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On Deformations of Generalized Complex Structures: the Generalized Calabi-Yau Case

Yi Li

TL;DR

This work proves an analog of the Tian-Todorov unobstructedness theorem for twisted generalized Calabi-Yau manifolds, showing that the moduli space of generalized complex structures is smooth with dimension $\dim H^2(d_E)$. It constructs the extended moduli space via a BV/ differential Gerstenhaber framework, establishes its representability and a Frobenius manifold structure with flat coordinates and a generating potential, and explains the physical relevance to generalized B-models as well as cohomological field theories. The results extend classical deformation theory beyond $SU(n)$ holonomy cases and provide a robust algebraic and geometric framework for deformations in generalized geometry. Collectively, the paper links intricate geometric structures to their moduli, periods, and physical interpretations in string theory and topological field theories.

Abstract

We prove an analog of the Tian-Todorov theorem for twisted generalized Calabi-Yau manifolds; namely, we show that the moduli space of generalized complex structures on a compact twisted generalized Calabi-Yau manifold is unobstructed and smooth. We also construct the extended moduli space and study its Frobenius structure. The physical implications are also discussed.

On Deformations of Generalized Complex Structures: the Generalized Calabi-Yau Case

TL;DR

This work proves an analog of the Tian-Todorov unobstructedness theorem for twisted generalized Calabi-Yau manifolds, showing that the moduli space of generalized complex structures is smooth with dimension . It constructs the extended moduli space via a BV/ differential Gerstenhaber framework, establishes its representability and a Frobenius manifold structure with flat coordinates and a generating potential, and explains the physical relevance to generalized B-models as well as cohomological field theories. The results extend classical deformation theory beyond holonomy cases and provide a robust algebraic and geometric framework for deformations in generalized geometry. Collectively, the paper links intricate geometric structures to their moduli, periods, and physical interpretations in string theory and topological field theories.

Abstract

We prove an analog of the Tian-Todorov theorem for twisted generalized Calabi-Yau manifolds; namely, we show that the moduli space of generalized complex structures on a compact twisted generalized Calabi-Yau manifold is unobstructed and smooth. We also construct the extended moduli space and study its Frobenius structure. The physical implications are also discussed.

Paper Structure

This paper contains 10 sections, 12 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Generalized Complex Manifolds and Generalized Kähler Manifolds
  3. Deformation Theory of Generalized Complex Structures
  4. Deformations on a Compact Twisted Generalized Calabi-Yau Manifold
  5. Some definitions
  6. The vanishing theorem
  7. Beyond the Classical Deformations
  8. The extended moduli space for a generalized Calabi-Yau manifold
  9. The Frobenius structure and flat coordinates
  10. Physical Interpretations

Key Result

Theorem 1

There exists an open neighborhood $U\subset H^2(d_E)$ containing zero that parameterizes a smooth family of generalized almost complex deformations of ${\mathcal{J}}$, and an analytic obstruction map $\Phi:U\rightarrow H^3(d_E)$ with $\Phi(0)=0$ and $d\Phi(0)=0$, such that the integrable deformation

Theorems & Definitions (16)

  • Theorem 1: Gualtieri, Gua1
  • Definition 1: Hitchin, Hitchin
  • Definition 2
  • Definition 3
  • Definition 4: Gualtieri, Gua1
  • Lemma 2
  • Corollary 3
  • Proposition 4
  • Theorem 5
  • Proposition 7
  • ...and 6 more