- Alain Bruguières: "Internal Reshetikhin-Turaev TQFTs"
A Reshetikhin-Turaev TQFT is associated with any modular category. On the other hand, a premodular category whose symmetric center is tannakian can be modularized, and the corresponding modular category gives rise to a TQFT. Can one construct the TQFT associated with the modularization of a premodular category already at the level of the premodular category. Following the recent thesis of Mickael Lallouche, I will show how a ribbon category C with coend gives rise to an internal TQFT, which takes values in the symmetric center T of C. In the case where C is premodular and T is tannakian, the internal TQFT, composed with the fibre functor $T \mapsto C$, yields the RT TQFT of the modularization of C. The construction of the internal TQFT does not require any linearity conditions. I will also consider the question whether, in the premodular case, such internal TQFTs arise in cases where the premodular category is not modular. If a premodular category is such that T is generated by an invertible object J with dimension -1 and twist 1, then it is not modularizable, and in that case the internal TQFT takes values in super vector spaces. But at this point we don't have an example of such a situation which is not 'split' (that is, where C is not the tensor product of a modular category with super vector spaces). We do have examples which are not ribbon (the twist is not self-dual).
- Jürgen Fuchs: "Universal correlators for logarithmic conformal field theories"
Given a modular finite ribbon category D, one can associate to any punctured closed surface M a functor Bl_M from a tensor power of D to finite-dimensional vector spaces. The so obtained vector spaces Bl_M(-) carry representations of mapping class groups and are compatible with sewing, in much the same way as the spaces of conformal blocks of rational conformal field theories.
I will present a universal construction which, given any object F of D, selects vectors in all spaces Bl_M(F,...,F) (i.e. when each
puncture is labeled by F). If and only if the object F carries a structure of a 'modular' commutative symmetric Frobenius algebra
in D, the so obtained vectors are invariant under the mapping class group actions and are mapped to each other upon sewing.
Thereby they are expected to describe the bulk correlators of a full CFT with bulk state space F. For non-semisimple D this CFT
is a logarithmic CFT.
For generalizing the construction to surfaces with boundary and with insertions of boundary fields one should take D to be the
enveloping category of a finite ribbon category C and make use of the central monad on C. I will discuss first steps towards such a
generalization.
(Based on arXiv 1604.01143 and 1712.01922)
- Rinat Kashaev: "Quantum dilogarithms and mapping class group representations"
A gaussian group is a Pontryagin self-dual locally compact abelian group together with a fixed gaussian exponential that is a symmetric second order character associated with a non-degenerate self-pairing. I will explain how a quantum dilogarithm over a gaussian group can be used for construction of (projective) unitary representations of the mapping class groups of punctured surfaces of negative Euler characteristic.
- Anna Lachowska: "The center of the small quantum group in type A and the diagonal coinvariants"
Understanding the structure of the center of the small quantum group u_q(g) at a root of unity is important for the study of the fusion categories and CFTs associated with this algebra. Until recently the dimension of the center was known only for g=sl_2. Based on the geometric description of the blocks of the small quantum group, we develop an algorithmic method for calculating the dimension of the regular and singular blocks of the center. The answers obtained in cases sl_3 and sl_4 allow us to formulate a conjecture relating the center of the principal block of u_q(sl_n) with Haiman’s diagonal coinvariant algebra of the symmetric group S_n. Applying the same method to the singular blocks leads to an intriguing combinatorial conjecture for the dimension of the whole center of u_q(g) in type A. This is a joint work with Qi You (Caltech).
- Marco de Renzi: "Renormalized Hennings invariants and TQFTs"
Non-semisimple constructions in quantum topology produce strong invariants and TQFTs with unprecedented properties. The first family of non-semisimple quantum invariants of 3-manifolds was defined by Hennings using finite-dimensional unimodular ribbon Hopf algebras satisfying a certain non-degeneracy condition. This enabled Lyubashenko to build mapping class groups representations out of every finite-dimensional factorizable ribbon Hopf algebra. Further attempts at extending these constructions to TQFTs only produced partial results, as the vanishing of Hennings invariants in many crucial situations made it impossible to treat non-connected surfaces. We will show how to overcome these problems. In order to do so, we will first renormalize Hennings invariants through the use of modified traces. When working with factorizable Hopf algebras, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a fully monoidal TQFT yielding mapping class group representations in Lyubashenko’s spaces. This is a joint work with Nathan Geer and Bertrand Patureau.
- Ingo Runkel: "The non-semisimple Verlinde formula and pseudo-trace functions"
One of the more surprising outcomes of the theory of vertex operator algebras is a relation between the braided monoidal structure on its category of representations and the modular properties of characters. While in the finitely semisimple setting, this is a theorem, the generalisation to the non-semisimple situation is only conjectured. I this talk I would like to explain this conjecture and how it can be verified in the example of symplectic fermions. This is joint work with Azat Gainutdinov.
- Katrin Wendland: "K3 theories and how to reflect some of them to vertex operator algebras"
Physics predicts that so-called non-linear sigma model constructions yield conformal field theories associated with any choice of complex structure and Ricci-flat metric on a compact Calabi-Yau manifold. One thus expects to obtain many-parameter families of such conformal field theories, surpassing rationality by far. In the case of K3 theories, these expectations can be made very concrete. In this talk, we will review the notion of K3 theories, and we will present a new technique, developed in joint work with Anne Taormina, which transforms certain conformal field theories into vertex operator algebras and their admissible modules, thus capturing a major part of the structure of the conformal field theory in terms of vertex operator algebra.
- Simon Wood: "Module classification through free fields and symmetric functions"
Abstract: It is well known that the module categories of rational conformal field theories and vertex algebras have rich structure and it is suspected that much of this structure generalises to suitable non-rational cases. However, it can be very hard in practice to classify the isomorphism classes of objects in these categories. In this talk I will present recent work on systematic approaches to classifying simple modules over certain simple quotients of universal vertex algebras.