Abstracts & slides

Adel Ben Moussa (IPhT Saclay)

A Macdonald theory functor inspired by quantum many-body systems

The trigonometric Calogero–Sutherland (tCS) model describes the dynamics of N interacting particles on the circle, where the potential decays as the inverse square of their distances. Although its hamiltonian and basic physical properties appear straightforward, they hide a deep algebraic structure. Employing the right formalism brings to light the quantum integrable nature of the tCS model and elegantly expresses the conserved quantities. This approach also paves the way to deriving other integrable quantum many-body systems featuring long-range interactions: more general boundary conditions than the circle, relativistic particles... For instance, the trigonometric spin Ruijsenaars–Schneider (tsRS) model is a relativistic version of tCS where the particles have spins. It is the dynamical model underlying the q-Haldane–Shastry spin chain, a long-range variant of Heisenberg XXZ.  This model's hamiltonian and conserved quantities can in all regards be considered as extensions of Macdonald operators to more general spaces than symmetric polynomials.

In this talk, I will introduce a surprisingly natural generalisation of Macdonald theory which captures the setting of tsRS. To an arbitrary (finite) Hecke algebra module M, I will functorially associate a space where Macdonald operators are defined. Using the setting of induced AHA-modules, I will express these operators in terms of ("Baxterised") R-matrices and construct eigenvectors via "partially symmetrised" Macdonald polynomials. Finally, in the particular case where M is the spin space of tsRS, I will briefly explain how this theory conveniently overlaps with Schur–Weyl duality, providing an action of the quantum toroidal algebra (of glr). 

Anastasia Doikou (Heriot-Watt)

Set-theoretic Yang–Baxter equation, twists & quandle Hopf algebras

The theory of the set-theoretic Yang–Baxter equation is established from a purely algebraic point of view. The derivation of solutions of the braid equation via certain self-distributive structures called racks and quandles is reviewed. Generic, non-involutive set-theoretic solutions of the braid equation are then obtained from rack solutions by a suitable Drinfel'd twist, whereas all involutive solutions are obtained from the flip map via a twist. The universal algebras associated to both rack and generic set-theoretic solutions are also studied as Hopf algebras and the corresponding universal R-matrices are derived.

Misha Feigin (Glasgow)

A subalgebra of DAHA and generalised Van Diejen's Hamiltonians

I would like to discuss an interesting subalgebra of DAHA of type GL(n) which flatly deforms the crossed product of the symmetric group with the image of the Drinfeld–Jimbo quantum group in the oscillator (Jordan–Schwinger) representation. The construction is naturally related to difference integrable systems introduced by Van Diejen and their generalisations.

* Misha kindly agreed last minute to give a talk to replace the one of Vidya.

Anna Puskás (Glasgow)

Pursuing a Coxeter theory for Kac–Moody affine Hecke algebras

Objects from representation theory and algebraic combinatorics such as an affine Weyl group, or a Hecke algebra, can be viewed in different ways: defined by generations and relations, or emerging from the structure of a reductive group over a nonarchimedean local field. These viewpoints lead to different presentations. For example, the affine Hecke algebra has one presentation that highlights that it is a deformation of a Coxeter group. The other (the Bernstein presentation) manifestly contains the group algebra of a lattice.

Replacing the reductive group with a Kac–Moody group, the situation becomes more subtle. One may define a Kac–Moody affine Hecke algebra. This was done by Braverman, Kazhdan and Patnaik for untwisted affine types, and Bardy-Panse, Gaussent and Rousseau in full Kac–Moody generality. This algebra again has two distinct bases, one of them containing the semigroup algebra of the Tits cone. The underlying double affine Weyl semigroup carries notions of a length function and a Bruhat order familiar from Coxeter Weyl groups.

This talk will introduce the Kac–Moody affine counterpart of the affine Hecke algebra. We will explain the differences and surprising similarities between their behaviour and present some conjectures in the double affine context.

D Muthiah, AP,  Pursuing a Coxeter theory for Kac-Moody affine Hecke algebras,  arXiv:2406.14447

Alexander Shapiro (Edinburgh)
Cluster structure on spherical DAHA and its applications 

Quantum cluster structure on an algebra L is certain combinatorial data that (among other things) supplies L with an action of a discrete group Γ and yields a Γ-equivariant L-module S. Moreover, it identifies a family of elements in L, the so-called A-variables, combined into "clusters" of the same size, such that any A-variable is expressed as a positive Laurent polynomial of A-variables from any other cluster. Finally, the cluster structure proved to be very useful in diagonalizing operators from L on the module S. In this talk we will discuss origins and applications of a cluster structure on the spherical DAHA and its t=0 specialization. 

Vidya Venkateswaran (CCR Princeton)
Quasi-polynomial generalizations of Macdonald polynomials

In joint work with Siddhartha Sahi and Jasper Stokman, we constructed (quasi-)polynomial generalizations of nonsymmetric Macdonald polynomials for arbitrary reduced root systems. We will introduce these objects and discuss how they arise from a family of cyclic Y-parabolically induced DAHA modules where the Ti  operators act via "truncated'' divided difference operators. We will also connect a limiting case of these polynomials to metaplectic Whittaker functions, and discuss recent duality results between (anti-)symmetric quasi-polynomials and partially (anti-)symmetric polynomials.

* Unfortunately, Vidya wasn't able to give this talk in the end.

Charles A S Young (Hertfordshire)
Higher current algebras and raviolo conformal blocks

The Bethe Ansatz for quantum Gaudin models of finite type takes a particularly elegant form -- due to  Feigin, Frenkel and Reshetikhin -- in which a key theme is the interplay between rational conformal blocks (which are global objects, defined on the Riemann sphere) and vertex algebras (which are local objects, associated to formal discs). 

Motivated in part by a desire to apply the same strategy to wider classes of quantum integrable models, one would like to generalize this story to "higher" settings. Here "higher" at first just means higher dimensions -- but turns out also necessarily to mean higher in the sense of higher/homotopy/differential graded (dg) algebras. 

The mildest such generalisation is perhaps the case of raviolo vertex algebras, recently introduced by Garner and Williams. They are associated to manifolds of real dimension three admitting a transverse holomorphic foliation; that is, roughly, manifolds having one complex-holomorphic and one topological direction.

I will describe this setting, and go on to show that raviolo vertex algebras too arise from the limiting behaviour of certain raviolo conformal blocks, which I will introduce in the talk, paralleling the usual situation for rational CFTs.

Maria Matushko (Steklov)

Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type

The Haldane–Shastry spin chain describes N particles on equal spaced sites on a ring with an internal spin degree of freedom. Many special properties of the Haldane–Shastry long-range spin chain naturally arise from a connection with the spin Calogero–Sutherland model. This connection is given by so-called Polychronakos freezing trick -- we put the particles of the dynamical model in their equilibrium positions in the corresponding classical mechanics and remove all terms with differential operators from the spin Calogero Hamiltonians.

We discuss a construction of a supersymmetric q-deformed Haldane–Shastry spin chain. We present a commuting set of matrix-valued difference operators in terms off trigonometric GL(N|M)-valued R-matrix. Next, we show how to obtain integrable long-range spin chain using the Polychronakos freezing trick from these models.

Bethan Turner (City, London)

Higher time derivative theories from integrable models

There has recently been some interest in theories that are both higher derivative in time and integrable, since it is thought intergability may play a role in regulating the instabilities associated with these systems. In these studies higher time derivative Theories are generated by considering space-time rotated KdV and mKdV systems and the stability of on the classical level of both analytic and numerical solutions are studied. How the integrability of these systems governs their dynamics is studied and found to be crucial in explaining the stability in the case of the rotated mKdV systems, although this is not the case for KdV.

Yegor Zenkevich (Edinburgh)

Elliptic integrable systems and spiralling branes

Using intertwining operators of quantum toroidal algebras we construct general elliptic deformations of the trigonometric Ruijsenaars–Schneider (RS) integrable system. We show that in special cases this general deformation reduces to the elliptic RS system and to the set of Hamiltonians recently introduced by Koroteev and Shakirov. We conjecture the form of the eigenfunctions of the system.