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Generalized complex geometry

Marco Gualtieri

TL;DR

Gualtieri develops generalized complex geometry as a unifying framework that encompasses complex and symplectic geometry via structures on $T\oplus T^*$ and the Courant bracket. He constructs the linear-algebra and spinor foundations, defines Dirac and Courant algebroids, and establishes $B$-field and gerbe symmetries, along with twisted variants. The work proves a generalized Darboux-type local normal form, develops an elliptic deformation theory with a Kuranishi-type moduli space, and relates generalized complex structures to generalized Kähler, bi-Hermitian geometry, and D-brane concepts. It also provides explicit exotic examples, interpolation results between complex and symplectic structures, and a framework connecting to mirror symmetry and string theory phenomena. Overall, the thesis lays a comprehensive mathematical foundation for generalized complex geometry and its deformation theory, with significant implications for geometry and mathematical physics.

Abstract

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Generalized complex geometry

TL;DR

Gualtieri develops generalized complex geometry as a unifying framework that encompasses complex and symplectic geometry via structures on and the Courant bracket. He constructs the linear-algebra and spinor foundations, defines Dirac and Courant algebroids, and establishes -field and gerbe symmetries, along with twisted variants. The work proves a generalized Darboux-type local normal form, develops an elliptic deformation theory with a Kuranishi-type moduli space, and relates generalized complex structures to generalized Kähler, bi-Hermitian geometry, and D-brane concepts. It also provides explicit exotic examples, interpolation results between complex and symplectic structures, and a framework connecting to mirror symmetry and string theory phenomena. Overall, the thesis lays a comprehensive mathematical foundation for generalized complex geometry and its deformation theory, with significant implications for geometry and mathematical physics.

Abstract

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.

Paper Structure

This paper contains 53 sections, 74 theorems, 388 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear algebra of $V\oplus V^*$
  3. Symmetries of $V\oplus V^*$
  4. Maximal isotropic subspaces
  5. Spinors for $V\oplus V^*$: exterior forms
  6. The bilinear form on spinors
  7. Pure spinors
  8. Complexification and the real index
  9. Functorial property of Dirac structures
  10. The spin bundle for $T\oplus T^*$
  11. The Courant bracket
  12. Lie algebroids
  13. The Courant bracket and Courant algebroids
  14. Symmetries of the Courant bracket; the B-field
  15. Dirac structures
  16. ...and 38 more sections

Key Result

Proposition 2.6

Every maximal isotropic in $V\oplus V^*$ is of the form $L(E,\varepsilon)$.

Figures (1)

  • Figure :

Theorems & Definitions (210)

  • Example 2.1: $B$-transform
  • Example 2.2: $\beta$-transform
  • Example 2.3: $GL(V)$ action
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 200 more