DE06FC864 UNIVERSITÄT BONN Physikalisches Institut Conformal Field Theory Between Supersymmetry and Indecomposable Structures von Holger Eberle Post address: BONN-IR-200&-08 Nussallee 12 Bonn University 53115 Bonn July 2006 Germany ISSN-0172-8741 UNIVERSITÄT BONN Physikalisches Institut Conformal Field Theory Between Supersymmetry and Indecomposable Structures von Holger Eberle Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaft- lichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn angenommen. Diese Dissertation ist auf dem Hochschulschriftenserver der Universitäts- und Lan- desbibliothek Bonn http://hss.ulb.uni-bonn.de/dissjonline elektronisch publiziert. 1. Referent: Prof. Dr. Werner Nahm 2. Referent: Prof. Dr. Hans-Peter Nilles Tag der Promotion: 12. Juli 2006 Erscheinungsjahr: 2006 Abstract This thesis considers conformal field theory in its supersymmetric extension as well as in its relaxation to logarithmic conformal field theory. Compactification of superstring theory on four-dimensional complex manifolds obey- ing the Calabi-Yau conditions yields the moduli space of N — (4,4) sup er conformal field theories with central charge c = 6 which consists of two continuously connected subspaces. This thesis is concerned with the subspace of K3 compactifications which is not well known yet. In particular, we inspect the intersection point of the Z2 and Z4 orbifold subvarieties within the K3 moduli space, explicitly identify the two corre- sponding points on the subvarieties geometrically, and give an explicit isomorphism of the three conformal field theory models located at that point, a specific Z2 and a Z4 orbifold model as well as the Gepner model (2)4. We also prove the orthogonality of the two subvarieties at the intersection point. This is the starting point for the programme to investigate generic points in K3 moduli space. We use the coordinate identification at the intersection point in order to relate the coordinates of both subvarieties and to explicitly calculate a geometric geodesic between the two subvarieties as well as its generator. A generic point in K3 moduli space can be reached by such a geodesic orig- inating at a known model. We also present advances on the conformal field theoretic side of deformations along such a geodesic using conformal deformation theory. Since a consistent regularisation of the appearing deformation integrals has not been achieved yet, the completion of this programme is still an open problem. Moreover, we regard a relaxation of conformal field theory to logarithmic confor- mal field theory. The latter allows the indecomposable action of the LQ Virasoro mode within a representation of the conformal symmetry. In particular, we study general aug- mented cPtq minimal models which generalise the well-known (augmented) cp^ model series. We calculate logarithmic nullvectors in both types of models. But most impor- tantly, we investigate the low lying Virasoro representation content and fusion algebra of two general augmented cp,q models, the augmented 02,3 = 0 model as well as the augmented Yang-Lee model at 02,5 = —22/5. These exhibit a much richer structure as the cP;i models with indecomposable representations up to rank 3. We elaborate several of these new rank 3 representations in great detail and uncover astonishing features. Furthermore, we argue that irreducible representations corresponding to the Kac table domain of the proper minimal models cannot be included into the theory. In particular, the true vacuum representation is rather given by a rank 1 indecomposable but not irreducible subrepresentation of a rank 2 representation. We generalise these generic examples to give the representation content and the fusion algebra of general augmented cP](? models as a conjecture. Finally, we open a new connection between logarithmic conformal field theory and quantum spin chains by relating some represen- tations of the augmented 02,3 = 0 model to the representation content of a c = 0 model which describes an XXZ quantum spin chain. Contents 1 Introduction 7 2 Conformal field theory 15 2.1 General structures in conformal field theory 15 2.2 Logarithmic conformal field theory 24 2.3 Important models 26 2.3.1 The Kac determinant and minimal models 26 2.3.2 Torus sigma models 28 2.3.3 Orbifold sigma models 35 2.3.4 Supersymmetric minimal models and Gepner models 40 3 Observations on K3 moduli space 47 3.1 Introduction to the geometry of the K3 moduli space 48 3.2 Geometric intersection of the Z2 and Z4 subvarieties 50 3.2.1 Reference lattices and four-planes for Z2 and Z4 orbifold models 50 3.2.2 Identification of lattice vectors at the intersection point 55 3.3 Identification of CFTs at the (2)4 Gepner point 58 3.3.1 The symmetry algebra of £(Z4,0) 59 3.3.2 The symmetry algebra of Z4(^D4,B*) 60 3.3.3 Non-twisted groundstate {\,\) fields 61 3.3.4 Twisted groundstate (|, |) fields in Z4(^=£>4,-B*) 62 3.3.5 Twisted groundstate {\,\) fields in/C(Z4,0) 65 3.3.6 Fields of the Gepner model (2)4 66 3.3.7 Explicit identification of the three theories 68 3.4 A geometric geodesic in K3 moduli space 70 3.4.1 Finding the transition matrix 70 3.4.2 Calculating the generator of the geodesic 73 3.5 Deforming CFT along a geodesic 77 3.5.1 Conformal deformation theory 77 3.5.2 Conformal deformations in K3 moduli space 80 3.5.3 Calculation of the two-loop integral 82 4 Higher rank indecomposable structures 97 4.1 What are augmented minimal models? 98 4.2 Exploring logarithmic nullvectors 102 4.2.1 Jordan cells on lowest weight level 103 4.2.2 Logarithmic nullvectors for cP)i 105 4.2.3 Possible logarithmic nullvectors for C2,3 = 0 108 4.3 How to calculate fusion products 112 4.3.1 The Nahm algorithm 113 4.3.2 Constraints for the fusion algebra 116 4.3.3 Implementation for the cVA models 117 4.4 Fusion analysis of the augmented 02,3 = 0 model 118 4.4.1 Higher rank representations 118 4.4.2 Explicit calculation of the fusion products 135 4.4.3 Consistency of fusion products 135 4.4.4 An unexpected connection to quantum spin chains 137 4.5 Fusion analysis of the augmented Yang-Lee model 138 4.6 Representations and fusion for augmented cVA models 142 4.6.1 The spectrum of representations 142 4.6.2 The fusion of irreducible representations 144 4.6.3 The fusion with higher rank representations 147 5 Conclusion 149 A Explicit Lo for rank 3 representations 155 A.I 7^3>(0,0,1,1) 155 A.2 7^(0,0,2,2) 156 A.3 n®(0,1,2,5) 157 B Fusion rules for two augmented models 159 B.I Fusion rules for C2,3 = 0 159 B.2 Fusion rules for c2]5 = -22/5 171 C Examples of logarithmic nullvectors 173 C.I An explicit nullvector for 03,1 = —7 173 C.2 Explicit nullvectors on the border of 02,3 = 0 174 C.3 Explicit nullvectors in the bulk of 02,3 =0 178 Bibliography 181 Chapter 1 Introduction Reality is a question of perspective; the further you get from the past, the more concrete and plausible it seems—but as you ap- proach the present, it inevitably seems more and more incredible. Salman Rushdie: Midnight's Children Conformal field theory (CFT) has become a very important and vibrant area of re- search since its break-through on the mathematical physics' stage in the mid 80's. The mathematical elaboration of the Virasoro algebra [1], the algebra of local conformal transformations, and its representation theory [2, 3] inspired Belavin, Polyakov and Zamolodchikov to present the first concise study of two-dimensional quantum field the- ory which exhibits local conformal invariance and, at the same time, to construct an important series of examples, the so-called minimal models [4]. This was the final step of a long development in which conformal symmetry was slowly incorporated into a quantum field theoretic framework. Indeed, already in 1958, Thirring had constructed the first quantum CFT model in two dimensions, the famous Thirring model [5], not realising its local conformal properties at that time. This model is known today as one of the continuous CFT torus models at central charge c = 1. For a very enlightening review of this exciting historical development we refer to [6]. From that time on, conformal field theory quickly became a key ingredient in nu- merous branches of mathematical physics. It is important for the description of two- dimensional phase transitions in statistical physics (see e.g. [7, 8, 9]) as well as for the description of the quantised string theory worldsheet (see e.g. [10, 11]). And even in the modern string theoretic approach in the context of D-branes the so-called bound- ary conformal field theory provides the microscopical description [12, 13, 14, 15, 16]. Furthermore, implications of CFT can be found in Seiberg Witten theory [17], in solid state physics as e.g. the quantum Hall effect [18, 19, 20], and certainly in the whole field of integrable systems. And perhaps even most importantly, CFT has inspired quite a lot of modern mathematical work with such topics as vertex operator algebras [21, 22, 23] or mirror symmetry [24, 25, 26]. 8 Chapter 1. Introduction But the pure Virasoro algebra of local conformal transformations does not seem to be sufficient for current applications. The direction of research has, hence, diversified in several directions. Either the conditions of local conformal symmetry are restricted further by the inclusion of more symmetries-, especially for theories with a lot of param- eters, which e.g.