The Kapustin-Li formula revisited

TL;DR

This work provides a direct, model-specific derivation of the Kapustin–Li duality pairing for morphism complexes in the matrix-factorization category $ ext{MF}(R,w)$ of an isolated hypersurface singularity. By combining local duality with the Basic Perturbation Lemma and Grothendieck residues, it produces an explicit chain-level formula for the pairing: $$(F,G) o (-1)^{inom{n+1}{2}}rac{1}{n!} ext{Res}igl[ ext{tr}(FG(dQ)^{ abla n}) abla_1 w, abla_2 w,\dots, abla_n wigr],$$ which is shown to be nondegenerate. The authors further lift this pairing to a Calabi–Yau structure on $ ext{MF}(R,w)$, enabling the construction of 2D topological quantum field theories with matrix factorizations as boundary conditions, and they derive an explicit boundary-bulk map consistent with the Kapustin–Li framework. Finally, they outline a Riemann–Roch formula arising from the induced field theory, linking chern characters in Hochschild (cyclic) homology to Euler characteristics of morphism spaces.

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.

Figures (4)

• Figure 1: The boundary-bulk map
• Figure 2: Twice punctured sphere
• Figure 3: First Decomposition
• Figure 4: Second Decomposition

Theorems & Definitions (25)

• Lemma 2.1: Basic Perturbation Lemma
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 4.1: polishchuk