Hodge theoretic aspects of mirror symmetry

TL;DR

This paper explores the extension of Hodge theory to non-commutative spaces, examining its relation to Calabi-Yau conditions and mirror symmetry, and developing new invariants and classification methods.

Contribution

It introduces an abstract framework for non-commutative Hodge structures, investigates their existence, variations, and interactions with mirror symmetry, and provides explicit construction techniques.

Findings

Development of non-commutative Hodge structures from symplectic and complex geometries

Identification of invariants related to mirror symmetry interactions

Analogues of classical Hodge theory consequences like unobstructedness

Abstract

We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some of the more interesting consequences of Hodge theory such as unobstructedness and the construction of canonical coordinates on moduli.