Conformal Field Theories

Session 10 Conformal Field Theories Chairs: Yasuyuki Kawahigashi (University of Tokyo, Japan) Liang Kong (Tsinghua University, China) Christoph Schweigert (Hamburg University, Germany)

Titles and Abstracts Tomoyuki Arakawa (Kyoto University, Japan) Title: Rationality and $C_2$-cofiniteness of W-algebras Abstract: We prove the rationality and the $C_2$-cofinitenss of all minimal series W- algebras associated with simple Lie algebras and their principal nilpotent elements. This has been a long standing problem since the work of Frenkel, Kac and Wakimoto in 1992. This talk partly is based on a joint work with Maria Gorelik.

David Evans(Cardiff University, UK) Title: The search for the exotic - subfactors and conformal field theories Abstract: Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories and almost Calabi-Yau algebras. I will describe some recent work which is also motivated by links with twisted equivariant K-theory through the K-theoretic realisation of Freed-Hopkins-Teleman of the Verlinde algebra of primary fields. This is joint work with Terry Gannon and Mathew Pugh.

Jens Fjelstad (Nanjing University, China) Title: Finite groups acting on conformal blocks Abstract: In any rational chiral CFT with a non-trivial simple current, we construct a finite group acting on the corresponding conformal blocks. Our groups are generalisations of the finite Heisenberg groups appearing in U(1) Chern-Simons theory and the chiral CFT of a compactified free boson, and of finite groups appearing in the SU(2) TFT's constructed by Blanchet et al. The action of these finite groups have implications for the mapping class group representations carried by conformal blocks, as will be illustrated by examples in the case of chiral SU(N) WZW models.

Jürgen Fuchs (Karlstads Universitet, Sweden) Title: All-genus correlators from factorizable Hopf algebras Abstract: I will describe elements of suitable morphism spaces in the category of bimodules over a factorizable ribbon Hopf algebra H that are invariant under a natural action of the mapping class group of m-punctured genus-g Riemann surfaces, for any m and g. These elements are candidates for the correlation functions in conformal field theories whose category of chiral sectors is equivalent to the category of H- modules.

Terry Gannon (University of Alberta, Canada) Title: Much ado about Mathieu Abstract: A couple years ago, Eguchi-Ooguri-Tachikawa observed that the elliptic genus of type II string theory on K3 surfaces appears to possess a Moonshine for the largest Mathieu group, the group which Conway calls the most remarkable. I'll review this observation, and explain how it can be proved.

Ching Hung Lam (Academia Sinica, Taiwan, China (Taipei)) Title: Classification of holomorphic framed vertex operator algebras of central charge 24 Abstract: We will discuss the classification of holomorphic framed vertex operator alegrbas of central charge 24 using certain binary codes and quadratic spaces over Z2.

Antun Milas (State University of New York at Albany, USA) Title: W-algebras and integral lattices Abstract: In the first part of the talk, I will introduce certain remarkable W-algebras coming from infinite order automorphisms of lattice vertex algebras. I will report on a recent progress obtained for the algebras of "rank one", whose representation theory has been worked out in many details. In the second part, I will discuss the modular properties of characters of their representations and the related combinatorics, including various q-series identities.

Michael Mueger (Radboud University Nijmegen, Netherlands) Title: The Witt group of modular tensor categories Abstract: I will give an overview of my recent joint work with A.Davydov, D. Nikshych and V.Ostrik (J.Reine Angew. Math. 2012). This work promises to be relevant for the classification of modular categories and has many applications to conformal field theory.

Ingo Runkel (University of Hamburg, Germany) Title: A braided monoidal category for symplectic fermions Abstract: Understanding the category of representations of the chiral symmetry algebra of a conformal field theory is an important step in investigating the properties of such theories. For logarithmic CFTs, these categories are non-semisimple and not many examples are known in detail. I would like to present a class of braided monoidal categories which includes the Ising model and symplectic fermions as special cases. Symplectic fermions form one of the most basic examples of logarithmic conformal field theories and I will point out some of the properties of the corresponding braided category. Rafał Roman Suszek (Uniwersytet Warszawski, Poland) Title: Gauge defect networks in 2d CFT Abstract: I shall give an interpretation of the gauge anomaly of the two-dimensional multi-phase sigma model in terms of an obstruction to the existence of a topological defect network implementing the action of the gauge group in the gauged sigma model. A link to the construction of principal bundles with a structural (action) groupoid over the world-sheet will be established, and a relation to the Courant- algebroidal interpretation of the small gauge anomaly will be discussed.

Feng Xu (University of California at Riverside, USA) Title: On questions related to intermediate subfactors Abstract: In this talk we discuss questions of intermediate subfactors motivated by some old questions on subgroups of finite groups and report some recent progress on examples related to Hopf algebras, fusion categories and conformal field theory.

Hiroshi Yamauchi (Tokyo Woman’s Christian University, Japan) Title: Extended Griess algebras, conformal designs and Matsuo-Norton trace formulae Abstract: We introduce a notion of the extended Griess algebra of a vertex operator superalgebra with an involution and derive formulae expressing the trace of the composition of several (up to 5) adjoint actions of elements of the Griess algebra based on conformal design structure. We also present its application to the Baby- monster vertex operator superalgebra.