TRINO

Speaker: Raphaël Belliard (Center for Quantum Mathematics, SDU, Denmark)

Title: Quantum correlators from local systems, WKB analysis, and cameral geometry.

When: July 9 @ 14h30

Where: UniTS - Seminar Room,  3rd floor, H2Bis Building, Vial Valerio 12/1.

Abstract: In this talk, I will introduce a viewpoint on certain families of G-local systems on complex curves, for G a reductive Lie group, that can in many ways be interpreted as a quantisation of the classical theory of Riemann surfaces. In corresponding semiclassical regimes, we will see the natural emergence of the geometry of corresponding families of G-Higgs bundles, with corresponding cameral geometry, namely a Borel structure-group reduction over a Weyl-equivariant Galois cover of the base.

From an analytical perspective, the construction relies on the existence of a non-Abelian, or quantum, counterpart to Fay’s twisted prime-form on Riemann surfaces, solving twisted d-bar problems associated to the local system on the base-curve. It is used to engineer non-perturbative correlators associated to local systems; quantum differentials with corresponding quantum periods. They have a quantum field bootstrap interpretation in terms of the representation theory of certain conformal algebras, and their semiclassical expansions are reconstructed recursively from the emergent cameral geometry.

This construction clarifies the geometry of the initial data of the spectral curve topological recursion, albeit restricted to the algebraic cases. Depending on time and interest, we may expand on specific use-cases of this framework, with choice ranging from singularity theory, as in FJRW intersection theory and oscillating integrals, to knot-invariants and topological quantum computing.