ReNewQuantum

I'll describe an old-fashioned way to quantize an energy level of a complexified 1d configuration space. A quantum energy level turns out to be a CP^1 structure on the configuration space, and its perturbations are deformations of that geometric structure. This offers a geometric perspective on one of the algebraic perturbation theory results that underpin the exact WKB method.

Airy structures - a notion introduced by Maxim Kontsevich and Yan Soibelman - are certain systems of linear ODEs admitting a unique solution (the so-called partition function) which is computed by a topological recursion. Airy structure therefore serve as initial data for the recursion. I will review the strategy to obtain Airy structures from the W(gl_r) vertex operator algebra (from joint work with Bouchard, Chidambaram, Creutzig, Noshchenko). This strategy depends on the choice of an element of the Weyl group of gl_r, used for "twisting". For a Coxeter element this is equivalent to Bouchard-Eynard topological recursion on spectral curves with higher order ramification. For other twisting we show with Aghaei, Kramer and Schueler that this defines a topological recursion on singular spectral curves. Schueler has classified which twisting do lead to Airy structures: surprisingly there are some restrictions, posing non-trivial conditions on the type of spectral curves allowed in this definition, and whose interpretation is still mysterious.

The quantum curve conjecture claims that one can associate a differential equation to a spectral curve, whose solution can be reconstructed by topological recursion applied to the original spectral curve. I will explain how starting just from loop equations, one can construct a system of PDEs which will annihilate the wave function built from topological recursion, solving the conjecture affirmatively for all hyperelliptic curves. The families of spectral curves can be parametrized with the so-called times, which can be defined as periods on second type cycles on the curves. These deformation parameters giving rise to families of spectral curves will play a key role when producing our system of PDEs. Our system of PDEs can be used to prove that the WKB solution of many isomonodromic systems coincide with the topological recursion wave function. If time permits, I will comment on how to generalize this procedure to spectral curves of arbitrary rank, but with simple ramifications, which is work in progress.