# Meromorphic connections in filtered $A_{\infty}$ categories

## TL;DR

This paper introduces a new framework for meromorphic connections in filtered $A_{
abla}$ categories, inspired by symplectic geometry, defining CH modules, morphisms, and connections to study their properties and compatibilities.

## Contribution

It develops a novel algebraic structure for meromorphic connections in $A_{
abla}$ categories, connecting symplectic geometry and algebraic properties of open-closed maps.

## Key findings

- Defined CH modules, morphisms, and connections in the context of $A_{
abla}$ categories.
- Clarified compatibility of meromorphic connections under CH module morphisms.
- Provided a framework aligning algebraic properties with symplectic geometric structures.

## 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.