DE05F5029
UNIVERSITÄT BONN Physikalisches Institut
Families and Degenerations of Conformal Field Theories
von Daniel Roggenkamp »DEO21241946*
In this work, moduli spaces of conformal field theories are investigated. In the first part, moduli spaces corresponding to current-current defor- mation of conformal field theories are constructed explicitly. For WZW models, they are described in detail, and sigma model realizations of the deformed WZW models are presented. The second part is devoted to the study of boundaries of moduli spaces of conformal field theories. For this purpose a notion of convergence of families of conformal field theories is introduced, which admits certain degenerated conformal field theories to occur as limits. To. such a degeneration of conformal field the- ories, a degeneration of metric spaces together with additional geometric structures can be associated, which give rise to a geometric interpreta- tion. Boundaries of moduli spaces of toroidal conformal field theories, orbifolds thereof and WZW models are analyzed. Furthermore, also the limit of the discrete family of Virasoro minimal models is investigated.
Post address: BONN-IR-2004-14 Nussallee 12 Bonn University- 53115 Bonn September 2004 Germany ISSN-0172-8741
Contents
1 Introduction 4
2 Moduli spaces of WZW models 8 2.1 Current-current deformations of CFTs 10 2.2 Some deformation theory 13 2.3 Deformations of WZW models 17 2.4 Sigma model description of deformed WZW models 20 2.4.1 Classical Action 20 2.4.2 Thedilaton 25 2.5 Explicit example: su(2)fc 31 2.6 Discussion 35
3 Limits and degenerations of CFTs 38 3.1 From geometry to CFT, and back to geometry 39 3.1.1 From Riemannian geometry to spectral triples 40 3.1.2 Spectral triples from CFTs 43 3.1.3 Commutative (sub)-geometries 47 3.2 Limits of conformal field theories: Definitions 52 3.2.1 Sequences of CFTs and their limits 52 3.2.2 Geometric interpretations 60 3.3 Limits of conformal field theories: Simple examples 64 3.3.1 Torus models 64 3.3.2 Torus orbifolds . . 69 3.3.3 WZW models 71 3.4 The m —* oo, c —> 1 limit of the unitary Virasoro minimal models M(m,m+ 1) 73 3.4.1 The unitary Virasoro minimal models A4(m, m + l)m_>oo 74 3.4.2 Geometric interpretation of M.(m,m + l)™-^ 81 3.5 Discussion 87
A Properties of conformal field theories 89
B c = 1 Representation theory 93
C Structure constants of the unitary Virasoro minimal models, and their c —> 1 limit 97
1 Introduction Moduli spaces of conformal field theories are important objects in the study of two dimensional quantum field theories because they describe critical subspaces in the space of coupling constants. In string theories, whose small coupling limits are described by conformal field theories,, these moduli spaces arise as parameter spaces of string vacua. The understanding of moduli spaces .of conformal field theories is thus a very important issue for string theory. However, although conformal field theories are quite well understood, there are only few examples of explicitly known moduli spaces at present, most of which correspond to free field theories as e.g. toroidal conformal field theories or orbifolds thereof. The reason for this is that a good conceptual understanding of deformations of conformal field theories beyond conformal perturbation theory is still lacking. Perturbation theory is usually technically involved, at least when one wants to obtain higher order contributions, and hence is only applicable to study CFT moduli spaces in small neighborhoods of explicitly known models. In particular it is in general not possible to obtain global information about the moduli spaces from perturbation theory, except in situations, where symmetry as e.g. super- symmetry is preserved by the corresponding deformations. In this work, the problem of understanding moduli spaces of conformal field theories is approached in two different ways. The first part, Sect. 2, which is based on joint work with Stefan Forste published in [39] is devoted to the study of a special class of deformations of conformal field theories, namely CURRENT- CURRENT DEFORMATIONS. All known deformations of conformal field theories are generated by per- turbations of the theory with exactly marginal fields, and it is widely believed that in fact all deformations of CFTs can be obtained in this way. Deforma- tions which are generated by perturbations with products of holomorphic and antiholomorphic currents are called current-current deformations. As was shown in [16], such deformations preserve the algebras of the cur- rents from which the perturbing fields are constructed. It turns out that this gives enough structure to determine global properties of the subspaces of mod- uli spaces corresponding to this kind of deformations. Namely we show that current-current deformations only affect the representation theory of the alge- bra generated by the currents involved in the perturbation, and not those of its commutant. In particular the coefficients of the operator product expansion of highest weight states with respect to this algebra are invariant. This allows us to explicitly construct the corresponding deformation spaces which have basis of the form O{d,d)/O{d)xO(d), where d and d are the numbers of holomorphic and antiholomorphic currents which generate the deformations. The corresponding moduli spaces are obtained by identifying isomorphic CFTs in these spaces which amounts to dividing out discrete "duality groups". The latter strongly depend on the structure of the CFTs under consideration. In fact, this generalizes the treatment of deformations of toroidal conformal field theories, whose moduli spaces have been constructed in [83, 15, 23], and it gives a new class of very explicitly known families of conformal field theo- ries. Namely, to each conformal field theory which possesses holomorphic and antiholomorphic currents, such families can be constructed. Important examples of non-free conformal field theories admitting current- current deformations are WZW models (see e.g. [54]). Moduli spaces of current- current deformed WZW models associated to compact semi-simple Lie groups are analyzed in detail. Furthermore a sigma model description of the deformed WZW models is derived. It is shown that like the undeformed WZW models, also the deformed ones admit representations as sigma models with Lie group target spaces. Sigma model-metric, B-field and dilaton are obtained explicitly as functions on the deformation spaces. For this, a representation of the deformed WZW models as orbifolds of products of coset- and toroidal conformal field theories is used, which is mimicked in a sigma model construction. Thus, we gain an explicit description of the moduli spaces of current-current deformed WZW models, in terms of conformal field theory as well as in terms of sigma models. The second part of this work, Sect. 3, which is based on joint work with Katrin Wendland published in [89] is devoted to the study of boundaries of CFT moduli spaces. In general, moduli spaces of conformal field theories, e.g. the moduli spaces discussed in Sect. 2 are not complete and it is an interesting question what happens to the CFT structures if one approaches the moduli space boundaries. Indeed, limits and degenerations of conformal field theories have occurred in various ways in the context of compactifications of moduli spaces of CFTs, in particular in connection with string theory. For example, zero curvature or large volume limits of CFTs that correspond to sigma models are known to give boundary points of the respective moduli spaces [3, 81]. These limits provide the connection between string theory and classical geometry which for instance is used in the study of D-branes. Also in the Strominger-Yau-Zaslow mirror construction [100, 95, 62], boundary points play a prominent role. In fact, Kontsevich and Soibelman have proposed a mirror construction on the basis of the Strominger-Yau-Zaslow conjecture which relies on the structure of the boundary of certain CFT moduli spaces [72], All the examples mentioned above feature interesting degeneration phenom- ena. Namely, subspaces of the Hubert space which are confined to be finite dimensional for a well-defined CFT achieve infinite dimensions in the limit. In fact, such degenerations are expected if the limit is formulated in terms of non- linear sigma models, where at large volume, the algebra of low energy observables is expected to yield a non-commutative deformation of an algebra ^4°° of func- tions on the target space. The algebra of observables whose energy converges to zero then reduces to A°° at infinite volume. An entire non-commutative ge- ometry can be extracted from the underlying CFT, which approaches the target space geometry in the limit [42]. By construction, this formulation should encode geometry in terms of Connes' spectral triples [19, 20, 21]. By the above, degeneration phenomena are crucial in order to single out an algebra which encodes geometry in CFTs. An intrinsic understanding of limiting processes in CFT language is therefore desirable. This will also be necessary in order to take advantage of the geometric tools mentioned before, away from those limits. Vice versa, a good understanding of such limiting processes in CFTs could allow to take advantage of the rich CFT structure in geometry. The main aim of the investigations is to establish an intrinsic notion of such limiting processes in pure CFT language and to apply it to some interesting examples. To this end, a definition of CONVERGENCE FOR SEQUENCES OF CFTs is given, such that the corresponding limit has the following structure: There is a limiting pre-Hilbert space H°° which carries the action of a Virasoro algebra, and similar to ordinary CFTs to each state in H°° we assign a tower of modes. Under an additional condition the limit even has the structure of a CFT on the sphere. This is the case in all known examples, and in particular, this notion of limiting processes is compatible with deformation theory of CFTs in the sense that limits of compact one-parameter families of CFTs are full CFTs, i.e. those defined on arbitrary surfaces. If the limit of a converging sequence of CFTs has the structure of a CFT on the sphere, but is not a full CFT, then this is due to a degeneration as mentioned above. In particular, the degeneration of the vacuum sector can be used to read off a geometry from such a degenerate limit. Namely, in our limits the algebra of zero modes assigned to those states in 7i°° with vanishing energy is commutative and can therefore be interpreted as algebra of smooth functions on some manifold M. The asymptotic behavior of the associated energy eigenvalues allows to read off a degenerating metric on M and an additional smooth function corresponding to the dilaton as well. Moreover, being a module of this commutative algebra, H00 can be interpreted as a space of sections of a sheaf over M as is explained in [72]. Simple examples which these techniques are applied to are the torus models, where the limit structure yields geometric degenerations of the corresponding target space tori ä la Cheeger-Gromov [17, 18]. In this case, Ti.°° is the space of sections of a trivial vector bundle over the respective target space torus. Similar statements are true for orbifolds of torus models, only that in this case the fiber structure of H°° over the respective torus orbifold is non-trivial. Namely, the twisted sectors contribute sections of skyscraper sheaves localized on the orbifold fixed points. Degenerations of CFTs in current-current deformation spaces constructed in Sect. 2 also give rise to geometric degenerations of tori. In this case, 7i°° correspond to spaces of sections of trivial vector bundles with fiber given by Hubert spaces of certain coset models. These coset models arise from the original CFTs in the deformation spaces by means of the coset construction with respect to certain subalgebras of the current algebras which generate the deformations. Another example, which is studied in detail and in fact was the starting point of these investigations is the family of unitary Virasoro minimal models. It strongly differs from the examples mentioned before in that it is a discrete family of CFTs, which even does not have constant central charge. This makes the construction of the sequence, the verification of convergence and the analysis of its limit much more involved. Nevertheless, it turns out that the A-series of unitary Virasoro minimal models indeed constitutes a convergent sequence of CFTs. All fields in its limit theory at infinite level can be constructed in terms of operators in thesu(2)i WZW model. The sequence degenerates, and the limit has a geometric interpretation in the above sense on the interval [0, TT] equipped 1 with constant metric 'g = dx® dx and dilaton <&(z) = In (•s/^sm~ x). A different limit for the A-series of unitary Virasoro minimal models at in- finite level was proposed in [61, 90, 91]. It is described by a well-defined non- rational CFT of central charge one, which bears some resemblance to Liouville theory. In particular, its spectrum is continuous, but degenerations do not occur. The techniques developed in Sect. 3 can also be used to describe this latter limit. The relation between the two different limit structures is best compared to the case of a free boson, compactified on a circle of large radius, where apart from the degenerate limit described above one can also obtain the decompactified free boson. While the degenerate limit focused on in this work has the advantage that it leads to a consistent geometric, interpretation, the one which corresponds to the decompactified free boson gives a new well-defined non-rational CFT.
Acknowledgements Ich danke meinem Doktorvater Werner Nahm und Matthias Gaberdiel, Andreas Recknagel und Katrin Wendland, die mich mit Rat und Tat unterstützt haben. Für die Zusammenarbeit an in die Arbeit eingegangen Artikeln bedanke ich mich bei Stefan Forste und Katrin Wendland. Ferner danke ich für Ihre Unterstützung dem DFG-Schwerpunktprogramm 1096, dem Marie-Curie-Trainingssite am Mathematics Department des King's College London, dem Institut für Theoretische Physik der ETH-Zürich und dem Institute for Pure and Applied Mathematics, UCLA sowie der "BIGS". 2 Moduli spaces of WZW models
In this section, special deformations of conformal field theories, namely those generated by perturbations with products of holomorphic and antiholomorphic currents are studied. As was shown in [16], such deformations preserve the alge- bras of the perturbing fields. Indeed, this allows to determine global properties of the subspaces of moduli spaces corresponding to this kind of deformations. Important examples of non-free conformal field theories admitting current- current deformations are WZW models (see e.g. [54]). The corresponding moduli spaces for WZW models associated to compact semi-simple Lie groups, will be discussed in detail. Since WZW models have descriptions as sigma models on Lie groups, it is a natural question if there are such descriptions for all conformal field theories from these moduli spaces. In fact, families of sigma models containing WZW models at special points have been discussed by many authors (e.g. in [63, 58, 94, 70, 55]). In particular one-parameter families of sigma models containing the su(2)fc-WZW models have been studied very explicitly by Giveon and Kiritsis [58], who also compared them to the families of current-current deformed su(2)fc- WZW models which were described in [101]. Ideas about a generalization of these considerations to arbitrary WZW models have also been presented in [58, 70, 71]. Here, WZW-like sigma model representations of current-current deformed WZW models will be explicitly constructed. These are sigma models with the same target space as the "undeformed" WZW model in the family, but with different (in general not bi-invariant) metrics, additional £f-fields and dilaton. The latter are constructed as functions on the deformation spaces. Thus, we obtain explicit descriptions of the moduli spaces of current-current deformed WZW models associated to compact semi-simple Lie groups in terms of conformal field theories as well as in terms of sigma models. In Sect. 2.1, exactly marginal current-current deformations of conformal field theories are discussed. We start from the facts obtained in [16], that perturba- tions of conformal field theories with products of holomorphic and antiholomor- phic currents are exactly marginal iff the holomorphic as well as the antiholo- morphic currents belong to commutative current algebras, and that in this case these holomorphic and antiholomorphic current algebras are preserved under the deformations. This can be used to reduce the problem of studying finite current- current deformations to first order deformation theory, which is carried out in Sect. 2.2, and from which it follows that the effect of these deformations on the CFT structures is completely captured by pseudo orthogonal transformations of their charge lattices with respect to the preserved commutative current algebras. The corresponding deformation spaces can thus be described by
V S 0{d, d)/O{d) x O(d). (2.1)
This generalizes the deformation results of toroidal conformal field theories [23]. The corresponding moduli spaces are obtained from the deformation spaces by taking quotients with respect to "duality groups". In Sect. 2.3 we discuss an important class of examples, namely WZW models corresponding to compact semi-simple Lie-groups. The general results from Sect. 2.1 are compared to a realization of deformed WZW models obtained from a representation of WZW models as orbifolds of products of generalized parafermionic and toroidal models given in [51]. In Sect. 2.4 various aspects of exactly marginal deformations of WZW models are analyzed from a sigma model perspective. This approach is best suited for a semiclassical treatment and in that sense less powerful than the algebraic one. However, it can illustrate the results and provide a picture for the class of deformed models. Exactly marginal deformations of WZW models from the sigma model perspective have been discussed in the past mainly for rank one groups [63, 58, 94] and for models where coordinates can be chosen such that the relevant set of chiral and anti-chiral currents follows manifestly from the equations of motion [64]. Mimicking the orbifold realization of deformed WZW models described in Sect. 2.3, we will consider an orbifold of a direct product consisting of a vectori- ally gauged WZW model and a d-dimensional torus model, where d is the rank of the group. Since a sigma model is not very well designed to accommodate orbifolds, we perform an axial-vector duality (generalized T-duality) to obtain a dual description without an additional orbifold action. To this end, we first implement the orbifold by gauging in addition an axial symmetry of the WZW model combined with shifts in the torus factor. We force the corresponding gauge connection to be flat but choose the zero modes of the corresponding La- grange multiplier such that the gauge bundle is twisted in a non-trivial way. It turns out that integrating out the gauge field instead of the Lagrange multiplier provides a sigma model without an additional orbifold action. The result is a "WZW-like" model, i.e. a sigma model with Lie group as target space, and a WZW-type action in which the bi-invariant metric is replaced by a more general bilinear form which is neither bi-invariant nor necessarily symmetric. The same sigma models can be obtained as coset models of a product of the original WZW model and d-dimensional torus models with gauge group U(l)d, embedded into both factors. All the sigma model manipulations described so far are carried out at a classical level. In full quantum field theory one has to replace the procedure of solving equations of motion by performing Gaussian functional integrals. These in general provide functional determinants, which in turn generate a non-trivial dilaton. In a pragmatic approach this can be computed by imposing conformal invariance, i.e. requiring vanishing beta functions. We will use a more elegant way consisting of a comparison of the Hamiltonian of the model and a generalized Laplacian, which depends on the dilaton [98]. Finally in Sect. 2.5, some of the results are illustrated in the example of the deformed su(2)fc-WZW model. 2.1 Current-current deformations of CFTs Although, not proven in general, it is widely believed that all deformations of con- formal field theories are generated by perturbations of the theories with marginal fields, i.e. fields Oi with conformal weights h(Oi) = 1 = h(Oi). The perturbed correlation functions on a conformal surface S of a combina- tion of operators X{p\,... ,Pk), Pi € £ are defined to be
:= (X(pu...,pk)exp fo^i/ 0iC^Ej>E, (2.2) where the integrals have to be regularized due to the appearance of singularities. If the perturbed correlation functions define a quantum field theory, which is a fixed point of the renormalization group flow, it is again a conformal field theory. This however is not the case in general. Indeed, preservation of conformal. invariance by perturbations gives non linear restrictions on the fields, which generate it (see e.g. [23]). This means that the set of exactly marginal fields, i. e. those which generate deformations of conformal field theories are not vector spaces in general. In particular the deformation spaces of conformal field theories need not necessarily be manifolds but may have singularities. (For more details on conformal deformation theory see e.g. [23, 73, 84, 85]). In the following, deformations of conformal field theories generated by a special class of marginal fields, namely products of holomorphic and antiholo- morphic currents will be discussed. These are simple enough to give a global description of the deformation spaces corresponding to them and to express the data of deformed CFTs explicitly in terms of the data of the undeformed ones. We consider conformal field theories, whose holomorphic and antiholomor- phic VF-algebras contain current algebras g^, g^ corresponding to Lie algebras g and g and k, k G N, i.e. for every j, f G g there exist holomorphic fields j{z),f(z) of conformal weight h = 1 in the theory, such that
m/ \ ., ^ j(w) dj(w) T{z)j(w) = JK ; + JK \ + reg, (z - w)z (z — w) where Kg(.,.) is a bi-invariant scalar product on g and [.,.] its Lie bracket. The holomorphic energy momentum tensor T{z) can be written as
9) aß
a with g the dual Coxeter number of g, (j )a a basis of g and TQ(Z)J(W) — reg. The same holds for the antiholomorphic current algebra replacing (g, k) by (g, k). In particular, there is a subspace of the CFT Hubert space isomorphic to g ® g of marginal fields. However, not all of these fields are exactly marginal. As was shown in [16], such fields are exactly marginal if and only if, under
10 the isomorphism above, they correspond to elements of a
KQQ®VQ®VQ. (2.5)
The set of charges A C a* x ä* forms a lattice equipped with bilinear pairing
K© (—«)• As is shown in Sect. 2.2, deformations corresponding to pairs (a, ä) only affect the representation theory of the W-algebras a, a, but not the OPE-coefncients of a © ä-highest weight vectors. More precisely, if one chooses suitable connections on the bundles of Hubert spaces over the deformation spaces [85], the effect of the deformations on the CFT structures is completely captured by transformations of the charge lattices A in the identity component O(d, d)o of the pseudo orthogonal group O(d,d), and all structures independent of the charges are parallel with respect to the chosen connection. That O(d, cQo-transformations of the charge lattice A indeed give rise to new modular invariant partition functions and also preserve locality is easy to see even without any perturbation theory. Namely, for O € O(d, d) define deformed operators L®, Lo on Ji by
: L + (Lo + Lo) lwe?i5!?<8>v<3<8>% = ( ° + (O* -
:= L (LQ-LO) I«QI3®V0®% ( 0 - + (O* - 1)(K© -S)((g,Q),
Locality is maintained because of the preservation of «©(—7c) by transformations O £ O(d,d). To show the preservation of modular invariance we consider the torus partition function depending on modular parameters r and r (q = e2mT, q = e~2mr) of the O(i)-transformed model along a smooth path O : [—1,1] —»
1From now on K = £K8|O, « =
11 O(d,d)0 with O(O) = 1, dtO(t)\t=0 = Te o{d,d)
(2.6)
and use the modular transformation properties of the unspecialized characters (see e.g. [66]):
(Z) (2-7)
27T2 —• 2irt where q = e~r~, g = e~?~. This shows that invariance under the modular trans- formation T h-> — i is preserved under O(d, (^-transformations of the charge lattices. A similar calculation proves the statement for THT + 1. Since charges (Qi Q) only characterize the o©a-modules up to automorphisms of the underlying Lie algebras, transformations of A by O(d) x O(d) C O(d,d) leave the conformal field theory invariant. Thus, the deformation spaces corre- sponding to pairs (a, a) are given by
P(o,s) = O(d, d)0/((O(d) x 0(5)) n O(d,5)0) = 0(d,5)/0(d) x 0(5). (2.8)
To get the respective moduli spaces from these deformation spaces, one has to identify points describing equivalent conformal field theories. In fact, the con- formal field theories are specified by the charge lattices marked with the Hubert spaces Hrq -Q. of a © ä-highest weight states (with all the structure they carry, as e.g. structure of modules of the Virasoro algebra etc.) and the coefficients of the operator product expansion of these highest weight states. Denoting these additional structures by S, and the automorphisms of A together with S by Aut(A, S), the components of the moduli spaces corresponding to (o,a) defor- mations can be written as
= Aut(A, S)\O(d,d)/O{d) x 0(5). (2.9)
If the action of Aut(A, S) has fixed points, M.(a$) has singularities and the Hilbert space bundle over it has non-trivial monodromies around them. More
12 precisely elements in Aut(A, 5) act on the Hubert space bundles over the defor- mation spaces, and monodromies around fixed points are given by the respective actions of the stabilizers. This gives a very explicit description of the components of moduli spaces of conformal field theories corresponding to current-current deformations. In particular conformal field theories as above come in £>(a,o) families of explicitly known CFTs, and the conformal field theory data at any point in these families can be easily reconstructed from the CFT data at one point. A well known example of this kind is the moduli space of toroidal conformal field theories. These models have holomorphic and antiholomorphic W-algebras, each of which contains a u(l)d current-algebra. They are completely character- ized by their charge lattices CHQQ are trivial for all charges), which for integer spin of the fields, locality and modular invariance of the torus partition function have to be even, integral, selfdual lattices of signature (d, d) in Rd'd [15, 83]. Hence, S is trivial and Aut(A,5) = Aut(A) = O(d,d,Z), such that the moduli spaces Md,d corresponding to the current-current deformations are isomorphic to the Narain moduli spaces [83]
^ Z)\O{d, d)/O{d) X 0{d) , (2.10) of even, integral selfdual lattices of signature (d, d) in M.d 2.2 Some deformation theory In this section, techniques from conformal deformation theory (see e.g. [23, 73, 84, 85]) are used to calculate the effect of current-current deformations on arbi- trary conformal field theories containing current algebras in their holomorphic and antiholomorphic W-algebras. In the following, a family of conformal field theories is regarded as a Hermi- tian vector bundle over a differentiate manifold2 parametrizing deformations of the conformal field theory structures in a smooth way, i.e. all CFT structures are smooth sections of corresponding vector bundles. Such families can be realized as perturbations (2.2) by exactly marginal fields. In this case, the tangent bundle of their base manifolds are subbundles of the Hermitian vector bundles. The choice of regularization method and renormaliza- tion scheme gives rise to connections on them [85]. Here connections D (called c in [85]) will be used, which restrict to the Levi-Civita connections on the tan- gent bundles of the base manifolds equipped with the respective Zamolodchikov metrics. These connections are defined by "minimal subtraction" of divergences in the regularization constant. Given a conformal field theory, it is in general quite hard to make global statements about the family of conformal field theories, generated by pertur- bation with a given set of exactly marginal fields. This is due to the fact that Singularities do not occur in our situation. 13 information on the CFT structures in points of the family have to be more or less completely reconstructed out of the structures at one point (the point corresponding to the CFT which is being perturbed), by means of perturbation theory. Thus, one gets perturbative results only, and perturbation theory usually becomes technically difficult at higher orders. However, in the case of perturbations by products of holomorphic and an- tiholomorphic currents, there is in fact enough structure to make exact global statements about the families of CFTs generated by them using first order per- turbation theory only. In [16] it was shown that tensor products of fields of holomorphic and an- tiholomorphic currents are exactly marginal, if and only if they form abelian current algebras ä, ä respectively, and that in this case the deformations gener- ated by them preserve the corresponding current algebras a and ci. Thus these deformations give rise to families of conformal field theories with ä and ö con- tained in their holomorphic and antiholomorphic W-algebras. Moreover, the tangent vectors to the families in every point are given by products of currents, whose CFT-properties are known independently of the actual CFT. Thus the derivatives of the CFT-structures can be calculated in every point of the families and can then be integrated up. Assuming that the Hubert spaces of the conformal field theories in the fam- ilies decompose into a © ä-highest weight representations as in (2.5) n ~ 0 ^QQ®VQ®VQ. it is shown in the following that these deformations only affect the a © ä- repre- sentations, while the OPE-coefficients of d © o-highest weight states are parallel with respect to the connection D. To be more precise, the only effect of the deformations will be O(d, ^-transformations of the charge lattices A € a* x a*. From this it follows in particular that the corresponding deformation spaces are given by (2.8) ,5) = O(d,d)o/(0(d) x O(3))0 = 0{d,d)/0{d) x O(d). (2.11) First of all, by the coset construction [60], the holomorphic W-algebra W of a conformal field theory in such a family contains the coset algebra W/d as a subalgebra, such that [W/d, a] — 0. In particular W/d is not affected by the deformation, i.e. the elements of W/d as well as their mutual OPE are covariantly constant, which implies that W/d is contained in the W-algebras of any conformal field theory in the family. Moreover, the holomorphic energy momentum tensor T of the CFT decomposes into a sum T = Ta + T", where Ta is the energy momentum tensor associated to the current algebra d and T' belongs to the coset algebra W/d and is therefore covariantly constant. Hence, the covariant derivative of T is given by the covariant derivative of Ta. The same statements hold for the antiholomorphic W-algebra. Let us now calculate the covariant derivatives of the modes of the d- and ö-currents and holomorphic and antiholomorphic energy-momentum tensors de- 14 fined by n T(z) = ^J z -*Ln, T(z) = neZ neZ a a where, as in Sect. 2.1 (j )a and {j )a are basis of a and ö respectively, and we denote the generators of the deformations by Oaa(z,~z) := ja(z)ja('z). By the definition of D, DOaajn can be expressed as ^ -8 ( T^ -B\ \ 1 dz _n f L 27r l ^ 'n /c(0) ^ JCP \D€{z) (2.13) where e is the regularization parameter and [X]€ means the regularization pa- rameter independent part of X3. Using the OPE (2.3) this can be expressed as L /c.(0) ^ C(w) l 2l = [fJce(o) ^ aß = -nkK ^5nfi. (2.14) The same kind of arguments lead to p-.z~n f C(0) Z7rz JCPi-\Dt(z) Je (2.15) and similar expressions for the modes of ä. Altogether we find aß Doc-sß = -iykK 5nfiJZ, Doa«Ln = -irj%j%, (2.16) Since the zero modes j%, jfi belong to the center of a © ä, the ä © ä-PF-algebra structure is parallel with respect to D, and only the a©a-charges4 change under these deformations. Their covariant derivatives can be read off from (2.16) 0 f , (2.17) 3 a X can be written as a sum of terms proportional to e for some a. [X]e denotes the term proportional to e°. 4As noted above, we assume the zero modes of the currents to be diagonalizable on H. 15 which are transformations in o( a* © ä*, K ©(—«)). This characterizes completely the deformations of the ä 0 ä-W-algebra structures of the CFTs. Moreover the a ©a-highest weight property and the decomposition of the Hubert space (2.5) are preserved under the deformations, which means in particular that we only have to assume (2.5) for one CFT in the family. To show that this is in fact the only effect of the deformation on the CFT structures, we have to show, that the OPE of a © a-highest weight vectors is not deformed. Now, the covariant derivative of correlation functions of those vectors is given by (2.18) But the ln-term in the last line of (2.18) is just the logarithm of the corresponding d-, o-conformal block. Thus, the correlation functions are deformed only through the conformal blocks and the OPE-coefficients of & © ä-highest weight states are parallel. Thus, the effect of current-current deformations on the conformal field the- ory structure is completely characterized by the deformations of the charges described above. In particular from (2.17) it follows that the base manifold of the family of CFTs generated by current-current deformations is indeed given by (2.11). Let us finish with a comment on another connection D. In the discussion above, we used connections D on the Hubert space bundles over the deforma- tion spaces, which were defined by minimal subtraction. With respect to these connections operators from the W-algebras are parallel except zero modes of the current algebra. For the discussion of e.g. boundary conditions other connections D will be useful. These are defined by5 5In fact they are nothing else than the connections f from [85], which are obtained by a regularization scheme consisting of cutting out radius one disks around the punctures of the surfaces, i.e.. in (2.13) instead of taking the regularization parameter independent part [,]t one sets e = 1. 16 They satisfy & ^f^, (2-20) ro+n > and thus for all n G Z, the parallel transport of (j£, j^n)a,ä is given by the vector representation of O(d,d). As the connection D, also D restricts to the Levi-Civita-connection on the tan- gent bundle of the deformation space equipped with the Zamolodchikov metric. 2.3 Deformations of WZW models WZW models are conformal field theories associated to Lie groups G with bi- invariant metrics (.,.) (see e.g. [54]). For simplicity we will only consider compact semi-simple G with bi-invariant metrics corresponding to the Killing forms on the respective Lie algebras here. So, let k G N, G a semi-simple Lie-group, g its Lie-algebra of rank d with Killing form K/k, roots A, weights Q, root lattice Q(g), coroot lattice Q(g) C Q(g), and weight lattice P(g). Furthermore denote by flk the set of integrable weights of the affine Lie-algebra gk at level k. The WZW models associated to (G,K,k) have the affine Lie algebras gk as holomorphic and antiholomorphic W-algebras. Its Hubert spaces decompose into tensor products of integrable highest weight representations V^, A G Cik of Qk- For simplicity only diagonal WZW models are considered in the following, i.e. those WZW models whose Hubert spaces are given by (2.21) Generically, the only marginal fields in WZW models are products of holomor- phic and antiholomorphic currents from the current algebras. From the general considerations in Sect. 2.1 it is clear that every pair of Cartan subalgebras f) C g, fj C g gives rise to deformations of the WZW models. However, all such pairs lead to equivalent deformations, because all maximal abelian subalgebras of a semi-simple Lie algebra are pairwise conjugated6 and inner automorphisms of g induce automorphisms of the corresponding WZW models. Thus the deforma- tion space of current-current-deformed WZW models is given by 2>wzw^O(d,d)/O(d)xO(d). (2.22) For a given Cartan subalgebra f) C g the integrable g^-highest weight represen- tations decompose into ^-highest weight modules VQ, Q G f)* as follows ^ ® V? * © V^ ® © V(ß+kS), (2.23) 6This is not true for non-semi-simple Lie algebras, where one gets more interesting moduli spaces. 17 where F^ := P(g)/fcQ(g) is a finite abelian group, V^ is a highest weight module of a generalized parafermionic W-algebra associated to the coset construction gfc/F), and V^** are highest weight modules with respect to an extended F) W- algebra [51] 7. Prom this, the charge lattice can be read off to be •Ao = {(/i,7*)€P(g)xP(fl)|AJ-Ä*eQ(fl)}. (2.24) The "duality group" is given by the automorphisms of Ao compatible with the additional structures S^, alluded to in the last section. In the case of diagonal WZW models discussed here, all these structure are determined by representa- tion theory, and the duality group is given by the semi-direct product Aut(A0, Sfc) S A(g) K W(g) (2.25) of the automorphism group A(g) of the root lattice Q(g) with the Weyl group W(g), where A(g) acts diagonally on Ao C P(g) x P(g), and W(g) acts on the second factor only (see [70] for a discussion of dualities of WZW models). Since A(g) = W(g) ix F(g), the "duality" groups can be written as Aut(A0, Sk) = (W(fl) x W(fl)) K F(g), (2.26) with the Weyl groups acting separately on the two factors of Üxfi and F(g) acting diagonally. Note that these groups are finite, as opposed to the "duality groups" of d-dimensional toroidal models for d > 1. For the special case g = su(2), the group (2.26) coincide with the toroidal "duality" group 0(1,1,Z) = Z2 x Z2. Given the duality group, the moduli space of current-current deformed WZW models can now be written as (2.10) •Mwzw = (W(g) x W(s)) x F(0)\O(d, d)/O(d) x O(d). (2.27) In fact, Ao has an even integral selfdual sublattice Ak := {(fi, Ji) e P(g) x P(g) \ii-pe fcQ(g)} C Ao (2.28) of signature (d, d), which can be regarded as charge lattice of d-dimensional toroidal conformal field theory. Let us denote by Aut(Afc,5fc) C Aut(Ao,5fc) the subgroup of the duality group fixing Afc. This group is also a subgroup of O(d, d, Z). 7In terms of characters xx(<7>w) corresponding to the gfc-highest weight representations, this is just the string function decomposition [67] with cji denoting the string functions. 18 Every even integral selfdual sublattice of Ao of signature (d, d), which is obtained from A& by applying a transformation of Aut(Ao,S&) gives rise to a representation of the WZW model as orbifold model 9k = (öfc/6 0 tAk)/rfc • (2.29) of a product of a coset model ßfc/f) and a toroidal conformal field theory tAA. with charge lattices A*. Such representations were presented in [51] and were actually used in [71] in the study of dualities of WZW and coset models. The torus partition function of coset and toroidal models (with the notation from footnote 7) are given by z»kl\q,q) = E EE 4 k^ where r){q) = qu rins-iC-"- ~ #n) denotes Dedekind's 77-function. acts on the Hubert spaces of the models by giving rise to the (a, /?)-twisted torus partition functions for a, ß € Ffc . (2-33) 2d Prom this, one can easily read off, that the orbifold partition function of (2.29) ,9) (2.35) agrees with the torus partition function of the diagonal g^-WZW model Q)- (2-36) The fact that the orbifold group F^ acts trivially on the W-algebras of coset and toroidal models makes this representation of the WZW models useful for the study of current-current deformations. Namely, for given (f), h) the holomorphic and antiholomorphic W-algebras t) C gk and f)CflJt can be identified with the 19 d d ü(l) , ü(l) W-algebras of tAfc, and deformations corresponding to (F),r)) are then just deformations of the toroidal factor tAfc in (2.29). Thus, for a point O G .Mwzw the corresponding conformal field theory can be represented as8 flfc(O) = (öfe/6 ® toAk)/rfe . (2.37) This provides concrete isomorphisms between the. current-current deformation spaces of WZW- and toroidal models. Note however that in general the "duality" groups of the WZW models gfc and the toroidal models i\k do not agree. On the one hand, there might be dualities in the WZW models which do not preserve Afc and thus correspond to a change of the orbifold representation, and on the other hand dualities in the toroidal model need not lift to automorphisms of Ao together with S^. As noted above, for g = su(2), the duality group of the WZW model and the corresponding toroidal models coincide. In fact, the conformal field theories in .Mwzw have WZW-like sigma model descriptions, which will be discussed in Sect. 2.4. 2.4 Sigma model description of deformed WZW models 2.4.1 Classical Action To show that the deformed WZW models discussed above in fact have WZW-like sigma model descriptions, i.e. they are described by WZW-type actions, however with metrics different from the bi-invariant metrics and additional S-fields, we write down a sigma model action, which on the one hand defines orbifold models as in (2.37) and on the other hand describes WZW-like models. Let us start by describing the ingredients used to construct this sigma model. As above, we denote by G a compact semi-simple Lie group of rank d with Lie algebra g, Cartan subgroup H C G, corresponding Cartan subalgebra f)Cg and k € N. Furthermore, S is a 2-dimensional surface, bounding the three manifold B. Then, an ASYMMETRICALLY GAUGED WZW MODEL on E is described by the following action 9,B) (2.38) where 9 X ~ IIB * ) is the WZW action with g : S —> G, (.,.) the Killing form on g, and x the three form on G associated to ([.,.],.) [54]. The integral over B in (2.39) is called WESS-ZUMINO TERM. (2.39) defines a quantum field theory if HziG) — 0 and G HS(G, 2?rZ), which we assume to hold. This WZW model is vectorially 8A realization of the torus partition function of the deformed su(2)fc-WZW models as a partition function of Zfc-orbifolds of tensor product of parafermionic models and free bosonic theories has already been given in [101]. 20 coupled (see e.g. [49]) to an H-gauge connection A — (A, A) € Q1'0 © fiOil(E, fj) by - (dgg-\Ä) - <(1 - Ad9) A,Ä)} , (2.40) and axially coupled (see e.g. [70, 57]) to an H-gauge connection B = {B,B) G B,B)} . (2.41) Adding both terms (2.40) and (2.41) to the WZW action (2.39), one obtains the general asymmetrically gauged model, provided we introduce the following coupling between the two gauge fields A and B which restores gauge invariance c S Gtk(g,A,B) = ^ Jj{(l + Adg)B,Ä) - ((1 + Adg)A,B)} . (2.42) This contains more terms than the interaction given in [6], where however a constraint relating the two gauge fields was imposed. We avoid such a constraint by adding a Lagrange multiplier Sb,k(KB) = - / {-(dX,B) + (B,dX)}, (2.43) with A : £ —> V(27rQ(s)')> where Q(g)' is the image of Q(g) under the isomor- phism between \j* and F) provided by (.,.). This model will be coupled to an axially gauged H-WZW model, i.e. an axially gauged d-dimensional toroidal model Sn,k,E (y, B) = ~^J ((y~ldy ~2B),E (y^dy - 2B)> , (2.44) where y : S —• H = U(l)d, and metric and B-field are parametrized by an invertible d x d-matrix with positive definite symmetric part E G Glps(d, R). . Below, it will be argued that the action m ß SGtk(g,A,B,X,y) := S^k (g,A,B,X)+Sn,k,E(y^ ) (2-45) describes the current-current deformed gVWZW models discussed in Sect. 2.3. e In fact the vectorially gauged G-WZW model part S™™ + SG k of the action (2.45) describes the parafermionic factor in the orbifold representation (2.37) of the deformed models, while the H-part (2.44) describes the toroidal one. The axial gauging which couples these two parts in (2.45) amounts to the orbifold- construction in (2.37). To see this, we will make use of the local symmetries of the model (2.45) under vector transformations g -> hgh-1 (2.46) . A -* A + hdh'1 Ä -» Ä + hdh'1 X —> A + AC, 21 with h = exp (in) : £ —> H, and axial transformations 9 -> fgf, (2.47) £? -» B + f~ldf, B -> B + f-^f, y -* with / = exp(i77),T7 : S —» l)/(iQ(g)'). Let us first integrate out the Lagrange multiplier field A. Performing a partial integration in (2.43) yields ik f Ud{\B) + (\F(B))) (2.48) <£ B) + - [ (\,F(B)). (2.49) where J1" = dß is the curvature of B, 7$ represent the non-trivial one-cycles of E and n$ are the "winding numbers" of A around the dual cycles. Thus, the Lagrange multiplier forces B to be flat with logarithms of monodromies around 7i9 &i := jf Be (2ArQ(0))* S ±P(0). (2.50) Now, axial gauge transformations can shift the bi by elements in ^Q(g). There- fore, the gauge equivalence classes of flat connections B satisfying (2.50) are com- pletely characterized by bi € (^fP(fl)) / (jQCß)) — Tfe, and the integration over the gauge field B reduces to summing over these b{. Note that Fk = P(&)/(kQ(g)) is nothing else than the orbifold group in the orbifold representation (2.37) of the deformed WZW models. Having integrated out A and B we end up with a product of an H-gauged G-WZW model, corresponding to the parafermionic coset model Qklh [49], and a d-dimensional toroidal model parametrized by E, which are coupled by F^-twists. These twists indeed correspond to the Ffc-action (2.32) on the Hilbert spaces of the conformal field theories gk/h®toEAk, and the sigma models (2.45) describe the deformed WZW models (2.37). The identification of parametrizations of deformation spaces is as usual in toroidal CFTs: O(d, d) acts on Glps(d, R) by fractional linear transformations, and we get 0(1) = E for O € O(d, d) parametrizing the orbifold models (2.37) and E G Glps(d, R) parametrizing the sigma models (2.45). Next, we construct the "T-dual" models by integrating out B first. Since B enters the action algebraically, this can be done by solving the equations of motion and plugging back the solution into the action. The calculation simplifies if we gauge fix y = 1. Solving the equations of motion for B yields T 1 1 1 B = (PAdg + l + 2E )~ {Pdgg- -{l + PAdg)A-2A- dA} , (2.51) 1 l 1 B = (PAdff-i + 1 + 2E)' {Pg- dg + (l + PAd9-i) Ä + 2A~ 9A} , 9A similar construction was used in [88]. 22 where P : g —> F) denotes the orthogonal projection on the Cartan subalgebra and A = exp (zA). This we plug back into the action (2.45). In the end, we will also integrate out A. Therefore, we gauge fix A = 1 already at this stage. Then we obtain g,A) (2.52) Next, we integrate out A by solving the classical equations of motion T 1 A = (PAd9 - 1 - (1 + PAdfl) (PAd5 + 1 + 2E )~ (1 + PAd9))~* 1 T l 1 x {Pdgg- - (1 + PAd9) (PAds + 1 + 2E )~ Pdgg~ } , (2.53) l 1 Ä = (PAdg-i - 1 - (1 + PAd^-i) {PAdg-i + 1 + 2E)~ (l + PAdfl-i))" 1 l l x {-Pflr- ^ + (1 + PAd5-0 (PAd^-i + 1 + 2E)~ Vg- dg} . Plugging this back into the action and performing some algebra yields So,t(s) = SgT + ^^tPAdj-JJ-^-'Pajj-'.P»-1»«) (2-54) with the abbreviations 1 1 1 Sg := (l-P)-(PAdp-J?" )~ (PAd5 + JR- ) (2.56) 1 = (1 - P) + (RPAdg - P)- (i?PAd9 + P) . Thus, we obtain the action of a WZW-like model with a deformed metric and additional B-field encoded in the choice of f) C g and of the matrix E. As expected from the comparison with the CFT considerations we recover the action of the original G-WZW model for E = 1. Note that in general deformed metric and B-field are not bi-invariant with respect to G. But they are bi-invariant with respect to H C G, which follows 1 from the identities £hg = Eg, Sgh = Ad^ 5sAd/l for all g € G, h € H. Moreover, as also expected from the CFT results, for generic E the model (2.45) only has a h © f) chiral symmetry algebra which is generated by 5g = ge- RTeg, (2.57) 23 with e, e : S —> fy. The corresponding chiral currents read J = kR-1 (1 - RRT) (PAdg - A"1)"1 Pdgg'1, (2.58) T l T T i 1 1 J = -k{R )- {l-R R){PAd9-1-(R y y P9- dg. Let us mention that the two degenerations of the model (2.45) E = Al, A —> 0, A —> oo correspond to the axially and the vectorially gauged WZW model respectively10 (c./. the discussion of limits of WZW models in Sect. 3.3.3). Thus, at the classical level, we achieved to construct a class of models connecting the axially gauged WZW model via the ungauged to the vectorially gauged one. Moreover, the response of our sigma model (2.54) to a variation of the de- formation parameters 6S = ^ = ~f^R(RTR-l)-1(SR-1)(l-RRT)-lRJ,j) (2.59) is bilinear in the conserved currents (2.58), suggesting that our family is indeed generated by current-current perturbations. So far, we have seen that the action of the deformed model looks like a WZW model action with deformed bilinear form, which is generically not bi-invariant anymore. This however is not the full story. When integrating out the gauge fields, one picks up Jacobians which usually give rise to a non-trivial dilaton. Hence, we expect that in addition to the deformed metric and B field, there will also be a non-trivial dilaton in the deformed model. By construction our model should be conformally invariant in the semiclassical limit. This means that the background should satisfy beta function conditions (see e.g. the review [96] and references therein). Checking these equations one will also observe that a non trivial dilaton (coupling with the power of fc° to the sigma model action) is needed. That would be one way to obtain the non-trivial dilaton. In the following subsection, we will use a different method to derive the expression for the dilaton. But before coming to that, let us comment on the relation of these deformed models to the families GxH)/H 4 (5, B, y) := S^{g) + S%k{g, B) + Sn^(y, B) (2.60) 10This can be easily seen by comparing the sigma model actions. Alternatively, it can be deduced by the following observation. For E = 0 the additional U(l)d factor decouples and integrating over B yields the "T-dualized" orbifold of the vectorially gauged model, i.e. the axially gauged model. On the other hand in the limit A —» oo the gauge field B is frozen to zero and we are left with the vectorially gauged model. In both cases there is an additional decoupled U(l)rf factor whose torus is of vanishing size or decompactified, respectively. In our previous discussion this additional factor appears, because in the decoupling limits the gauge fixing conditions need to be altered, i.e in those limits one should gauge fixcoordinate s on G such that the metric on the coset does not degenerate. 24 of gauged WZW models of type (G x H)/H with varying embedding of the gauge group in the symmetry group of the G- and H-WZW models, parametrized by E G Glps(d, R). For the special case of symmetric E these models were described in [99]. Integrating out the gauge fields B in these models, one obtains11 (2.61) _\ —l \ PAd5 + 1 + 2£ This in fact coincides with the action (2.54), which was obtained by integrating out B in the sigma model (2.45) we started with, if one relates the parameters E = -E(E- I)"1 . (2.62) Hence, for E such that E in (2.62) is well-defined and positive definite, the sigma model (2.45) we started with has a coset realization given by (2.60). This is the case e.g. for E G Glps(d, R) whose eigenvalues lie in (0,1). Note however that e.g. for E = 1, E in (2.62) is not well-defined and thus, the original G-WZW model does not have a proper coset realization as in (2.60). It only corresponds to a degeneration thereof. Nevertheless, for convenience, we will use the coset model realization in the following subsection to calculate the Hamiltonian of the model (2.45). Although the realization we use is only defined on part of the actual moduli space, we suppose that the results we obtain are actually valid on the whole moduli space. This is supported by the observation that they reproduce the correct results for the WZW model at E = 1. 2.4.2 The dilaton Next, we would like to derive the Hamiltonian of the coset models (2.60). For this, we choose E = R x S1 to be the cylinder with coordinates (r, a) and complex structure dT >-» da, da •-> — dT. In order to perform the Legendre transform, we return to Minkowskian worldsheet signature using coordinates x± :— r±a. Let us first discuss the ungauged G-WZW model (2.39). The conjugate momenta are given by (2-63) where f2 denotes the contribution from the Wess-Zumino term. Since it contains exactly one r derivative, its final effect drops out in the Hamiltonian. The Hamiltonian is obtained by performing the Legendre transform T l H = IdaTr(w g- dTg)- IdoL aiig-'d^g-^ + ig-'d^g-^g}), (2.64) "The result can be read off from (2.51) by setting A = A = 0, E<-+E and A = 1. 25 where L stands for the Lagrangian belonging to (2.39). Introducing the currents J+ = -kd+gg'1 , J_ = kg^d-g (2.65) we arrive at the classical Sugawara expression J+) + (J-,J-)). (2.66) For later use, we also give these currents expressed as phase space functions T l J+ = 2irAdgzu - 2-n-kAdgÜ - ^dagg~ , (2.67) J- = -2-nkwT + 2-Kkn - -g^d^g. (2.68) In order to construct the Hamiltonian of the deformed G-WZW models we follow the prescription given in [10], where our starting point will be the coset model (2.60) before integrating out the gauge fields, but after gauge fixing y — 1, i.e. B+,g-id-g) (2.69) The additional terms as compared to the ungauged model modify the conjugate momenta according to ±-B+ + ±A4,B-. (2.70) The corresponding Hamiltonian reads T -2 ((l + Ad5 + 2JB )ß+, S-)), (2-71) where H denotes the Hamiltonian (2.66) of the ungauged model. Next, we modify the currents J±, such that the modified currents J+ = J+ + kPAdgB+ + kB_, (2.72) J- = J--kB+-kPAdg-iB-. (2.73) obey a gauge invariant Poisson algebra (see [10] for more details). In terms of the new currents the Hamiltonian of the deformed model is given by 2 2 2 2 +2k (B+,B+) + 2kk (B-,B-) - Ak (ß+, (l + 2E^B^. (2.74) 26 The additional constraints due to the vanishing of the conjugate momenta of B± can be used to eliminate the gauge fields by solving their algebraic equations of motion +-B- = ~PJ+, (2.75) _ - B+ = \?J- (2.76) Since for E invertible12 these equations allow a unique solution for B±. Before giving this it is useful to employ (2.75) and (2.76) to simplify the expression (2.74) slightly. For the last term we write and obtain n = --^ Jda((J+,J+) + (JL,J_) + 2k{B+,J-) -2k(B-,J+)). (2.77) Finally, we plug in the solutions B+ = i(£ + £T + 2££T)1(j_-(l + 2E)j+), (2.78) B- = ^{E + ET + 2ETE} "1 ( (l + 2^T) J- - 2 J+) . (2.79) The result for the Hamiltonian of the deformed model is T T H = --^^ JdalJdal /(l+(^E + E + 2E Ey^J+,J+\ (2.80) E + ET + 2EETy1\ J-,J- E + ET +2EETY1 ( T 2E) (E + E + 2 For symmetric E this expression agrees with the one given in [99]. Now, the zero mode part of the Hamiltonian restricted to a certain subspace of the Hubert space should "match" with the generalized Laplacian on the space of smooth functions on G [98] A* := / f df,e-^^/d^(G)G^du , (2.81) 12The degeneration E = 0 for example describes the coset model G/H which is discussed in [10]. 27 which takes into account the dilaton $, i.e. there is an isomorphism t between C°°{G) and a subspace of the Hubert space, such that tA^i"1 is equal to the zero mode part of the Hamilton operator restricted to this subspace. This can be explained (for more details see [98]) by the correspondence of the constraint (Lo + L0-a) Iphysical) = 0 (2.82) (LQ+LQ is the zero mode of the Hamiltonian and a is a normal ordering constant) with the mass shell condition, which in location space reads ~™2* • (2-83) Here \? denotes the target space field associated to the physical state in (2.82). This will be used in the following to determine the dilaton in the deformed G-WZW models. The subspace of the Hubert space which should correspond to the space of functions on G consists of highest weight vectors only (see [42] for a discussion of this point), and thus the Hamiltonian (2.80) restricted to this subspace can be written completely in terms of zero-modes of left and right currents. Since zero-modes of left and right chiral currents are the generators of left- and inverse right- multiplication by G on itself, they can be identified with the respective sections JL and JR of TG ® g*. Choosing a basis of g*, one obtains from these sections the vector fields j£, j^, A € {1,... ,dim(G)} on G. In every point p£G, ({JL)P)A} {{3R)P)A are two basis of the tangent space TPG of G in p. Hence, A* can be written in terms of jf; or j^, <& and the target space metric G. G can be read off from the kinetic term of the action (2.54), such that we can compare A* with the Hamilton operator (2.80) to obtain 3>. This is the general strategy presented in [98]. But before applying it to the the models (2.45), let us illustrate it in the example of the "undeformed" G-WZW model. For notational convenience, we will use the sections ji and JR of TG A = ^(JLJL) = -J^URJR) = 2J; (ULJL) + (3R,3R)) , (2.84) where the second equality reflects the bi-invariance of the metric. Observing that under the identification JQ ~ j£, J£ ~ 2R 1 °f zero-modes of the holomorphic and antiholomorphic currents with generators of the left and right multiplication the zero-mode part of the Hamiltonian (2.66) "matches" the Laplacian A, we deduce that the dilaton $ is constant. Now let us come to the discussion of the deformed models (2.45). The Hamil- tonian corresponding to the coset representation of these models has been cal- culated above (2.80). The target space metric can be obtained from (2.61) by 28 symmetrization13 GßV/2irk = {MjLß,jLv} = (NjRß,JRU), (2.85) with T T T M = (l+2E +PAdp)~ (l-P+2E +2E +4E &) x (l+2£+Ad«riP)~\ (2.86) T x (l+2£ +Ad,?p)~ . (2.87) Now, the generalized Laplacian can be written as 47rfcA* = {f-l3L, fM~lJL) + (rl3R, fN-'jn) , (2.88) where derivatives act on everything appearing to their right, and / = e-2*V/det(G)/Vdet(Go), (2.89) with Go denoting the "undeformed", i.e. the Killing metric on G. The inverses of M and iV are given by y T x (l + 2E + PAd9) (1-P) (2.90) T T T ^ + 2E + Adff-iP) (E + -E + 2E E) ^ (l + 2E + PAd5) , (2-91) T + E + 2EEF^ "' (l + 2E + PAds-i) , where we used the block diagonal structure of (1 - P + 2 (ET + E + 2ETE\\ and (\ - P + 2 (ET + E + 2EETX\ and the relations 13As noted before the actions (2.45) and (2.61) agree under the identification (2.62), and we will use the parametrization by E for the calculation of A*, because the Hamiltonian (2.80) was obtained in the coset representation. This however does not restrict the region of validity of the results. 29 Substituting (2.90) and (2.91) into (2.88) we obtain the generalized Laplacian. Employing furthermore the identities ( ** \ / **** • • • *~ "** m "* \ / ~ m *~ ** (TI ^* \ I "* /T^ \ -1 i^ O 771 I / T7>J. 1^ IT1 I f) IPJ T7> 1 I TTIJ I U1 1 O IP-i TJ1 1 I 1 I O ET'J 1 -I ~r ^-ti I i S-J ~~T i-i v ^-£> -L> ] — I JZ/ ~\~ i~t ~T~ ^-C/ ill I I JL —I— Zu J and AdpjL = -JR , Adg-iJR = -JL , the result can be brought into the form (2.92) jR This expression matches with the Hamiltonian (2.80) provided that / is equal to a constant, which can be taken to one by performing a constant dilaton shift. Thus, we conclude that along the deformations there is a non trivial dilaton such that \/det(G)e~2* is independent of E. (2.93) This result does not come as a surprise however. Namely, deforming the kinetic term in the WZW action to the one in (2.54) while keeping the norm of the states in the Hubert space fixed leads to a Hamiltonian which represented on L2(G) (i.e. the space of ground states of the theory) takes the form (2.94) Comparison with (2.81) HQ = A* leads to from which (2.93) follows. As a byproduct of this discussion we see from (2.93) that eigenfunctions of A* at a given E, which are also eigenfunctions with respect to the H-actions 30 induced by the left and right multiplication of H on G are also eigenfunctions of A* at all E, with different eigenvalues however. This is expected from the CFT considerations above, where the only effect of the deformations were changes of the {j, f) charges, and f)-, f)-highest weight states remained highest weight states under the deformations. To complete the discussion we should recall that the coset description we used here to construct the Hamiltonian (2.80) is only valid in the part of the parameter space of the model (2.45), where E is positive definite. In particular, it breaks down, when one of the eigenvalues of E becomes 1. Nevertheless, the Hamiltonian, we obtained, can be continued to the region where E has eigenvalues 1, as can be read off from T 7 1 -1)[E + E )' {E + 1) J+, J_) ) , (2.96) and in fact for E = 1 it coincides with the one of the G-WZW model (2.66). This suggests that (2.96) is indeed the Hamiltonian on the whole parameter space of (2.45) and all the results from this section also apply to the entire moduli space. 2.5 Explicit example: su(2)k In this subsection we would like to illustrate our previous discussion very ex- plicitly in the simplest example, namely G = SU(2). We do not want to go through all the details, since much of the discussion for SU(2) (or SL(2, M))14 can be found in the literature (e.g. in [63, 58, 94]), but we will briefly describe the deformed CFT and compare it with the sigma model construction. For this, we will firstly derive the deformed sigma model by T-dualizing the model (SU(2)fc/U(l)xU(l)) /Zfc, and secondly compute the spectrum of the general- ized Laplacian. From the discussion of current-current deformed WZW models correspond- ing to arbitrary compact semi-simple Lie groups in Sect. 2.3, we know that the deformed su(2)fc-WZW models can be realized as orbifold models (compare (2-37)) (su(2)fc/ü(l) ® ü(l)^ß) /Zfc , (2.97) where R € (0, oo) parametrizes the u(l)-factor15. That the deformed su(2)fc- WZW model can be written in this way has first been suggested by Yang in [101], where a one-parameter family of modular invariant partition functions corresponding to such orbifold models with varying radius in the ü(l)-factor 14The discussion of deformation is often presented for the non compact version of Ai because in that case the interesting phenomenon of smooth topology change is observed. 15 As alluded to above, R H-> i is a duality. 31 was presented, noting that the partition function at R = 1 coincides with the partition function of the sti(2)fc-WZW model [53]. Let us start by describing the orbifold realization (2.97) of the deformed sii(2)fc-WZW model. First of all the model u(l)r of a free boson compactified on a circle of radius r has central charge c = 1 and the Hubert space decom- poses into irreducible highest weight modules VQ and VQ of holomorphic and antiholomorphic u(l) algebras with charges Q and Q: where the charges and the conformal weights of the corresponding highest weight states \(p,w)) are given by Q(P;U)) = ^(^+ti)B), Q{PiW) = -^(%-wR), h{p,w) = hQlpw) and h(p,w) — \Q{p,w)- Thus, the partition function of the model can be written as . /„\ E On the Hubert space there is a Zfc-action given by The parafermionic models pffc := su(2)fc/u(l) with central charge c = j^^ — 1 have been classified in [53]. Here we only need the diagonal models, in which the Hilbert spaces decompose into highest weight representations of the holomorphic and antiholomorphic W-algebras as follows %B)®VütB)l (2.101) where Jk = {(j,n)\j G \Z, 0 < j < |, n € Z2fc, 2j + n = 0(mod2)}/~, (j,n) - (f — 3,n + A;), denotes the set of irreducible highest weight representations of the parafermionic W-algebra. The corresponding highest weight vectors \(j, n)) have conformal weights fy^re) = ^(j,rc) — (k+2) ~ Ik' ^or ^ — lnl — ^J- There is also a Zfc-action on this Hilbert space Together with (2.100) this defines a Zfc-action on the product model sit(2)fc ® {1(1)/^-^, which is divided out to obtain the su(2)^(i?)-models. For r, s 6 Z& the (r, s)-twisted partition functions of the u(l)^Ä and the sit(2)fc theories are given by16 . (2-104) 16In the following we set (f1 = 0, V(j,m) = {0} for 21 - m 32 leading to the following partition function of the orbifold model Thus the Hubert space of the orbifold theory is given by17 (2.106) The summand with r — 0 is the untwisted sector of the theory. In the following we will compare this with explicit sigma model analysis. We start with the (SU(2)/U(1) xU(l))/Zfc gauged sigma model (the parametrization is taken from [58] )18 S=Y Id2z Id+xd-x + tan2 xd+6d-9 + ^d+yd-y j , (2.107) where for the time being we omitted the dilaton term coupling to the Gauss- Bonnet density. Now, we redefine coordinates according to 9 = a + ß , y = a-ß (2.108) In these coordinates the relevant components of the target space metric can be written as 2 Gaa = Gßß = k (tan x + j^ J (2.109) (2.110) Now we T-dualize the a direction. The T-dual metric and f?-field follow from the Buscher fromulse [1.2]. We obtain r l R2cos2x k (cos2 x + R2 sm^ x) G^ ksin2x 7; = ö , no • 2 (2.112) 2 2 Gaa COS^ X + R Sin" X 17The feictor | is due to the identification of parafermionic highest weight representation. 18In order to make contact with our general discussion in subsection 2.4.1 we note that in [58] an SU(2) group element is written as exp \i [6 — 0J 0-3/2 exp [ix 33 ^=0 (2.113) C B*ß = ^ = -2 2 ^.2 +1 (2.114) 2 2 Gaa cos x + R sm^ x Now we define ä = k9, ß = 9 and drop the constant term 1 in the B-field. This amounts to SK = -^ / d*z\ d+xd-x + :—^d+ed-9 (2.115) 2TT J I cos2 x + Rl sin x R2 cos2 x cos2 x COS2X cos2 x + R? sin2 x »)}• It remains to discuss the periodicity of 9. In order to obtain the deformed model 9 should be a 2?r periodic coordinate, i.e. ä should be a 2-7rfc periodic coordinate. If we perform the T-duality according to the prescription given in [88], the intermediate gauged sigma model on a worldsheet E will contain a term (compare (2.48)) YLf A> (2.116) where 7* are one cycles of E and rii are the winding numbers of the Lagrange multiplier ä around one cycles of S dual to 7*. As in the discussion around (2.48), summing over the fcZ valued windings of ä yields a Kronecker delta, which is non-vanishing if § A takes values in 2irZ/k. Since the original model is obtained by absorbing a pure gauge A± = d±p into a redefinition of the original coordinate, a parametrizes actually an orbifold 51/Zfe where S1 is the unit circle. Thus for the case G = SU(2) we explicitly obtained the deformed WZW action (2.54) from the orbifold representation (SU(2)fc/U(l) x U(l)) /Zk. Next, we would like to discuss the generalized Laplacian A* in this example. For notational simplicity we set k = 1 during the calculation and reinstall it in the end. After a coordinate change p = sinx (2.117) the metric of the deformed model takes the form Up to a constant shift, the dilaton is given by the relation e~2 Vdet(G) = p. (2.119) Thus, we find explicitly „2# ^L (2.120) 34 which shows that eigenfunctions of the Laplace operator corresponding to the Killing metric on SU(2) which are also eigenfunctions of 8$, 8$ are in fact eigen- functions of the generalized Laplacian A* for all R19, however with different eigenvalues. For an eigenfunction of the Laplace operator in the irreducible SU(2) representation labelled by j 6 {0,..., |} with left and right U(l)-quantum numbers n and n, the difference of the eigenvalues of A* at R and R = 1 is given by = ^ (V - 1) "2 + Lj^»2 where we have reintroduced the level fc. Now, let us compare this with the CFT description (2.37). The difference of LQ + Lo-eigenvalues of a highest weight state in the r-twisted sector for R and R — 1 can be read off from (2.106) to be ^ = h (^(m ~r)2+{R2 ~l) r2)' (2-122) where we already identified the moduli spaces according to our general discus- sion. We observe that (2.121) and (2.122) agree, if we identify n = m-r , n = r, (2.123) and this identification is actually the expected one. 2.6 Discussion Having shown that the effect of current-current deformations of a conformal field theory on its structure is completely captured by deformations of a charge lattice, we obtained a description of the subspaces of CFT moduli spaces correspond- ing to these deformations as moduli spaces of certain lattices with additional structure. This generalizes the case of deformations of toroidal conformal field theories [23]. The general considerations were applied to WZW models, where they were compared with a realization of the deformed models as orbifolds of products of coset models with varying toroidal models. This realization was well suited for the construction of sigma models corresponding to the deformed WZW models. For this purpose we employed axial-vector duality to transform the orbifold of the (G/H x H)-sigma model into a WZW-like model with (in general) non-bi- invariant metric, additional B-field and non-trivial dilaton. This provides a very explicit description of the sigma models associated to deformed WZW models. It would be interesting to investigate further the geometry of metric and 5-field we obtained for the deformed models. Since the sigma models correspond to conformal field theories, they should for example satisfy some nice differential equations, namely the beta-function equations (see [96] and references therein): 19Note that at R = 1 A* = A as discussed in the previous section. 35 Apart from the general CFT considerations, we focused the discussion on the example of WZW models associated to compact, semi-simple Lie groups. There are however other interesting conformal field theories admitting current-current deformations, which deserve an analysis of their moduli spaces. These are for example WZW models corresponding to non-compact Lie-groups (see e.g. [5, 36, 4, 77]), which could provide time dependent exact string backgrounds and thus might give hints about how string theory deals with cosmological singularities (see e.g. [74, 31, 32] for a discussion of this point). But also an investigation- of moduli spaces of e.g. coset models, and in particular Kazama-Suzuki-models [69] should be of interest, because the latter would provide examples of explicitly known moduli spaces of N = 2 super confer mal field theories. A discussion of the relation between mirror symmetry and gauge symmetry in this setting has been presented in [59]. The analysis of the "behavior" of D-branes (i.e. conformal boundary condi- tions) on moduli spaces of conformal field theories is in fact a very important sub- ject in string theory. As our considerations show, current-current deformations are in fact easily tractable, and hence provide a good setup to study structural questions concerning "bulk-deformations" of boundary conformal field theories. Some semi-classical aspects of D-branes in deformed su(2)fc-WZW models were presented in [38, 37], but the general conformal field theory analysis of boundary conditions in deformed WZW models is an interesting open problem. In particu- lar, an investigation of D-branes in deformed Kazama-Suzuki models could lead to a better understanding of D-branes in moduli spaces N = 2 superconformal field theories. 36 37 3 Limits and degenerations of CFTs In this section, techniques are developed to analyze limits and degenerations of conformal field theories, occurring e.g. at the boundaries of CFT moduli spaces. Notions of SEQUENCE OF CONFORMAL FIELD THEORIES and CONVERGENCE of such sequences are introduced. To a convergent sequence of CFTs, a LIMIT can be associated which in general is not a well-defined conformal field theory but a degeneration thereof. It possesses a pre-Hilbert space H°° carrying two commuting actions of the Virasoro algebra, and as in CFTs, to each state in "H00 can be associated a tower of modes. Under slightly stronger conditions, which are fulfilled in all examples, it even has the structure of a conformal field theory on the sphere. This notion of convergence (also the stronger one) is compatible with de- formation theory. Namely a sequence of points in a moduli space of conformal field theories, which converges in this space gives rise to a sequence of conformal field theories converging in the sense which will be defined below to the CFT specified by the limit point, i.e. in this case, the limit structure is indeed a full CFT. In general however, degeneration phenomena will occur in the limiting process, which lead_ to infinite degeneracies of certain subspaces of the Hubert space. In particular correlation functions on higher genus surfaces may diverge in the limit. These degenerations however provide new interesting structures. Namely, it turns out that the states whose conformal dimensions vanish in the limit generate a commutative *-algebra .4°°, which can be regarded as function algebra on a "target space" M. Moreover, the degenerating conformal dimensions can be used to equip the "target space" with a degenerating metric and an additional smooth function, the dilaton. Furthermore, A°° acts on the pre-Hilbert space "H°° associated to the limit, which can thus be interpreted as the space of sections of a sheaf of vector spaces on M. It even seems that the limit CFT structures are .A00-homogeneous, such that they can be localized to corresponding structures on the fibers. Thus, to a degenerate limit of conformal field theories can be associated the structure of a sheaf on M of conformal field theories defined on the sphere.. The fiber vector spaces in turn do not show any degenerations. Structures of this kind have actually already been conjectured in [72], where they were used to motivate a version of mirror symmetry which only relies on the limits of certain CFT moduli spaces. Having set up the notions of convergence of sequences of CFTs and deriv- ing the desired properties of the limit structures, some classes of examples are studied in detail. In the case of toroidal conformal field theories, degenerations give rise to trivial fibrations of torus models with smaller dimensions over de- generating subtori. Similarly orbifolds of toroidal CFTs can be analyzed. In this case we again find the respective torus orbifolds as degenerating "target space" geometries. However, the fiber structure of the limit CFT is more interesting. Namely, the twisted sectors contribute sections of skyscraper sheaves over the fixed points of the orbifold action, which gives a nice geometric interpretation to the twisted sectors. Furthermore, degenerations at the boundaries of current- 38 current deformation spaces, which were constructed in Sect. 2 are studied in the special case of WZW models. In this case degenerations give rise to fibrations of certain coset-models over degenerating tori. Moreover, also a somewhat more involved example is studied, namely the A-series of unitary Virasoro minimal models, which in contrast to the examples mentioned before is a discrete family of CFTs with varying central charge. This makes the construction of the sequence, the verification of convergence and the analysis of the limit structures more difficult. As is expected from gauged- WZW model considerations, we obtain a degenerating interval as "target space" geometry with a non-trivial dilaton. In fact, the construction of the degeneration algebra A°° can be motivated by sigma-model considerations. In large-volume sigma models, one expects the algebra of low-energy observables to be isomorphic to a non-commutative de- formation of the algebra of smooth functions on the target space, which should become commutative in the limit of infinite volume. This can be viewed as a classical limit of "quantum geometries" associated to the respective conformal field theories. In [42] a method was proposed how to extract non-commutative geometries out of CFTs. In fact, the algebra A°° can be regarded as the limit of certain subalgebras of these non-commutative geometries. This, on the one hand confirms that the construction of our limit structures is natural. On the other hand, it gives the possibility to identify "non-commutative geometric structures" in CFTs. These could be used to carry over the powerful geometric tools appli- cable to the study of sigma models in large volume limits to regions inside the CFT moduli spaces. This suggests that non-commutative geometry in terms of Connes' spectral triples [19, 20, 21] seems to be a valuable tool in the study of degenerations of conformal field theories. This chapter is organized as follows: In the first Sect. 3.1 we explain how non-commutative geometries can be extracted from CFTs, after giving a brief overview of some of the basic concepts. Sect. 3.2 contains our definitions of sequences, convergence, and limits, and is the technical heart of this part of the work. Moreover, the geometric interpretations of degenerate limits are dis- cussed. Sect. 3.3 is devoted to the study of torus models, orbifolds thereof, and degenerations of current-current deformed WZW models, where we exemplify our techniques. In Sect. 3.4 we present our results on the A-series of unitary Virasoro minimal models. We end with a discussion in Sect. 3.5. Several appen- dices contain background material and lengthy calculations. 3.1 From geometry to CFT, and back to geometry String theory establishes a natural map which associates CFTs to certain, some- times degenerate geometries. Conversely, one can associate a GEOMETRIC IN- TERPRETATION to certain CFTs, and the latter construction is made precise by using Connes' definition of spectral triples and non-commutative geometry. In Sect. 3.1.1 we very briefly remind the reader of SPECTRAL TRIPLES, ex- plaining how they encode geometric data. Somewhat relaxing the conditions on spectral triples we define SPECTRAL PRE-TRIPLES which will be used in Sect. 39 3.1.2. There, we recall the basic structure of CFTs and show how to extract spectral pre-triples from them. If the spectral pre-triple defines a spectral triple, then this will generate a non-commutative geometry from a given CFT. In Sect. 3.1.3 we explain how in favorable cases we can generate commutative geometries from CFTs. In the context of string theory, this prescription gives back the original geometric data of the compactification space. Much of this Sect. 3.1 consists of a summary of known results [20, 42, 21, 87, 72], but it also serves to introduce our notations. 3.1.1 From Riemannian geometry to spectral triples For a compact Riemannian manifold (M,g), which for simplicity we assume to be smooth and connected, the spectrum of the associated Laplace-Beltrami operator Ag:C°°(M) —> C°°(M) encodes certain geometric data of (M,g). However, in general one cannot hear the shape of a drum, and more information than the set of eigenvalues of Ag is needed in order to recover (M, g). By the Gel'fand-Naimark theorem, the point set topology of M is completely encoded in C°(M) = C°°(M): We can recover each point p G M from the ideal of functions which vanish at p. In other words, given the structure of C°°(M) as a C*-algebra and its completion C°(M), M is homeomorphic to the set of closed points of Spec(ÖM), where OM is the sheaf of regular functions on M. Connes' dual prescription uses C*-algebra homomorphisms x: C°°(M) —• C, instead, such that pgM corresponds to Xp: C°°(M) —> C with xp(f) := f(p)'> the Gel'fand-Naimark theorem ensures that for every commutative C*-algebra A there exists a Hausdorff space M with A = C°(M). M is compact if A is unital. Example 1.1 + 1 Let /JeR , then M = §R = R / ~ with coordinate x ~ x + 2TTR has the Laplacian A = — J^. Its eigenfunctions \m)R,m € Z, obey Z: \m)R: x -> e ; \*\m)R &\m)R; Vm,m' eZ: \m)R • \m')R = |m + m')ß, and they form a basis of C°(M) and C°°{M) with respect to the appropriate norms. Any smooth manifold is homeomorphic to §R, equipped with the Zariski topology, if its algebra of continuous functions has a basis fm, m G Z, which obeys the multiplication law fm • fm> = fm+m>. To recover the Riemannian metric g on M as well, we consider the SPECTRAL 2 TRIPLE (H :=L (M,dvolg),iI := ^A9,^:=C°°(M)), where H is viewed as self-adjoint operator which is densely defined on the Hubert space HI, and A is interpreted as algebra of bounded operators which act on elements of H by pointwise multiplication. Following [20, 42, 21], we can define a distance func- tional dg on the topological space M by considering 2 2 T:={feA I Gf:=[f,[H,f]] = -(f oH + Hof )+2foHof obeys \/heC°°{M): \Gfh\<\h\}. 40 One now sets Vx,yeM: dg{x,y):=swp\f{x)-f{y)\. (3.2) l 2 In Ex. 1.1 with M = § R one checks that for all f,h £ C°°{M):Gfh = (f') h, and in general Gfh = g(Vf,Vf)h. In fact, by definition [8, Prop. 2.3], any second-order differential operator O satisfying [/, [|O, /]] = V/ € A : V/ := [P, /]: H' -> HI'; V/i € A : [V/, h] = 0, (3.3) where in the above examples V/ acts on HI' by Clifford multiplication, and that A gives smooth coordinates on an "orientable geometry"; furthermore, there are fmiteness and reality conditions as well as a type of Poincare duality on the K-groups of A. If all these assumptions hold, then by (3.2) the triple (HI', V, A) defines a non-commutative geometry ä la Connes [19, 20, 21]. If the algebra A is commutative, then the triple {W,V, A) in fact defines a unique ordinary Riemannian geometry (M,g) [21, p. 162]. The claim that the differentiable and the spin structure of (M, g) can be fully recovered is detailed in21 [87]. Following [42], instead of studying SPECTRAL TRIPLES (E.',V,A), we will be less ambitious and mainly focus on triples (H, H, A), somewhat relaxing the defining conditions: Definition 1.2 We call (H, H, A) a SPECTRAL PRE-TRIPLE if H is a pre-Hilbert space over C, H is a self-adjoint positive semi-deßnite operator on M with Hop '•= ker(if) = C, and A is an algebra of operators acting on H. Since ?io,o - C 3 1, we can map A —>mbyAi-*A-l. 20In local coordinates a generalized Laplacian can thus be written as in (3.6) below. 21We thank Diarmuid Crowley for bringing this paper to our attention. 41 If additionally the eigenvalues of H have the appropriate growth behavior, i.e. for some 7 G R and V eR: N(E) := dime 0 W € H | Hip = \ . V /, h E A: (V/, V^EndH' = 2{/, Hh)u (3.5) and such that (H', V, A) obeys the seven axioms of non-commutative geometry, then we call (H, H, A) a SPECTRAL TRIPLE or a NON-COMMUTATIVE GEOMETRY OF DIMENSION 7. Remark 1.3 Note that in the geometric setting our condition (3.5) for the operator H does 2 not imply H — \ As on L (M, dvols). In fact, H will in general be a generalized Laplacian with respect to a metric 'g = (gy) in the conformal class of g. More 2 1 precisely, we will have dvol5 = e~ *dvolö with $ G C°°(M), and with 'g~ = 2H = -e^x/detgr1^ ^e"2*Vciet|^dj (3.6) with respect to local coordinates, in accord with (3.5). We call g the DILATON CORRECTED METRIC with DILATON $. Note that 42 3.1.2 Spectral triples from CFTs We do not attempt to give a complete definition of CFTs in this section; the in- terested reader may consult, e.g. , [7, 80, 56, 78, 40, 46]. Some further properties of CFTs that are needed in the main text are collected in App. A. A UNITARY TWO-DIMENSIONAL CONFORMAL FIELD THEORY (CFT) is spec- ified by the following data: • a C-vector space H of STATES with scalar product (-|-). This scalar product is positive definite, since we restrict our discussion to unitary CFTs; • an anti-C-linear involution * on 7i, often called CHARGE CONJUGATION; • an action of two commuting copies Virc, Virc of a Virasoro algebra (A.I) 22 23 with central charge c € R on W, with generators Ln, Ln, n 6 Z, which commutes with *. The Virasoro generators LQ and LQ are diagonalizable on 7i, such that H. decomposes into eigenspaces24 «fc,*. (3-7) h-heZ and we set Tih ^ := {0} if h — h $ Z. The decomposition (3.7) is orthogonal with respect to (-|-); + a GROWTH CONDITION for the eigenvalues H, h in (3.7): For some V G K :