Quantum spectra, integrable systems, and variation of Hodge structures

Abstract

A classical theorem due to Birkhoff states that on a real or a complex symplectic manifold a function near its Morse critical point can be transformed by formal symplectomorphism into a power series in the pairwise sums of squares of the local coordinates, called its "Birkhoff normal form". A result of Sjöstrand says that, given certain conditions, one can compute the eigenvalues of the Schrödinger operator using a quantum version of this normal form. In my talk, I will explain how to interpret the quantum Birkhoff normal form geometrically by relating it to the quantization of integrable systems and to formal deformations of the variation of Hodge structures. This is joint work in progress with Maxim Kontsevich.