# Nordita-Uppsala joint seminar on Theoretical Physics

- Date: –14:15
- Location: Nordita, Stockholm
- Lecturer: Rebecca Lodin (1) and Bertrand Eynard (2)
- Contact person: Vladimir Procházka
- Seminarium

(1) Solving q-Virasoro constraints

(2) Integrable system constructed from the geometry of a spectral curve

(1) The Virasoro constraints - arising from Ward identities - are a key component in understanding the relation between matrix models and conformal field theories; they provide the set of equations constraining the generating function which can then be solved using CFT methods. These Virasoro constraints can be derived either using differential operators or by using the so-called free field representation of the Virasoro algebra. In this talk I will discuss what happens when these constraints are q-deformed. In particular, I will outline how such q-Virasoro constraints can be derived for a large class of deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral. These q-Virasoro constraints can then be solved recursively and they also have applications for gauge theories.

(2) One usual way of defining an integrable system is in terms of a Tau-function obeying Hirota equations.

The Tau-function (example KdV) is usually defined as a function of an infinite set of times $t=(t_0,t_1,t_2,t_3,...)$.

Here instead we shall define Tau as a function on the moduli space of spectral curves (plane analytic curves with extra structure), and the "times" can be viewed as local coordinates (but not global in general). The tangent space (i.e. the span of all $\partial/\partial t_k$, i.e. Hamiltonians) to the moduli space of spectral curves, is isomorphic to the space of meromorphic 1-forms on the curve, and by form-cycle duality is isomorphic to a Lagrangian in the space of cycles. In other words, we reinterpret Hamiltonians as cycles, and the symplectic Poisson structure as the intersection of cycles.

The topological recursion (TR) defines invariants of the spectral curve, and we show how to get a Tau-function from the TR-invariants. This is an efficient method, which gives new insights on integrable systems.