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UNIVERSITДT BONN Physikalisches Institut 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.
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