Formulae in noncommutative Hodge theory

TL;DR

This paper establishes that the cyclic homology of saturated A-infinity categories can be endowed with a polarized variation of Hodge structures, providing detailed proofs and explicit formulas to support the connection between homological and enumerative mirror symmetry.

Contribution

It offers complete proofs and explicit formulas for the Hodge structure on cyclic homology of saturated A-infinity categories, advancing the understanding of mirror symmetry.

Findings

Cyclic homology admits a polarized variation of Hodge structures.

Provides explicit formulas for the relevant structures.

Supports the link between homological and enumerative mirror symmetry.

Abstract

We prove that the cyclic homology of a saturated $A_\infty$ category admits the structure of a `polarized variation of Hodge structures', building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.