Matrix factorizations and Cohomological Field Theories

TL;DR

This paper constructs a new algebraic cohomological field theory linked to hypersurface singularities, extending Gromov-Witten theory concepts to Landau-Ginzburg models using matrix factorizations and moduli of curves.

Contribution

It provides a purely algebraic construction of a cohomological field theory for hypersurface singularities, connecting matrix factorizations with moduli of curves and Hochschild homology.

Findings

Constructs algebraic cohomological field theory for singularities.

Relates matrix factorizations to moduli of curves.

Specializes to known theories for simple singularities.

Abstract

We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.