A microlocal category associated to a symplectic manifold

Boris Tsygan

doi:10.1007/978-3-030-01588-6_4

A microlocal category associated to a symplectic manifold

Boris Tsygan^*

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

4 Scopus citations

Abstract

For a symplectic manifold subject to certain topological conditions a category enriched in ∞ local systems of modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov’s deformation quantization algebra that have an additional structure, namely an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it motivated by the theory of Lagrangian distributions and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the Tamarkin microlocal category and the Fukaya category as computed by Polishchuk and Zaslow.

Original language	English (US)
Title of host publication	Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013
Editors	Michael Hitrik, Dmitry Tamarkin, Boris Tsygan, Steve Zelditch
Publisher	Springer New York LLC
Pages	225-337
Number of pages	113
ISBN (Print)	9783030015862
DOIs	https://doi.org/10.1007/978-3-030-01588-6_4
State	Published - 2018
Event	Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013 - Evanston, United States Duration: May 20 2013 → May 24 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	269
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013
Country/Territory	United States
City	Evanston
Period	5/20/13 → 5/24/13

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-030-01588-6_4

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Tsygan, B. (2018). A microlocal category associated to a symplectic manifold. In M. Hitrik, D. Tamarkin, B. Tsygan, & S. Zelditch (Eds.), Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013 (pp. 225-337). (Springer Proceedings in Mathematics and Statistics; Vol. 269). Springer New York LLC. https://doi.org/10.1007/978-3-030-01588-6_4

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author = "Boris Tsygan",

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Tsygan, B 2018, A microlocal category associated to a symplectic manifold. in M Hitrik, D Tamarkin, B Tsygan & S Zelditch (eds), Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. Springer Proceedings in Mathematics and Statistics, vol. 269, Springer New York LLC, pp. 225-337, Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013, Evanston, United States, 5/20/13. https://doi.org/10.1007/978-3-030-01588-6_4

A microlocal category associated to a symplectic manifold. / Tsygan, Boris.
Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. ed. / Michael Hitrik; Dmitry Tamarkin; Boris Tsygan; Steve Zelditch. Springer New York LLC, 2018. p. 225-337 (Springer Proceedings in Mathematics and Statistics; Vol. 269).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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AB - For a symplectic manifold subject to certain topological conditions a category enriched in ∞ local systems of modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov’s deformation quantization algebra that have an additional structure, namely an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it motivated by the theory of Lagrangian distributions and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the Tamarkin microlocal category and the Fukaya category as computed by Polishchuk and Zaslow.

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A microlocal category associated to a symplectic manifold

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