Bernd Siebert
Schriftzug: Fachbereich Mathematik 
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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.


 
  Seitenanfang  Impressum 2017-12-05, Bernd Siebert