IPMU Colloquium

Speaker: Alexis Roquefeuil (Kavli IPMU)
Title: Gromov--Witten invariants and Givental's formalism: counting curves with differential equations (Posdoc Colloquium)
Date: Fri, Nov 27, 2020, 15:00 - 15:30
Place: Zoom
Related File: 2596.pdf
Abstract: Gromov--Witten invariants are numbers that, in some circumstances, count the number of complex curves on a projective variety that satisfy some incidence conditions.
In algebraic geometry, these invariants have a very technical definition given by the integration of some cohomological classes constructed on Kontsevich's moduli space of stable maps.
In the study of Fano varieties, these invariants can be encoded in a system of differential equations, called quantum D-module, which has proven to be a key ingredient to the computation of Gromov--Witten invariants through mirror symmetry.

In 2004, Y.P. Lee defined new invariants, called K-theoretic Gromov--Witten invariants, as the Euler characteristic of some vector bundles on the same moduli space of stable maps.
In the case of these new invariants, the analogue of the quantum D-module turns out to be defined not with differential equations, but with q-difference equations.

In this talk, after motivating and discussing the definitions of these invariants, my main goal will be to explain how one can compare these two technical invariants by seeing differential equations as formal limits of q-difference equations. Such a principle is called the "confluence" of a q-difference equation.
This point of view motivates new questions that I will then discuss, the first regarding Stokes phenomena of such systems, the second regarding the relation between Gromov--Witten invariants and some oscillatory integrals involving the Gamma class.
Remarks: IPMU Postdoc Colloquium Series 15
Registration necessary from here: https://ipmu.zoom.us/webinar/register/WN_SiQNqIjNSfCUiA0na9o58g