A Note on Semi-Infinite Non-Commutative Hodge Theory and LG-Models

TL;DR

This paper explores semi-infinite invariants in Landau-Ginzburg models, connecting them to classical twisted de Rham theory and the chiral de Rham complex, and suggests their broader relation to dg-categories and non-commutative Hodge theory.

Contribution

It introduces semi-infinite invariants for Landau-Ginzburg models and links them to established theories, proposing a broader categorical framework.

Findings

Semi-infinite invariants specialize to twisted de Rham theory.

In the vanishing potential case, they relate to the chiral de Rham complex.

Evidence suggests these invariants can be associated with dg-categories.

Abstract

We study some semi-infinite invariants associated to Landau-Ginzburg models. These specialize classically to the usual twisted de Rham package and in the case of vanishing potential to the chiral de Rham complex of Malikov, Schechtman and Vaintrob, \cite{MSV}. Further we offer some small evidence that these semi-infinite invariants can be associated to dg- categories more generally, so that in the classical limit they reproduce the usual non-commutative Hodge theory associated to the category.