Title: Exact solutions of the non-stationary elliptic Calogero-Sutherland equation
Abstract: The non-stationary elliptic Calogero-Sutherland (eCS) equation is a non-stationary Schrodinger equation with the Hamiltonian defining the eCS model, with known relations to the Knizhnik-Zamolodchikov-Bernard equation and the quantum Painleve VI. In this talk, I will present exact solutions of the non-stationary eCS equation given by explicit integrals that generalize the integral representations of Jack polynomials: I will start by reviewing trigonometric Calogero-Sutherland model and its exact eigenfunctions given in terms of the Jack polynomials, then explain the construction of integral representations of Jack polynomials. This construction has a natural generalization to the elliptic version, and I will show how the elliptic version yields a two-parameter generalization of the Jack polynomials that satisfy the non-stationary eCS equation.
Based on joint work with E. Langmann (KTH Royal Institute of Technology): J. Integrable Syst. 2020 (arXiv:1908.00529).
Title: New integrable systems related to elliptic Calogero-Moser models
Abstract: Connections between integrable many-body systems of Calogero-Moser type and integrable partial differential equations have been studied since the inception of soliton theory. In this talk, I will focus on elliptic Calogero-Moser systems, in both the scalar and spin cases, and their relations to a variety of novel integrable integro-differential equations. These include generalizations of the Benjamin-Ono and Heisenberg ferromagnet equations and certain vector nonlinear Schrödinger equations. This talk is based on collaborations with Alexander Fagerlund, Rob Klabbers, Edwin Langmann, and Jonatan Lenells.
Title: Determinantal expressions for Ohyama polynomials
Abstract: The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present two different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice.
Title: Symmetry resolution of entanglement measures in excited states of (1+1)dimensional massive integrable quantum field theories
Abstract: In this talk I will present the results obtained with my supervisor Dr. Olalla Castro-Alvaredo and other collaborators on the symmetry resolution of entanglement measures in excited states of (1+1)d massive integrable quantum field theory (IQFT). This work generalises the results known to hold for excited states of massive IQFT with no internal symmetry to the case in which the theory enjoys a U(1) symmetry and the entanglement entropy (EE) admits a charge decomposition. Specifically, we looked at a complex free boson and a complex free fermion on a circle. The results are obtained through a field-theoretic approach that makes use of a "replica trick" and generalised twist fields. Becuse of the universality features of our formulae, we found that these can be derived also in a much simpler setup, in which the excitations are multi-qubit states. Our theoretical predictions perfectly matches the numerical analysis performed on two different lattice Hamiltonians. Finally, I will briefly present some further generalisations of this work to higher dimensional theories, interacting/non integrable theories and to other entanglement measures.
Title: Integrable deformations of sigma models
Abstract: I will review some aspects of integrable eta- and lambda- deformations of two-dimensional sigma models. These deformed models are in principle exactly solvable, they generically break all the symmetries of the original theory, have a hidden quantum group symmetry, and are related to each other through a type of worldsheet duality. Their construction can also be extended to the integrable superstring theories considered in the context of the AdS/CFT correspondence.