Abstract
We present an improved construction of the fundamental matrix factorization in the FJRW-theory given in Polishchuk and Vaintrob (J Reine Angew Math 714:1–22, 2016). The revised construction makes the independence on choices more apparent and works for a possibly nonabelian finite group of symmetries. One of the new ingredients is the category of dg-matrix factorizations over a dg-scheme.
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References
Ciocan-Fontanine, I., Kapranov, M.: Virtual fundamental classes via dg-manifolds. Geom. Topol. 13, 1779–1804 (2009)
Ciocan-Fontanine, I., Favero, D., Guéré, J., Kim, B., Shoemaker, M.: Fundamental factorization of a GLSM, part I: construction (2018). arXiv:1802.05247
Efimov, A.I., Positselski, L.: Coherent analogues of matrix factorizations and relative singularity categories. Algebra Number Theory 9, 1159–1292 (2015)
Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry, and quantum singularity theory. Ann. Math. (2) 178, 1–106 (2013)
Fan, H., Jarvis, T., Ruan, Y.: A mathematical theory of the gauged linear sigma model. Geom. Topol. 22, 235–303 (2018)
Lunts, V.: Categorical resolutions, poset schemes, and Du Bois singularities. IMRN 19, 4372–4420 (2012)
Polishchuk, A., Vaintrob, A.: Matrix factorizations and cohomological field theories. J. Reine Angew. Math. 714, 1–122 (2016)
Acknowledgements
I am grateful to Felix Janda and Yongbin Ruan for organizing the RTG Conference on Witten’s r-spin class and related topics in January 2017, where the results of this note were first presented. I also thank Institut Mathematique Jussieu and Institut des Hautes Etudes Scientifiques for hospitality and excellent working conditions during preparation of this paper.
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To Rafail Kalmanovich Gordin, with gratitude.
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Supported in part by the NSF Grant DMS-1700642 and by the Russian Academic Excellence Project ‘5-100’.
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Polishchuk, A. Fundamental Matrix Factorization in the FJRW-Theory Revisited. Arnold Math J. 5, 23–35 (2019). https://doi.org/10.1007/s40598-019-00100-3
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DOI: https://doi.org/10.1007/s40598-019-00100-3