Matrix factorizations and Cohomological Field Theories

By Alexander Polishchuk, A. Vaintrob. 2011

88 citations recorded by the source corpus.

We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c12c60c96c7d570f0814230a7cf9b0ecc34f629

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Citations in the corpus

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, methodology): “ing) can be identiﬁed with the state space of the cohomological ﬁeld theory (in the sense of [29]) constructed by Fan, Jarvis and Ruan in [13]. The results of the present paper are used in the sequel [39] to construct a purely algebraic version of the Fan-Jarvis-Ruan theory. 1 The category MF(w) for”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, methodology): “ould like to thank them for their time and patience. Furthermore, the authors are deeply grateful to Alexander Polishchuk and Arkady Vaintrob for providing us with a preliminary version of their work [PV11] and for allowing us time to prepare the original version of this paper in order to synchroniz”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “The starting point for this is the observation that GrMF(S,w) is a Zn-equivariant version of MF(S,w) (compare [41, 10]).”