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Hodge theoretic aspects of mirror symmetry

L. Katzarkov, M. Kontsevich, T. Pantev

TL;DR

The paper develops a comprehensive framework for noncommutative Hodge structures (nc-Hodge) that extends classical Hodge theory to nc spaces encoded by derived categories. It introduces rigorous data and axioms for nc-Hodge structures and their exponential type variants, explains how to glue such data from regular pieces, and connects these invariants to cyclic and Hochschild homology via Gauss-Manin type connections. The authors then relate nc-Hodge theory to mirror symmetry by analyzing A-model and B-model constructions, including Landau-Ginzburg models, and derive generalized Tian-Todorov type results that yield canonical coordinates for Calabi-Yau variations in the nc setting. The work provides a unifying deformation theoretic and Hodge theoretic perspective on nc spaces, offering new invariants and a path toward a 2D CohFT framework tied to Calabi-Yau categories. Overall, the framework aims to deepen understanding of mirror symmetry beyond classical geometry by exploiting nc-Hodge data to capture obstructions, deformations, and moduli in a homological context.

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.

Hodge theoretic aspects of mirror symmetry

TL;DR

The paper develops a comprehensive framework for noncommutative Hodge structures (nc-Hodge) that extends classical Hodge theory to nc spaces encoded by derived categories. It introduces rigorous data and axioms for nc-Hodge structures and their exponential type variants, explains how to glue such data from regular pieces, and connects these invariants to cyclic and Hochschild homology via Gauss-Manin type connections. The authors then relate nc-Hodge theory to mirror symmetry by analyzing A-model and B-model constructions, including Landau-Ginzburg models, and derive generalized Tian-Todorov type results that yield canonical coordinates for Calabi-Yau variations in the nc setting. The work provides a unifying deformation theoretic and Hodge theoretic perspective on nc spaces, offering new invariants and a path toward a 2D CohFT framework tied to Calabi-Yau categories. Overall, the framework aims to deepen understanding of mirror symmetry beyond classical geometry by exploiting nc-Hodge data to capture obstructions, deformations, and moduli in a homological context.

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.

Paper Structure

This paper contains 59 sections, 27 theorems, 143 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Non-commutative Hodge structures
  3. Hodge structures
  4. Notation
  5. Meromorphic connections on the complex line
  6. Stokes data
  7. The definition of a nc-Hodge structure
  8. Variations of nc-Hodge structures
  9. Relation to other definitions
  10. Relation to usual Hodge theory
  11. nc-Hodge structures of exponential type
  12. Hodge structures in nc geometry
  13. Categorical nc-geometry
  14. The main conjecture
  15. Cyclic homology
  16. ...and 44 more sections

Key Result

Lemma 2.9

The functor $\mathfrak{N}$ is fully faithful and its essential image consists of all nc-Hodge structures that have regular singularities and monodromy $= \operatorname{id}$ on $H^{0}$ and $= - \operatorname{id}$ on $H^{1}$.

Figures (8)

  • Figure 1: The system of labels for the Deligne-Malgrange-Stokes filtration.
  • Figure 2: A system of paths for $S\subset \mathbb{C}$.
  • Figure 3: The half-plane $\mathfrak{H}_{\varphi,\lambda}$.
  • Figure 4: A system of thickened loops for $S\subset \mathbb{C}$.
  • Figure 5: A sector in $\boldsymbol{\Delta}$.
  • ...and 3 more figures

Theorems & Definitions (91)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • Remark 2.10
  • ...and 81 more