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.