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Homogeneity of cohomology classes associated with Koszul matrix factorizations

Part of: Quantum field theory; related classical field theories Local theory Homological methods

Published online by Cambridge University Press:  20 July 2016

Alexander Polishchuk
Alexander Polishchuk*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97405, USA email [email protected]
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Abstract

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In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

Keywords

matrix factorizationFan–Jarvis–Ruan–Witten theoryHochschild homology

MSC classification

Primary: 81T70: Quantization in field theory; cohomological methods
Secondary: 14B05: Singularities 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 2071 - 2112
Copyright
© The Author 2016 

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