Quantised Painlevé monodromy manifold, Sklyanin and Calabi-Yau algebras

In recent years, studying non-commutative rings through the methods of quantum algebraic geometry has sparked enormous interest due to its applications in mirror symmetry. The work by Gross–Hacking and Keel associates to Looijenga pairs on the A-side, i.e. pairs \((Y,D)\) where \(Y\) is a smooth projective surface and \(D\) is an anti-canonical cycle of rational curves, a mirror family on the B-side constructed as the spectrum of an explicit algebra structure on a vector space. The elements of the basis of global sections uniformise such a spectrum and are called theta functions. In this course we will study a certain class of del Pezzo surfaces that can be put on either side of the mirror construction. In particular, we study the quantisation of a family of Poisson manifolds defined by the zero locus \(M_\phi\) of a degree \(d\) polynomial \(\phi \in \mathbb{C}[x_1, x_2, x_3]\) of the form \(\phi(x_1,x_2,x_3) = x_1x_2x_3 + \phi_1(x_1) + \phi_2(x_2) + \phi_3(x_3)\) where \(\phi_i(\xi)\) for \(i = 1,2,3\) is a polynomial of degree \(\leq d\) in the variable \(\xi\) only. We will give its quantisation and explain how this fits into the scheme proposed by Etingof and Ginzburg.

Geometric realisations of Bethe equations

I will give an introduction to recent results in geometric representation theory, concerning the geometric realization of Bethe ansatz equations for quantum integrable models based on quantum groups. After a brief introduction to quantum integrable models and the Bethe ansatz approach, I will focus on two topics.

The first topic is devoted to the relation between quantum K-theory of Nakajima quiver varieties and Bethe ansatz equations, which was originally proposed by Nekrasov and Shatashvili in the study of the 2d and 3d gauge theories. Using A. Okounkov’s quasimap approach to quantum K-theory, I will give an introduction to the mathematical interpretation of the results of the mentioned authors. In particular, I will show that the Bethe ansatz equations arise as the relations for the quantum K-theory ring of the Nakajima variety. I will also talk about the benefits of this geometric approach to the theory of quantum integrable systems.

The second topic centers on another geometric interpretation of Bethe equa- tions. Since the 1990s, from the work of E. Frenkel et al, it is known there is a correspondence between certain (oper) connections with trivial monodromy and regular singularities on the projective line and solutions of the certain limit of the Bethe equations (Gaudin model Bethe equations). Such a relation between integrable models and oper connections is particularly important for represen- tation theory, since it serves as an example of the geometric Langlands correspondence. I will talk about more recent results, namely, the \(q\)-deformation of oper connections and their description using Bethe ansatz equations, as well as their interrelations with the results from the first topic.

Double ramification hierarchies

The double ramification cycle in the moduli space of smooth algebraic curves with marked points is the locus of curves whose marked points form the support of a principal divisor. A natural Chow class representing the compactification of such locus is the push-forward of the virtual fundamental class of the space of rubber maps to \(\mathbb{P}^1(\mathbb{C})\) relative to \(0\) and \(\infty\). In a series of papers, the two lecturers (partly in collaboration with B. Dubrovin and J. Gur), have studied the intersection theory of such cycle with other tautological classes. In particular, inspired by ideas from Eliashberg, Givental and Hofers Symplectic Field Theory, A. Buryak introduced a construction using the DR cycle to associate to any cohomological field theory on the moduli space of stable curves an integrable system (of Hamiltonian PDEs) called the DR hierarchy. In collaboration with P. Rossi this construction was further developed and generalized to include a quantization of Buryak’s classical hierarchy.This turns out to have a deep relation, via a more classical construction of Dubrovin and Zhang, with Gromov-Witten theory, mirror symmetry and the structure of the tautological ring.

In this course, after a brief reminder on moduli spaces of stable curves and cohomological field theories, we will present the main construction of the classical and quantum double ramification hierarchies and prove their integrability. We will then study their properties (recursion equations, tau-structure, etc.) and present some examples. In the second part we will explore the relation of the classical DR hierarchy with the Dubrovin-Zhang hierarchy, introduce the DR/DZ equivalence conjecture and study its applications to the study of the tautological ring of the moduli space of stable curves. Finally, some recent applications and generalizations (to infinite rank partial CohFTs and F-CohFTs) will be discussed.

Spin structures and cohomology classes on the moduli space of curves

The first part of this course will describe the moduli space of spin curves and its compactification. The moduli space of spin curves is a finite cover of the usual moduli space of curves and its compactification given by the moduli space of stable curves. Natural bundles over the compactified moduli space of spin curves will be introduced and used to give rise to cohomology classes on the moduli space. The push-forward of these natural cohomology classes to the moduli space of stable curves will be shown to have rather nice properties.

The moduli space of Higgs bundles will be used to relate the algebraic construction of the spin moduli space in the first part of the course to a construction of the spin moduli space using hyperbolic surfaces. The moduli space of spin curves has a very nice description using hyperbolic surfaces with spin structure, building on the description of the moduli space of curves via the moduli space of hyperbolic surfaces.

The third part of the course will relate the first two parts to the moduli space of super Riemann surfaces. The moduli space of super Riemann surfaces can be identified with the total space of a natural bundle over the moduli space of spin curves. Hitchin’s proof of uniformisation for the existence of a unique hyperbolic metric in the conformal class of any Riemann surface using Higgs bundles will be shown to naturally produce super Riemann surfaces.

Geometric recursion

I will give an introduction to the theory of geometric recursion, which describes functorial constructions out of the category of surfaces that can be made recursive in the Euler characteristic. In particular, one can produce in this way mapping class group invariant functions on Teichmüller space, whose integration on the moduli spaces can also be computed recursively on the Euler characteristic. Many problems in enumerative geometry are solved by this topological recursion. After giving an overview, we will focus on two topics.

Firstly, we will describe the application of topological recursion to intersec- tion theory on the moduli space of curves, in particular for the Hodge class, more generally Chiodo classes and Witten’s \(r\)-spin class.

Secondly, we will see the geometric recursion at work to study enumerative geometry problems in the Teichmu ̈ller space and in Thurston’s space of measured foliations, stressing the role of the recursive partitions of unity originating from the work of Mirzakhani and McShane. Three different flavors of integration on the moduli space will give access to Weil–Petersson volumes (Mirzakhani’s topological recursion), \(\psi\)-classes intersections (Witten conjecture/Kontsevich theorem), and lattice point counting (Norbury’s topological recursion). Two different integrations on the moduli space of measured foliations will give rise to statistics of multicurves and to Masur-Veech volumes of the moduli space of meromorphic quadratic differentials.