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A Construction of Formal Frobenius Manifold from Deformation of Complex Structure

Jian Han

TL;DR

The thesis develops a program to construct a formal Frobenius manifold from extended moduli of complex structures on compact Calabi–Yau manifolds by following Barannikov–Kontsevich. It casts deformation theory in the differential graded Lie algebra framework, uses Maurer–Cartan solutions modulo gauge to define (graded) moduli spaces, and proves unobstructedness for Calabi–Yau manifolds via quasi-isomorphisms to abelian DGLAs. Building on the extended moduli space, it shows how a graded Frobenius algebra structure emerges and yields a formal Frobenius manifold with a deforming product ⊛ on the total cohomology H, encoded by a potential Φ. This construction links deformation theory, extended moduli, and Frobenius manifold geometry, with implications for mirror symmetry and 2D topological quantum field theories. The work also situates Barannikov–Kontsevich’s framework within the larger context of dGBV algebras and graded moduli, highlighting the formal equivalence between deformational data and Frobenius structures on extended moduli spaces.

Abstract

This thesis studies Frobenius manifolds arising from extended deformations of complex structures on compact Calabi-Yau manifolds, following the construction by Sergey Barannikov and Maxim Kontsevich. The work is based on the investigation of formal moduli spaces of solutions to the Maurer-Cartan equations modulo gauge equivalence. We provide a foundational overview of deformation theory from the perspective of differential geometry and prove the equivalence between gauge-equivalent deformations and isomorphic deformations. Based on this framework, we construct a differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebra associated to the deformation of Calabi-Yau manifolds and then construct the corresponding Frobenius manifold.

A Construction of Formal Frobenius Manifold from Deformation of Complex Structure

TL;DR

The thesis develops a program to construct a formal Frobenius manifold from extended moduli of complex structures on compact Calabi–Yau manifolds by following Barannikov–Kontsevich. It casts deformation theory in the differential graded Lie algebra framework, uses Maurer–Cartan solutions modulo gauge to define (graded) moduli spaces, and proves unobstructedness for Calabi–Yau manifolds via quasi-isomorphisms to abelian DGLAs. Building on the extended moduli space, it shows how a graded Frobenius algebra structure emerges and yields a formal Frobenius manifold with a deforming product ⊛ on the total cohomology H, encoded by a potential Φ. This construction links deformation theory, extended moduli, and Frobenius manifold geometry, with implications for mirror symmetry and 2D topological quantum field theories. The work also situates Barannikov–Kontsevich’s framework within the larger context of dGBV algebras and graded moduli, highlighting the formal equivalence between deformational data and Frobenius structures on extended moduli spaces.

Abstract

This thesis studies Frobenius manifolds arising from extended deformations of complex structures on compact Calabi-Yau manifolds, following the construction by Sergey Barannikov and Maxim Kontsevich. The work is based on the investigation of formal moduli spaces of solutions to the Maurer-Cartan equations modulo gauge equivalence. We provide a foundational overview of deformation theory from the perspective of differential geometry and prove the equivalence between gauge-equivalent deformations and isomorphic deformations. Based on this framework, we construct a differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebra associated to the deformation of Calabi-Yau manifolds and then construct the corresponding Frobenius manifold.
Paper Structure (28 sections, 53 theorems, 203 equations, 5 figures)

This paper contains 28 sections, 53 theorems, 203 equations, 5 figures.

Table of Contents

  1. Acknowledgements
  2. Introduction
  3. Motivation
  4. Contents of this Thesis
  5. $G$ structure on Manifold
  6. Definition of $G$ structure on Manifold
  7. Complex Manifold and Integrability of Almost Complex Structures
  8. Deformation of Complex Structure
  9. One Parameter Deformation of Complex Structure
  10. First Order Deformation and its Obstruction
  11. Isomorphism between Deformations
  12. Formal Deformation, Obstruction of First Order Deformation
  13. Deformation theory via Differential Graded Lie algebra
  14. Differential Graded Lie Algebra and Maurer-Cartan Equation
  15. Gauge Action on Differential Graded Lie Algebra
  16. ...and 13 more sections

Key Result

Proposition 2.3

Given a smooth manifold $M$ with dimension $n$, there is one to one correspondence between Riemannian metrics on $M$ and $O(n)$ structures on $M$, where $O(n)$ is the orthogonal group in dimension $n$.

Figures (5)

  • Figure 1: A family of complex structures on torus
  • Figure 2: Gluing
  • Figure 3: Commutativity
  • Figure 4: Associativity
  • Figure 5: Disk as a morphism from $S^1$ to $\emptyset$

Theorems & Definitions (144)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 134 more