Project Details
Projekt
Mirror Symmetry and Irregular Singularities coming from Physics
Subject Area
Mathematics
Term
from 2013 to 2023
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 242588615
The overall aim of this project is to make important progress on the interplay between mirror symmetry and singularities of linear differential equations in any dimension. The following topics will be considered both separately and in connection to each other:- Mirror symmetry as an effective tool for the computation of Gromov-Witten invariants of various kinds of smooth algebraic varieties or orbifolds,- Irregular singularities of linear differential systems in any dimension, either from the point of view of holonomic D-modules or from that of isomonodromy deformations, - Hodge theoretic aspects of such differential systems.One of the original aspects of the project consists in obtaining results in each topic by exhibiting the interplay between these topics through the use of various tools and methods (algebraic geometry, non-commutative Hodge theory, singularity theory and D-modules, symplectic geometry) with, in the background, motivations and conjectures formulated by physicists.A central object of interest will be the generalized hypergeometric systems of linear differential equations (GKZ systems) as models for the quantum D-module of toric manifolds or orbifolds. These GKZ systems also provide a large class of examples of holonomic D-modules with irregular singularities, where conjectures and preliminary results can be tested.The understanding of the geometry of different types of moduli spaces like those for isolated hyper surface singularities, for curves, or more generally for stable mappings (entering in the very definition of Gromov-Witten invariants), and for meromorphic connections on vector bundles, is one of the most important motivations of the whole project. Although the first ones are known to be essential for mirror symmetry, a basic question will be to make sense/fully understand the notion of mirror symmetry for the moduli spaces of irregular singular connections on Riemann surfaces.The Stokes phenomenon, which is a fundamental property of irregular singularities of differential equations, is a basic object to be understood in the context of either Gromov-Witten theory or Landau-Ginzburg models and their extensions to singularity theory. Its relationship with Hodge-theoretic properties (in particular their non-commutative aspects) will allow the analysis of moduli spaces of singularities.
DFG Programme
Research Grants
International Connection
France
Co-Investigators
Professor Dr. Michael Dettweiler; Professor Dr. Marco Hien; Privatdozent Dr. Thomas Reichelt
Cooperation Partner
Professor Dr. Claude Sabbah
