## New Trends in Geometry and Mathematical Physics

##### Session Organizers

Vincent Bouchard, University of Alberta

Norman Do, Monash University

### 8 December, 2022

##### Abstract

Hurwitz numbers count covers of Riemann surfaces with given ramification data. The case of covers of $\mathbb{P}^1$ with one variable ramifications and a uniform ramification elsewhere is particularly interesting, as these numbers are related to the KP integrable hierarchy. A large family of such Hurwitz problems can be computed using topological recursion: a universal procedure, recursive on the Euler characteristic of the surfaces involved, which requires a spectral curve as input. The case of Atlantes Hurwitz numbers evaded this approach up to now, as it seemed to have the same spectral curve as completed cycles Hurwitz numbers. I will introduce all these notions, and explain how Atlantes Hurwitz numbers do satisfy a variant of topological recursion, with a transalgebraic spectral curve, and this distinguishes it from the completed cycles case. This is joint work with Vincent Bouchard and Quinten Weller.

##### Abstract

BPS structures were introduced by Bridgeland to describe certain outputs of both Donaldson-Thomas theory and four-dimensional N=2 supersymmetric quantum field theory. To solve a totally different problem, the so-called loop equations in the theory of matrix models, Chekhov and Eynard-Orantin introduced the topological recursion, which takes initial data called a spectral curve and recursively produces an infinite tower of geometric invariants, often with an enumerative interpretation. $\newline$

We will describe recent work and ambitions connecting these two different theories. In the simplest cases, we describe an explicit formula for the “free energies” of the topological recursion in terms of a corresponding BPS structure constructed from the same initial data. We will sketch the relation to the WKB analysis of quantum curves, and gesture towards a wide range of generalizations.

##### Abstract

This is the first of two talks about a “negative” spin analogue of the Witten $r$-spin theory. In the first talk, we will present the construction and properties of a cohomological field theory (without a flat unit) that parallels the famous Witten r-spin class for negative spin. The class for $r=2$ was constructed and studied by Norbury in 2017. By studying certain deformations of this class, we use Teleman’s reconstruction theorem to prove relations in the tautological ring, and in the special case of $r=2$ they reduce to relations involving only kappa classes which were recently conjectured by Norbury–Kazarian.

This is based on the following joint work with Elba Garcia-Failde (who will give the second talk) and Alessandro Giacchetto: https://arxiv.org/pdf/2205.15621.pdf

##### Abstract

This is the second talk of the series started by Nitin Chidambaram on a “negative” spin analogue of the Witten r-spin theory. In this second talk, we will exploit the relation between cohomological field theories and topological recursion to prove W-algebra constraints satisfied by the descendant potential of the class (constructed in Nitin’s talk). Furthermore, we conjecture that this descendant potential is the r-BGW tau function of the r-KdV hierarchy, and prove it for r=2 (confirming a conjecture of Norbury) and r=3.

This is based on the following joint work with Nitin Chidambaram (who gave the first talk) and Alessandro Giacchetto: https://arxiv.org/pdf/2205.15621.pdf

### 9 December, 2022

##### Abstract

Recent work has been done to relate ideas between topological recursion and Higgs bundles, such as studying the relationship between topological recursion and the geometry of the moduli space of Higgs bundles, and the relationship between quantum curves and quantization. In this talk I will discuss my current work, exploring these ideas in in a more general, twisted setting where the coefficient line bundle of the Higgs field is arbitrary.

##### Abstract

Exact WKB analysis consists in the Borel-resummation of the divergent series appearing in the context of singular perturbation theory of differential equations. However, beyond the obvious perks for those interested in genuine solutions, e.g. of some equivariant differential systems, it also provides a way to study asymptotic properties of certain enumerations through the generating function method. Applications of these procedures include for instance constructing physical solutions to Schrödinger equations in finite-dimensional Hilbert spaces, as well as obtaining large-genus asymptotics of generalised Catalan numbers.

##### Abstract

In this talk, motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform of Fuchsian groups, that we call the hyperbolic Bloch transform. I will prove that the hyperbolic Bloch transform is injective and “asymptotically unitary” and I will introduce a modified, geometric, Bloch transform, that transforms wave functions to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan.

##### Abstract

Geometric quantisation has proven an effective approach to many problems in mathematical physics. Many examples have been shown of theories whose classical solutions form geometric spaces with rich and interesting structures, which may then be used for quantisation. Sometimes, however, there is just too much structure, and it can become difficult to pick on to use. This is the case, for example, for hyper-Kähler manifolds, which come with infinite families of symplectic forms.

In a recent work with J.E. Andersen and G. Rembado, we proposed a new paradigm for quantisation of hyper-Kähler spaces assuming sufficient symmetry, which opens the way to exploration in many different directions. There are many examples of spaces to which this quantisation can be applied, including several from mathematical physics, and there are many famous results in “ordinary” quantisation that should be tested for this new version, notably the famous statement that quantisation commutes with reduction. In this talk I will give a panoramic of known results and possible research directions, including ongoing work with M. Mayrand.