Beijing Institute of Mathematical Sciences and Applications

Grothendieck-Verdier categories are monoidal categories endowed with a form of duality that generalizes rigidity. After introducing and motivating this notion I will discuss module categories over Grothendieck-Verdier categories, Frobenius algebras internal to Grothendieck-Verdier categories, and the connections between them. An important role is played by lax and oplax module functor structures on the action functors of a Grothendieck-Verdier module category. The talk is based on work [2306.17668 & 2405.20811] with G. Schaumann, C. Schweigert and S. Wood.

We will introduce the n-sphere function on n-dimensional lattice models and derive a spherical n-category from it. Then we introduce the alterfold construction of the n+1 topological quantum field theory from an n-sphere function with reflection positivity and finiteness. The alterfold TQFT has two colors A and B for the n+1 cell. The A-color theory contains a higher analogue of the Turaev TQFT and the B color theory contains a higher analogue of the Drinfeld center and the Reshetihkin-Turaev TQFT. In particular, we construct a 3-sphere function and a 3+1 alterfold TQFT of Ising type from it.

The conjectural bulk-boundary correspondence says that certain topological invariants of gapped lattice systems are equal to 't Hooft anomalies of their boundary field theories. Usually, this is argued by assuming that the bulk is described at low energies by a TQFT. The simplest example is a 2d system exhibiting the quantum Hall effect, which is assumed to be described by a Chern-Simons field theory. In this talk I will explain how to formulate the bulk-boundary correspondence as a purely bulk statement and then prove it. The bulk counterpart of 't Hooft anomaly is an obstruction to promoting a global symmetry of a gapped state to a local one. To define what this means, I will introduce the notion of a local Lie algebra which encodes the physical notion of a gauge symmetry and can be useful in other contexts too.

For C2-cofinite (self-dual) rational VOAs, the sewing-factorization theorem and the Verlinde formula is now a rigorous mathematical theorem by many people (Zhu, Huang, Damiolini-Gibney-Tarasca, G., etc.). Much less is know if the VOA is C2-cofinite but irrational. In this talk, I will present a version of sewing-factorization theorem for C2-cofinite VOAs. I will explain how this framework leads to a non-semisimple Verlinde formula (assuming that certain VOA modules are rigid). Our results confirm some of the conjectures by Gainuditnov-Runkel. This is joint work with Hao Zhang.

Given a 3-manifold M and a branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M, we define a map from the HOMFLYPT skein module of M to that of L. The definition is by counting holomorphic curves, but the theory of Morse flow graphs gives a more combinatorial prescription, which we make completely explicit in the case of branched double covers. After specializing to the case where M is a surface times an interval, and additionally specializing the HOMFLYPT skein to the gl(2) skein on M and the gl(1) skein on L, we recover the existing prescription of Neitzke and Yan, and the resulting map is a close cousin of the quantum trace map of Bonahon and Wong. When M is a surface times an interval, we also show that changing the branched double cover by disk surgery changes the map by skein-valued cluster transformation. This is a joint work in progress with T. Ekholm, P. Longhi, and V. Shende.

Given a braided Hopf algebra endowed with an automorphism, one can construct an R-matrix over the underlying vector space of this braided Hopf algebra. In the case of Nichols algebras, we obtain R-matrices which lead to multivariable knot polynomials generalising those related to Borel parts of small quantum groups. In the case of a generic Nichols algebra of rank 1, which is a polynomial algebra of one indeterminate, our construction reproduces the sequence of R-matrices underlying the coloured Jones polynomials.This is a joint work with Stavros Garoufalidis.

The A-polynomial is a knot invariant build from character varieties. When it was first constructed, it was believed to be independent of Jones-type invariants. Now we understand that the close relationship between these invariants only manifests after they're quantized. This talk will introduce the A-polynomial and explain a procedure for quantizing it that relies on defects in skein theory.

The Kauffman bracket skein algebra S of a punctured surface quantizes the SL_2(C )-character variety. Understanding representations of S is an important problem. Poisson order theory relates the representations of S to the Poisson geometry of the character variety. We introduce the sliced skein algebras, which quantize the symplectic leaves of maximal dimensions, and use them to study representations of S. In particular we show that each sliced skein algebra is a domain and calculate its center. This is joint work with C. Frohman and J. Kania-Bartoszynska.

Electron in a 2d square lattice immersed in a perpendicular and fractional magnetic field is known to exhibit a fractal energy spectrum known as the Hofstadter butterfly. To account for this interesting energy spectrum nonperturbative corrections to perturbative energy series must be taken into account. We reveal a surprising connection between Hofstadter butterfly and supersymmetry field theory, with the help of which, we construct the full trans-series for the energy series that includes all the non-perturbative corrections when the magnetic flux through a lattice plaquette is 2\pi/Z.

In this talk, I will discuss the alterfold theory of dimension three, where the input data is a spherical fusion category. We could define a partition function over 3 manifold with certain decoration. I will explain how it is related to other famous combinatorial quantum invariant, and how it can be applied to study the tensor categories.

Starting from a possibly non-semisimple ribbon category together with a modified trace on a tensor ideal, one can define surgery invariants of three-manifolds with certain embedded ribbon graphs. If the ribbon category is modular and the ideal is that of projective objects, these invariants extend to a three-dimensional TFT on so-called admissible bordisms. This TFT produces Lyubashenko's modular functor, and it agrees with the Reshetikhin-Turaev TFT if the input category is semisimple. This is joint work with J. Berger, M. De Renzi, A. Gainutdinov, N. Geer, and B. Patureau-Mirand.

Recently, Kashaev and the speaker defined a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, coming from the $R$-matrix of a rank 2 Nichols algebra, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$ polynomials for $n=1,2,3,4$ and discover applications and emerging patterns, including unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology. Joint work with Shana Yunsheng Li.

We have shown before that generalized quantum 6j-symbols appearing in the Turaev-Viro type TQFT based on triangulations of 3-manifolds, bi-unitary connections producing subfactors of finite depth, and 4-tensors appearing in recent studies of two-dimensional topological order are all the same. Alpha-induction is a tensor functor giving a fusion category from a Frobenius algebra in a braided fusion category, which has been studied in the context of extensions of chiral conforal field theory. We present recent advances on alpha-induction for generalized quantum 6j-symbols in the context of operator algebras with emphasis on the role of commutativity of the Frobenius algebra.

We present a skein theoretical construction based on a graphical calculus for pivotal bicategories. The graphical calculus for a pivotal monoidal category and the bicategory of Frobenius algebras internal to it are related by a Frobenius monoidal functor. It induces a relation between the skein modules that encodes information on CFT correlators and their mapping class groups.