The Kapustin-Li formula revisited

TL;DR

This paper offers a new perspective on the Kapustin-Li formula, clarifying its role as a local duality isomorphism in matrix factorizations and connecting it to Calabi-Yau structures and topological quantum field theories.

Contribution

It revisits the Kapustin-Li formula, providing an explicit local duality isomorphism and demonstrating its lift to a Calabi-Yau structure in matrix factorization categories.

Findings

The pairing is shown to be non-degenerate.

The pairing lifts to a Calabi-Yau structure.

Enables defining TQFTs with matrix factorizations as boundary conditions.

Abstract

We provide a new perspective on the Kapustin-Li formula for the duality pairing on the morphism complexes in the matrix factorization category of an isolated hypersurface singularity. In our context, the formula arises as an explicit description of a local duality isomorphism, obtained by using the basic perturbation lemma and Grothendieck residues. The non-degeneracy of the pairing becomes apparent in this setting. Further, we show that the pairing lifts to a Calabi-Yau structure on the matrix factorization category. This allows us to define topological quantum field theories with matrix factorizations as boundary conditions.