Konstantin Wernli - Quantum Chern-Simons Theory, 2023

Quantum Chern-Simons Theory, July 4 & July 2023, Aalborg, Denmark

Talks and Slides:

Pavel Putrov, "Analytically continued Chern-Simons theory on plumbed 3-manifolds". Slides

I will present a finite-dimensional model for analytically continued Chern-Simons theory on closed 3-manifolds that are described by plumbing trees. From this model, one can define a collection of topological invariants labeled by pairs of flat connections and valued in formal power series with integral coefficients. I will also comment on a possible categorification, which can be interpreted as a finite-dimensional model of Fukaya-Seidel category of Chern-Simons functional on the space of SL(2,C) connections.

Nezhla Aghaei, "Combinatorial Quantization of Chern-Simons Theory GL(1|1)". Slides

Chern-Simons topological field theories with gauge supergroups appear naturally in higher dimensional quantum field and string theory. They also possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. Here we quantize Hamiltonian Chern-Simons theory with gauge supergroups on punctured surfaces of the arbitrary genus by extending the framework of combinatorial quantization to graded modular Hopf algebras. In particular, we construct the algebra of Chern-Simons observables and study their rich representation theory as well as their restriction to representations of mapping class groups. We showed that our result in representations of mapping class groups is isomorphic with the result constructed by Lyubashenko.
As one key application, we consider the construction of 3-manifold invariants through Heegaard splitting. This involves a conjectured extension of a formula by Kohno to the graded (and non-semisimple) context. All our constructions are illustrated through the simplest nontrivial example, namely Chern-Simons theory with gauge supergroup GL(1|1). In this case, we recover the invariants of lens spaces found by Rozansky and Saleur using Dehn surgery.
We will conclude with some comments on possible applications in the context of topological phases of fermionic matter and graded Kitaev lattice models. These will be explored more thoroughly in future publications.

Ming Zhang, "Quantum K-invariants of Grassmannian via Quot scheme". Slides

Quantum K-theory of Grassmannians was first studied by Buch and Mihalcea, and it has rich combinatorial structures. In this talk, I will present an approach to studying the quantum K-theory of Grassmannians using Quot schemes. I will first introduce the K-theoretic Quot scheme invariants, which correspond to the invariants of a 3d $\mathcal{N}=2$ Chern-Simons-matter theory on $S^1\times S^2$ . Then I will provide bialternant-type formulas for these invariants and explain their relations to quantum K-invariants. As an application, I will present a vanishing result for the K-theoretic Quot scheme invariants, which implies an identity for quantum K-invariants of Grassmannians that resembles the divisor equation in Gromov-Witten theory. This is joint work with Shubham Sinha.

Gabriele Rembado, "Towards wild/irregular conformal blocks in the WZNW model". Slides

The conformal blocks of the Wess--Zumino--Novikov--Witten model (WZNW) can be related to nonabelian theta functions, i.e. the geometric quantisation of moduli spaces of flat unitary connections in (quantum) Chern--Simons theory. Importantly, this leads to projectively flat vector bundles over spaces of deformations of Riemann surfaces, and corresponding (quantum) representations of mapping class groups.
In this talk we will aim at reviewing part of this story, and describe recent work (joint with P. Boalch, J. Douçot, G. Felder, and M. Tamiozzo) about wild/irregular generalisations.In brief, on the semiclassical side we consider meromorphic connections with arbitrary poles, which lead to much bigger spaces of deformation parameters, and to `wild' mapping class groups: the latter control the braiding of Stokes data for irregular connections, and should again find projective representations after quantisation.

Veronica Fantini, "Resurgent structure and quantum modularity". Slides

In this talk, I will discuss the resurgent structure of the formal power series $\tilde{S}\in\mathbb{Q}[\![\hbar]\!]$ , $\tilde{S}(\hbar):=\hbar e^{\hbar/24}(1+e^\hbar-e^{2\hbar}(1-e^{\hbar})+e^{3\hbar}(1-e^\hbar)(1-e^{2\hbar})...)$ associated with the q-series $\sigma(q)=1+\sum_{n=0}^{\infty}(-1)^nq^{n+1}(1-q)(1-q^2)...(1-q^n)$ . Despite its simplicity, it conjecturally encodes the modularity property of the quantum modular form introduced by Zagier and associated with the same q-series, namely $f\colon\mathbb{Q}\to\mathbb{C}$ , $f(\tau):=q^{1/24}\sigma(q)$ where $\tau\in\mathbb{Q},\,\, q=\exp(2\pi i\tau)$ . In fact, the same resurgent structure appears when considering formal power series associated with other q-series, such as the Kontsevich--Zagier q-series for trefoil and the q-series coming from the fermionic spectral traces of quantum-mechanical operators related to the quantization of the mirror curve of toric CY 3-folds (recently studied by C. Rella arXiv:2212.10606).

Tudor Dimofte, "Braided tensor categories in 3d N=4 theories". Slides

Topological twists of 3d supersymmetric gauge theories are expected to provided extended TQFT's that are (typically) derived, non-finite, non-semisimple analogues of Chern-Simons theory. Much physical and mathematical work in the past few years has focused on deciphering the structure of such TQFT's. In this talk, I'll present some recent results on the categories of line operators in abelian 3d supersymmetric gauge theories, how they can be accessed via vertex operator algebras, and a (so far) conjectural Kazhdan-Lusztig-type correspondence relating them to modules for certain quantum groups.(Joint with A. Ballin, T. Creutzig, and W. Niu.)

Nils Carqueville, "Extended defect TQFTs". Slides

According to the cobordism hypothesis with singularities, fully extended topological quantum field theories with defects are equivalently described in terms of coherent full duality data for objects and (higher) morphisms as well as appropriate homotopy fixed point structures. We discuss the 2-dimensional oriented case in some detail and apply it to truncated affine Rozansky-Witten models, which are under very explicit computational control. This is joint work with Ilka Brunner, Pantelis Fragkos, and Daniel Roggenkamp.

Marco de Renzi, "Homological construction of quantum representations of mapping class groups". Slides

Non-semisimple TQFTs have deeply generalized the original approach to quantum topology of Witten, Reshetikhin, and Turaev, and have produced topological invariants that are significantly more sensitive than their semisimple counterparts. In this talk, I will focus on representations of the mapping class group $\mathrm{Mod}(\varSigma)$ of a surface $\varSigma$ . I will explain how the family of non-semisimple quantum representations of $\mathrm{Mod}(\varSigma)$ associated with the small quantum group of $\mathfrak{sl}_2$ can be recovered using twisted homology groups of configuration spaces of $\varSigma$ . This model sheds new light on these representations: indeed, it naturally pinpoints integral bases for these actions, while also setting up the analogue of the tools that already enabled Bigelow to prove linearity of braid groups.

Rinat Kashaev, "Generalized 3d TQFTs from local fields". Slides

Based on a particular quantum dilogarithm associated to a local field $F$ and by using the similar techniques as for the Teichmüller TQFTs, one can construct at least three different generalised distribution valued 3d TQFTs two of which are of Turaev-Viro type. The associated 3-manifold invariants are expected to be enumerative invariants counting with specific weights representations of $\pi_1$ into $PL_2F$. This is the work in collaboration with Stavros Garoufalidis.

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