SISYPH

Description of the project (January 2014 - June 2017)

The SISYPH project consists of a French partner and a German partner. The following topics will be considered both separately and in connection to each other:

1. Mirror symmetry as an effective tool for the computation of Gromov-Witten invariants of various kinds of smooth algebraic varieties or orbifolds,

2. Irregular singularities of linear differential systems in any dimension, either from the point of view of holonomic \(\mathcal D\)- modules or from that of isomonodromy deformations,

3. 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 \(\mathcal D\)-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 \(\mathcal D\)-module of toric manifolds or orbifolds. These GKZ systems also provide a large class of examples of holonomic \(\mathcal D\)-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 hypersurface 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.