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Brackets, forms and invariant functionals

Nigel Hitchin

TL;DR

This work develops a unified framework where differential forms are treated as spinors on the generalized tangent bundle T ⊕ T^*, with the Courant bracket encoding connections with skew torsion and B-field twists. It then builds five- and six-dimensional geometries from invariant functionals derived from open orbits in prehomogeneous vector spaces, yielding a five-dimensional structure defined by a Courant-commuting triple and a six-dimensional, Nahm-type evolution that produces a Spin(3,4)-related geometry with abelian Courant-integrable pieces. The approach links generalized geometry, gerbes, and invariant polynomials to explicit normal forms and a Hamiltonian picture, suggesting connections to string theory and higher-holonomy structures, including generalized G_2- and Calabi–Yau-type geometries. The results illuminate how invariant-theoretic methods can generate rich geometric structures in low and high dimensions, with concrete local normal forms and a clear Hamiltonian interpretation.

Abstract

In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant bracket, and we investigate the six-dimensional geometrical structure this describes.

Brackets, forms and invariant functionals

TL;DR

This work develops a unified framework where differential forms are treated as spinors on the generalized tangent bundle T ⊕ T^*, with the Courant bracket encoding connections with skew torsion and B-field twists. It then builds five- and six-dimensional geometries from invariant functionals derived from open orbits in prehomogeneous vector spaces, yielding a five-dimensional structure defined by a Courant-commuting triple and a six-dimensional, Nahm-type evolution that produces a Spin(3,4)-related geometry with abelian Courant-integrable pieces. The approach links generalized geometry, gerbes, and invariant polynomials to explicit normal forms and a Hamiltonian picture, suggesting connections to string theory and higher-holonomy structures, including generalized G_2- and Calabi–Yau-type geometries. The results illuminate how invariant-theoretic methods can generate rich geometric structures in low and high dimensions, with concrete local normal forms and a clear Hamiltonian interpretation.

Abstract

In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant bracket, and we investigate the six-dimensional geometrical structure this describes.

Paper Structure

This paper contains 17 sections, 8 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Generalized geometry
  3. The basic setting
  4. Gerbes
  5. The generalized tangent bundle
  6. Connections with skew torsion
  7. Invariant functionals
  8. Generalized geometry from open orbits
  9. Generalized $G_2$-structures
  10. A five-dimensional functional
  11. Basic algebra
  12. The variational problem
  13. The geometrical structure
  14. A normal form
  15. The six-dimensional geometry
  16. ...and 2 more sections

Key Result

Proposition 1

Let $V\subset E$ be a generalized metric, then

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Proposition 6
  • Proposition 7
  • ...and 1 more