Meromorphic connections in filtered $A_\infty$ categories

Fumihiko Sanda (Graduate School of Mathematics, Nagoya University, Nagoya, Japan; and Department of Mathematics, Kyoto University, Sakyo, Kyoto, Japan)

Abstract

In this note, introducing notions of CH module, CH morphism and CH connection, we define a meromorphic connection in the “$z$-direction” on periodic cyclic homology of an $A_\infty$ category as a connection on cohomology of a CH module. Moreover, we study and clarify compatibility of our meromorphic connections under a CH module morphism preserving CH connections at chain level. Our motivation comes from symplectic geometry. The formulation given in this note designs to fit algebraic properties of open-closed maps in symplectic geometry.

Keywords

filtered $A_\infty$ category, CH structure, mermorphic connection, Euler connection, Getzler–Gauss–Manin connection, cyclic homology, primitive form

2010 Mathematics Subject Classification

Primary 53D37, 53D45. Secondary 18G55.

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The first-named author was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No 15H02054 (PI. H. Ohta).

The second-named author was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No 15H02054 and for Young Scientists (B) No 17K17817 (PI. A. Kanazawa).

Received 4 April 2019

Accepted 20 December 2019

Published 11 November 2020