Schedule for: 18w5131 - Geometric and Categorical Aspects of CFTs

Factorization algebras, and factorization homology, began in the work of Beilinson-Drinfeld, as an algebro-geometric/coordinate-free approach to vertex algebras and conformal blocks, respectively. They were re-interpreted by Costello-Gwilliam as a framework for algebras of observables in quantum field theory. A special class, the so-called "locally constant" factorization algebras received special attention from Lurie, Ayala-Francis, and Scheimbauer in the context of fully extended topological field theories. In the first lecture I shall recall this history, define factorization homology in the mold of Ayala-Francis, and recall the key property of excision, which both uniquely determines factorization homology as a functor, and gives an effective mechanism for its computation. In the second lecture, I will turn to examples in geometry and representation theory, following Ben-Zvi-Francis-Nadler, and our works with Ben-Zvi-Brochier and Brochier-Snyder. Specializing the "coefficients" to lie in presentable k-linear categories (the natural home of algebraic geometry and representation theory), one recovers character varieties, and their canonical quantizations, as a computation in factorization homology.

There are three intertwined schools of thought in the world of factorization algebras. First, chronologically, is the theory of Beilinson-Drinfeld in their work on chiral algebras. Next, there is the Lurie, Francis-Ayala approach which is primarily the setting in which David Jordan’s talks are in. Finally, there are factorization algebras in the style of Costello-Gwilliam. Each of these approaches have their own advantages. In this talk, I will focus on the third option. In the topological case, the theory agrees with that of Lurie/Francis-Ayala. The primary advantage of this approach is that it is more intrinsic to the underlying geometry. In complex dimension one, for instance, there is the theory of *holomorphic* factorization algebras. We will see how this notion encodes the operator product expansion (OPE) for chiral CFT, while also providing some geometric examples. We will also see how factorization homology appears in this approach to factorization.

In this talk, I will give a positive answer to the following question: given a modular tensor category C, is there a mathematical structure such that its center is C? This question is crucial to the question of how to extend Reshetikhin-Turaev TQFT’s down to points. The idea comes from physics, more precisely, from the boundary-bulk relation of 2d topological orders with a chiral gapless edge. It was long believed that such an edge is described by a chiral conformal field theory. The key is to make this statement mathematically precise. This is a joint work with Hao Zheng.

We propose a new link between quantum invariants of 3-manifolds and the geometry of wild Hitchin moduli spaces. The construction goes through a class of four-dimensional quantum field theories known as Argyres-Douglas theories. Every such theory realizes a wild Hitchin space as its Coulomb branch and defines a VOA on the Higgs branch. The latter can be used to construct a non-unitary modular tensor category, which leads to 3d TQFTs that are generically semisimple but non-unitary. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye.

Rastelli et. al have constructed a map from 4d $N=2$ SCFTs to VOAs in such a way that the Schur index of a 4d $N=2$ SCFT coincides with the character of the corresponding VOA. Later, Rastelli and Beem have further conjectured that the Higgs branch of a 4d $N=2$ SCFT should coincide with the associated variety of the corresponding VOA. In my talk we confirm the conjecture of Rastelli and Beem for the theory of class S.

Many properties of an algebraic variety X can be expressed in terms of the derived category of coherent sheaves on X (or its differential-graded enhancement). Kontsevich proposed to view arbitrary smooth and proper dg-categories as non-commutative analogs of smooth projective varieties. I will show how holomorphic functions with isolated singularities fit into this picture. In the first part of the lecture we will talk about Chern characters and the Hirzebruch-Riemann-Roch theorem for dg-categories and will see how classical invariants of singularities appear via dg-categories of matrix factorizations. Then we will turn to quantum invariants and cohomological field theories - an algebraic structure underpinning formal properties of the Gromov-Witten invariants. For a quasi-homogeneous singularity W and a finite group G of its symmetries we will describe a CohFT whose state space is the equivariant local algebra (Milnor ring) of W and whose correlators can be viewed as analogs of Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). The role of the virtual fundamental class from the Gromov-Witten theory is played here by a "fundamental matrix factorization" over a certain moduli space.

In this talk, we will review a proposal by B. Eynard and my self to solve W-symmetric Ward identities on the sphere at generic values of the background charge (arXiv:1801.03433v2). It will consist in the introduction of a hierarchy of so-called generalized Ward identities, their solution in topological regimes by topological recursion for quantum spectral curves and eventually obtaining the conformal blocks by integrating special geometry relations. This can also be interpreted as a computation in refined topological string theory where the refinement parameter is identified with the background charge.

To any vertex algebra, one can attach in a canonical way a certain Poisson variety, called the associated variety. When the associated variety has only finitely many symplectic leaves, the vertex algebra is called quasi-lisse. In this talk, I will give various examples of quasi-lisse vertex algebras. Using the notion of chiral symplectic leaves, one can show that any quasi-lisse vertex algebras is a quantization of the arc space of its associated variety. If time allows, I will also give an application to the arc space of Slodowy slices and $W$-algebras.

Among the invariants that have a double life in geometry and in conformal field theory, there are the Euler characteristic and its refinements to complex elliptic genera. We argue that superconformal field theory motivates further refinements, resulting in a choice of several new invariants, the so-called Hodge-elliptic genera. At least for K3 surfaces and K3 theories, higher algebra and conformal field theory select the same refinement as the most natural one, allowing insights into generic features of K3 theories.

By definition, the endomorphism spaces of tensor powers of objects of a braided tensor category carries a representation of the braid group. For Lie types A and C, this can be used to classify all braided tensor categories whose fusion ring is the one of the representation category of the related Lie algebra. We also discuss the situation for other classical Lie types and some exceptional types. There are several different ways how to construct TQFTs and modular functors. One of the motivations for these categorical questions was to decide when these constructions yield the same results.

In the past ten years, an inductive procedure called topological recursion proved to provide a very efficient way of solving many problems of enumerative geometry through the computation of intersection of tautological classes over the Deligne-Mumford compactification of the moduli space of Riemann surfaces. In order to understand the mysterious geometric origin of this procedure, we developed a new inductive procedure called geometric recursion together with Andersen and Borot. This geometric recursion is a machinery defining all sorts of mapping class group invariant objects attached to surfaces. In this lecture, I will first review the original topological recursion formalism together with its application to the study of Cohomological Field Theories before presenting the formalism of geometric recursion. I will finally present some examples of application including some generalisation of Mirzakhani-McShane identities and the computation of some closed forms on Teichmuller space.

In this talk I will survey some results on and techniques to study actions of (finite-dimensional) Hopf algebras on noncommutative algebras. Many examples will be provided and the categorical context for such results will be emphasized.

In this introductory lecture I would like to discuss vertex operator algebras which allow for non-semisimple representations. There will be two parts, which will roughly relate to genus 0 and genus 1 properties. In the first part I will try to motivate why one might want to look at such VOAs and give the basic example, symplectic fermions. We will briefly look at VOA modules and intertwiners to get an idea of how the category of VOA modules can become a braided monoidal category. For the second part we look at certain traces over VOA modules and modular properties of conformal blocks on the torus. These have an intriguing relation to tensor product multiplicities, known as the Verlinde formula, which is a theorem for finitely semisimple theories and a conjecture without the semisimplicity assumption.

Via a Lego-Teichmüller game, the correlators of a conformal field theory can be obtained from sewing a small number of fundamental correlators. For the case of a full CFT based on a (not necessarily semisimple) modular finite ribbon category D, this construction is completely under control for correlators on closed world sheets. In particular, it can be shown that the possible consistent systems of correlators are in bijection with modular Frobenius algebras F in D, and there is a universal formula expressing each correlator through the data of D and F. I will present the main ideas that lead to these results. In order to obtain also correlators for surfaces with boundary and with insertions of boundary fields, one has to take D to be the center of a finite ribbon category C and make use of the connections between C and Z(C) via the central monad and comonad. I will discuss some aspects of ongoing work that will eventually lead to full-fledged analogues of the results for closed world sheets.

Unitary vertex operator algebras and conformal nets are mathematical axiomatizations of roughly the same physical idea: a two-dimensional unitary chiral conformal field theory. From a mathematical perspective, the axiomatizations are quite different in nature, and physical ideas which have been rigorously proven in one framework sometimes remain quite difficult in the other. In this talk I will explain a positivity conjecture for unitary VOAs which arose in the course of ongoing work to compare the theory of tensor products (i.e. fusion of sectors) as it appears in both settings. The conjecture is not limited to rational VOAs, and it suggests a general construction of unitary tensor products of modules. As time allows, I will present ongoing work to relate the representation theory of conformal nets and VOAs, as well as work in progress to establish the positivity conjecture in a broad class of examples.

Modifed trace theory allows construction of invariants using non trivially the non semisimple structure of quantum groups at root of $1$. We discuss the invariants and TFTs which are obtained using modified trace on several versions of quantum $\mathfrak{sl}(2)$.

In recent years, subfactor methods have constructed hundreds of new fusion categories, with strong evidence these live in infinite families. Taking their doubles, we obtain hundreds (and probably infinite families) of new modular tensor categories. These modular tensor categories have a distinctive appearance. Abstracting this appearance, it is natural to guess that there is a new construction, which we call the smashed-sum, combining old modular tensor categories into new ones. It is tempting to guess the smashed-sum construction also lives in the VOA world, in fact this may be its natural home. An example of such a VOA would be the mythical Haagerup VOA.