Research Seminar: Complex Geometry
15.11.2017 Jie Zhou (Köln)
Calabi-Yau/Landau-Ginzburg correspondence for elliptic orbifold curves via modularity
Abstract:
The Calabi-Yau/Landau-Ginzburg correspondence states that the enumeration of
holomorphic curves in a Calabi-Yau variety (solutions to the d bar equation) is
equivalent to the enumeration of solutions to a seemingly completely different
differential equation on a singularity (called Witten's equation in
Landau-Ginzburg model). The former is rigorously established by Gromov-Witten
theory and the latter by Fan-Jarvis-Ruan-Witten theory.
In this talk, I will explain the proof of the Calabi-Yau/Landau-Ginzburg
correspondence for special examples—quotients of elliptic curves and the
corresponding simply elliptic singularities. The proof mainly uses
Witten-Dijkgraaf-Verlinde-Verlinde equations and some basics of modular forms.
We first solve the Witten-Dijkgraaf-Verlinde-Verlinde equations in Gromov-Witten
theory, and prove that the generating series of Gromov-Witten invariants are
modular forms, which are then automatically globally defined on a certain
modular curve. Then we analytically continue the modular forms to elliptic fixed
points on the modular curve, and show that the differential equations and
boundary conditions satisfied by these analytical continuations match those in
the Fan-Jarvis-Ruan-Witten theory. This approach of proving the
Calabi-Yau/Landau-Ginzburg correspondence does not rely on any ingredient from
mirror symmetry or Givental's formalism. The talk will be based on a joint work
with Yefeng Shen.
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