University of Leeds Workshop on Geometry, Topology and Integrability

Friday 11th/Saturday 12th May, 2018

Abstract

Topological recursion associates to a spectral curve $S$ (an algebraic plane curve, with some extra structure), a sequence of invariants $F_g(S)$. Out of these, we define (as a formal $\hbar$ series): $\tau(\hbar^{-1} S) = \exp{ \sum_{g=0}^\infty \hbar^{2g-2} F_g(S)}$. In many examples, this is the Tau function of some integrable system, for instance for the plane curve $0=y^2-\sum_{k=0}^{m} t_k x^{2k+1}$, this is the KdV tau function. For more general spectral curves, the definition needs to be slightly modified in a way that we shall explain, and we claim that $\tau(S)$ is then a Tau function in the sense that it satisfies Hirota equations. We also claim that all classical Tau functions (in fact their asymptotic WKB series and transeries $\hbar$ expansion) can be obtained in this way.

The formalism is geometric, intrinsic (there is no need to choose a family of "times", times are just local coordinates in the space of spectral curves), and shows the link to modular invariance, and conformal symmetries (Virasoro and W algebras).

Abstract

The counting of exact matchings or dimer configurations on the honeycomb lattice is a combinatorial problem with a long history. The partition function of such configurations can be computed with the methods of quantum integrable systems, the algebraic Bethe ansatz. The new aspect explored here is that the underlying Yang-Baxter algebra has a purely geometric construction in terms of convolution algebras. In particular, Baxter’s commuting transfer matrices generate a ring which for various choices of the parameters of the dimer model describes either the equivariant quantum cohomology or equivariant quantum K-theory of Grassmannians. The Bethe vectors are identified as the idempotents of these rings, geometrically they correspond to the fixed point basis in quantum Schubert calculus and the Bethe wave functions are the localised Schubert classes.

This talk will focus mostly on the combinatorial and algebraic aspects and the connection with integrability. This talk is based on joint works with Vassily Gorbounov (Aberdeen) and Catharina Stroppel (Bonn).

Abstract

We will associate a tau function to the Riemann-Hilbert problem set on a union of non-intersecting smooth closed curves with generic jump matrix. The main focus will be on the one-circle case, relevant to the analysis of Painlevé VI equation and its degenerations to Painlevé V and III. The tau functions in question will be defined as block Fredholm determinants of integral operators with integrable kernels. They can also be represented as combinatorial sums over tuples of Young diagrams which reduce to Nekrasov functions in problems of isomonodromic origin.

Abstract

The Askey–Wilson polynomials are a special case of Macdonald polynomials that are basic hypergeometric polynomials, i.e. polynomials expressed in terms of the q-Hypergeometric series (the q-difference version of the classical hypergeometric series). The basic hypergeometric polynomials are organised into a hierarchy called q-Askey scheme, in which different families of polynomials are connected by directed arrows, each arrow representing a degeneration in which the parameters and the variable (or in come cases the exponentiated parameters and variables) are rescaled by some appropriate powers of $\epsilon$ and then the limit $\epsilon\to 0$ is taken.

In this talk the speaker will give a new approach to study degenerations as well as dualities of basic orthogonal polynomials. Based on the fact that all families of polynomials in the q-Askey scheme admit a certain algebra of symmetries (the Zhedanov algebra and its degenerations) and in the limit as $q\to 1$ this algebra becomes the coordinate ring of a certain (decorated) character variety.

Abstract

One can associate Gaudin models to any symmetrizable Kac-Moody algebra. In the finite case, they describe a wide class of finite-dimensional quantum integrable systems. In particular, finite Gaudin models are known to admit a large commutative algebra of integrals of motion which is isomorphic to the algebra of functions on a certain space of differential operators, known as opers’, associated with the Langlands dual algebra. In the affine case, Gaudin models are expected to describe a broad family of quantum integrable field theories. An important open problem is to extend the above picture to the affine setting.

In this talk I will present recent joint work in this direction with Sylvain Lacroix and Charles Young. In particular, we show in arXiv:1804.01480 that functions on the space of affine opers’, a natural extension of opers to the affine case, are given by certain hypergeometric-type integrals. This leads us to conjecture the existence of higher Gaudin Hamiltonians in affine Gaudin models which should also be given by certain hypergeometric-type integrals. I will provide some evidence in support of this conjecture, based on arXiv:1804.06751, and explain its possible relation to the ODE/IM correspondence.

Abstract

We prove the Hodge-GUE correspondence conjecture on explicit relationship between the special cubic Hodge integrals over the moduli space of stable algebraic curves and enumeration of ribbon graphs. We also discuss conjectural relationship between the cubic Hodge integrals satisfying the local Calabi-Yau condition and the fractional Volterra hierarchy. Based on a series of joint work with B. Dubrovin, S.-Q. Liu, Y. Zhang, and C. Zhou.