δ ′ ,but instead it gives transformations of the form X ˜ ǫ as well, for certain ˜ ǫ . In fact, suchtransformations are of the type ˜ δ that we introduced in (2.28), and (A.6) may be writtenabstractly as[ δ ′ , δ ′ ] = δ S + ˜ δ ( M − ǫ † M − ǫ ′ , t ) + ˜ δ ( M − ǫ † M + ǫ ′ , t ) . (A.7)Of course, the extra ˜ δ terms on the right-hand side was not seen in Schwarz’s calculations,because he calculated the commutator only on M , for which we know ˜ δM = 0, but not on X . The conclusion from (A.7) is that if all the δ ′ transformations (A.3) are included in thesymmetry algebra, then it is necessary to extend the algebra further by including the ˜ δ transformations too, in order to achieve closure. As may be seen from (A.3), Schwarz’s δ ′ transformations are themselves a combination of our δ and ˜ δ transformations; in fact, onehas δ ′ ( ǫ ) = δ ( ǫ ) − ˜ δ ( M − ǫ † M ) . (A.8)The upshot is that once one has extended Schwarz’s transformations to comprise not only δ ′ but also ˜ δ , one has, equivalently, extended to the full set of δ and ˜ δ transformations thatwe considered in section 2.2. These, as we showed, generate the complete affine Kac-Moodyextension ˆ G of the original G algebra. 30ne can, alternatively, take a more restrictive viewpoint, which is effectively the onethat was adopted by Schwarz in [16]. Namely, the commutation relations (A.7) implythat it is only if either δ ′ or δ ′ is a level-0 transformation that the ˜ δ transformations aregenerated. (This follows from the fact that the second term on the right-hand side of (A.7)is independent of t , and the third term is independent of t .) Thus, we have[ δ ′ ( m ) ( ǫ ) , δ ′ ( n ) ( ǫ )] = δ S ( m + n ) ( ǫ ) , for m > , n > , (A.9)[ δ ′ (0) ( ǫ ) , δ ′ ( n ) ( ǫ )] = δ S ( n ) ( ǫ ) + ˜ δ ( n ) ( M − ǫ † M + ǫ ′ ) , n > . (A.10)(We have taken δ ′ to be a level-0 transformation in the second equation, for definiteness.)One can therefore avoid generating any ˜ δ transformations if one restricts the level-0 trans-formations in δ ′ to be such that M − ǫ † M + ǫ = 0 . (A.11)This equation is essentially the condition that ǫ should belong to the denominator algebra H of the coset model. This is most immediately clear if one chooses, as one may, the “gauge”in which M = 1. Equation (A.11) then implies that ǫ is anti-Hermitean, which is preciselythe standard condition for it to lie in the denominator algebra H . If some other gaugechoice is made for M , then ǫ is again required to be in the denominator algebra, in a basisconjugated by M . The upshot of this discussion is that the necessity of including all the˜ δ symmetries as well in order to achieve closure of the algebra (A.7) can be avoided if onetruncates to that subset of the δ ′ transformations in which the K transformations at 0-levelare omitted.This, therefore, accounts for the symmetry algebra that was found by Schwarz in [16].The full Kac-Moody symmetry algebra ˆ G is generated by our δ and ˜ δ transformations, whilstSchwarz’s subalgebra, which he denoted by ˆ G H , corresponds to the transformations δ ′ givenin (A.8), with the further restriction that at level-0 the K transformations are omitted.Omitting these particular transformations is precisely what is needed in order to maintaina fixed boundary condition for M (such as M = 1). In the gauge choice M = 1, we seefrom (A.8) that δ ′ ( ǫ ) = δ ( ǫ ) ± ˜ δ ( ǫ ), with the plus sign occurring when ǫ lies in H and theminus sign when ǫ lies in K . The generators J ′ ni of the Schwarz subalgebra are thereforegiven in terms of the Kac-Moody generators J in by J ′ ni = J in + J i − n , for i ∈ H ,J ′ ni = J in − J i − n , for i ∈ K . (A.12)One sees immediately that the level-0 generators J ′ i vanish if t lies in K . It can easily beverified directly that the generators J ′ ni form a closed subalgebra of the full Kac-Moodyalgebra (2.41). 31he Schwarz subalgebra of the Kac-Moody algebra can be interpreted as follows. Bywriting X ( t ) as in (A.1), a choice has been made to set X ( t ) = 1 at the point x in thetwo-dimensional spacetime. This can be viewed as a gauge-fixing that is achieved by usingthe ˜ δ transformations. Furthermore, as we remarked below (A.5), M must be held fixed,which is a further gauge fixing (of the original sigma-model fields), achieved by using the K part of the original G Lie algebra transformations. In other words, only the H partof the original G symmetry survives. If we wish instead to retain the full algebra G oforiginal symmetries, then Schwarz’s subalgebra will necessarily have to be extended to thefull Kac-Moody algebra ˆ G .It is instructive to look at this truncated subalgebra in the concrete example of the SL (2 , R ) /O (2) sigma model that we studied in section 4. Especially, it is interesting to lookat the transformations of the original SL (2 , R ) symmetry and the dual SL (2 , R ) symmetry,to see which are retained and which are truncated out in the subalgebra.The combinations of Kac-Moody generators J in that lie in K and in H are given, respec-tively, by K : J n , ( J + n + J − n ) , H : ( J + n − J − n ) . (A.13)It then follows from (A.12) that the generators J ′ ni that are retained in the truncated algebraof [16] are K : J ′ n (1) = J n − J − n , J ′ n (2) = J + n + J − n − J + − n − J −− n , H : J ′ n (3) = J + n − J − n + J + − n − J −− n . (A.14)Since the SL (2 , R ) transformations correspond to the Kac-Moody generators J i , andthe SL (2 , R ) transformations correspond to the generators J + − , J and J − , it suffices toconsider just the levels m = 0 and m = 1 in (A.14). These give the four following non-vanishing generators: n = 0 : J ′ (3) = 2( J +0 − J − ) ,n = 1 : J ′ (1) = J − J − ,J ′ (2) = J +1 + J − − J + − − J −− ,J ′ (3) = J +1 − J − + J + − − J −− . (A.15)32e see that just two of the five inequivalent transformations in SL (2 , R ) and SL (2 , R ) areretained within the truncated algebra: J ′ (3) ↔ ( ǫ + − ǫ − ) , ( J ′ (2) − J ′ (3) ) ↔ (¯ ǫ + − ¯ ǫ − ) . (A.16)Thus, the infinite-dimensional subalgebra of the full Kac-Moody algebra that is retained inthe truncation (A.12) omits not only the K generators in the original SL (2 , R ), but alsothe K generators in the dual symmetry algebra SL (2 , R ). If one wants to have a symmetryalgebra that at least contains all the generators of the original and the dual SL (2 , R )algebras then, as we showed in section 4.2, this will necessarily be the full Kac-Moodyalgebra. B The Virasoro-type Symmetry and the Schwarz Approach
In section 2.3 we obtained a Virasoro-like symmetry of the symmetric-space sigma models,with generators K n satisfying the algebra (2.57). Our construction was closely related toone given in [16] but there were significant differences, which we shall elaborate on here.The first respect in which our discussion diverges from that in [16] is that in that paper,the quantity ξ ( t ) appearing in the our transformation δ V ( t ) V = V ξ ( t ) (see (2.45)) is replacedby ˜ ξ ( t ) = ( t −
1) ˙ X ( t ) X ( t ) − + I , (B.1)where I = Z ∗ A . (B.2)One can see from the path-ordered integral expression (A.1) for X ( t ) that X ( t ) = 1 + t Z ∗ A + O ( t ) , (B.3)and so in fact I = ˙ X (0) = ˙ X (0) X (0) − . Thus from (B.1) we see that Schwarz’s ˜ ξ and our ξ are related by ˜ ξ ( t ) = 1 − t t ξ ( t ) − (cid:20) − t t ξ ( t ) (cid:21) t =0 . (B.4)Thus the lowest mode in our transformation is excluded in the PCM analysis in [16].The lowest mode had to be excluded in [16] for the principal chiral model, as opposedto the symmetric-space sigma model, in order to ensure that the transformation was asymmetry of the equations of motion. In brief, the transformation of A under ξ (defined as33n (2.45)) in the PCM case is simply δ V A = Dξ , rather than (2.47) of the SSM case, andso using (2.48) one finds δ V A = ∗ d ( t − ξ ) + 11 − t A + t − t ∗ A . (B.5)This means that d ∗ δ V A = t/ (1 − t ) dA , and so the equation of motion d ∗ A = 0 is notpreserved. However, if the lowest-order term in δ V ( t ) is subtracted out, as is done in (B.4),then the resulting transformation ˜ δ V does give a symmetry.Although Schwarz carried over the assumption that the lowest mode should also besubtracted out when he then considered the SSM case, it is actually no longer necessary todo so, as we explained in section 2.3. As we showed there, with the transformation δ V A now given by (2.47), one finds using (2.48) and (2.49) that the contributions to δ V A of theform ∗ A coming from the two terms in (2.47) cancel out, and so d ∗ δ V A = 0 automatically,without the need to subtract the lowest mode term. The upshot is that the set of Virasoro-like symmetries that we find for the symmetric-space sigma models is actually larger thenthe set obtained by Schwarz in [16], by virtue of the inclusion of the lowest mode in δ V ( t ).A second difference between our results and those in [16] is concerned with the preciseform of the Virasoro-like algebra in the two cases. We were able to make a convenient choiceof − t as the prefactor of ˙ XX − in (2.45) which gave the algebra in the form (2.57), whichis very close in structure to the Virasoro algebra. On the other hand, in [16] the choiceof t -dependent prefactor was apparently constrained by certain requirements of matchingbetween left and right acting transformations on the group manifold of the PCM (a consid-eration that does not apply in the SSM case). This led to the choice of ( t −
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