Speaker: Sergey Galkin

Title: Homotopy quantization of real plane curve enumeration

Date: Nov 13, 2018, 15:30

Place: IMPA, Sala 236, Differential Geometry seminar

Abstract:
Observe that an integral Heisenberg group has a topological incarnation as a second homotopy group of a particular space relative to a two-torus. Similar construction and computation assigns Heisenberg-like groups (generated by broken lines) to toric surfaces. We construct a natural integer-valued functional on the center of this group. By re-embedding a southern hemi-sphere via a second Frobenius map, we can associate an element in this center to every real rational curve on a projective plane (or a toric surface), that passes through Menelaus-generic collection of real points on a toric boundary. Thus to every such real rational curve we associate an integer invariant, a quantum index. We show that it coincides with Mikhalkin's logarithmic area, and consequently it coincides with a tropical refinement introduced by Block and Goettsche. Time permits I will explain another motivation for our constructions: a non-commutative deformation of mirror symmetry for symplectic del Pezzo surfaces.

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