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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

Online
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1435-5345
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Volume 2016, Issue 714

Issues

Matrix factorizations and cohomological field theories

Alexander Polishchuk / Arkady Vaintrob
Published Online: 2014-04-18 | DOI: https://doi.org/10.1515/crelle-2014-0024

Abstract

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.

About the article

Received: 2011-09-16

Revised: 2013-12-31

Published Online: 2014-04-18

Published in Print: 2016-05-01


Funding Source: NSF

Award identifier / Grant number: DMS-1001364

The first author was partially supported by the NSF grant DMS-1001364.


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 714, Pages 1–122, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0024.

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Citing Articles

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[1]
Alexandr Buryak, Boris Dubrovin, Jérémy Guéré, and Paolo Rossi
Communications in Mathematical Physics, 2018
[2]
Huijun Fan, Tyler Jarvis, and Yongbin Ruan
Geometry & Topology, 2017, Volume 22, Number 1, Page 235
[3]
Alexander Polishchuk
Compositio Mathematica, 2016, Volume 152, Number 10, Page 2071
[4]
Alexander I. Efimov and Leonid Positselski
Algebra & Number Theory, 2015, Volume 9, Number 5, Page 1159

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