The Kapustin-Li formula revisited

We provide a new perspective on the Kapustin-Li formula Tor 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 its to define topological quantum field theories with matrix factorizations as boundary conditions. (C) 2012 Elsevier Inc. All rights reserved.