How I learned to stop worrying and to love both instantons and anti-instantons (video) Nikita Nekrasov, Professor of Physics, Simons Center for Geometry and Physics, Stony Brook University

I will talk about the topological renormalization group at the example of quantum integrable systems. In quantizing classical mechanical systems to get (non-perturbative in hbar corrections to) the eigenvalues of the Hamiltonian one often sums over the classical trajectories as in localisation formulas, but also take into account the contributions of the so-called "instanton-antiinstanton gas". The latter is an ill-defined set of approximate solutions of equations of motion. The talk will attempt to alleviate some of the frustrations of this 40+ yrs old approach by making use of honest solutions of equations of motion of complexified classical mechanical system. The examples will include algebraic integrable systems, from the abstract Hitchin systems to the well-studied anharmonic oscillator. If time permits, I will explain the origin of these ideas in the Bethe/gauge correspondence of Nekrasov-Shatashvili.