Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variations on a theme of MacDowell-Mansouri

P. D. Alvarez, K. Krasnov

Abstract

Inspired by the MacDowell-Mansouri formulation of four-dimensional General Relativity, we study a class of four-dimensional gauge-theoretic functionals obtained from the Pontryagin density of a G-connection by inserting, under the trace, a matrix that breaks the gauge group G to a subgroup H. Concretely, we study the model with the pair (G,H) given by (SU(3), U(2)). We show that the critical points of the resulting functional are constant scalar curvature almost-Kahler 4-manifolds. On compact 4-manifolds, a stronger conclusion holds under the additional assumption that the scalar curvature is non-negative and the first Chern class is such that an Einstein metric can exist. In this case, results in the literature imply that the critical points are Kahler-Einstein 4-manifolds.

Variations on a theme of MacDowell-Mansouri

Abstract

Inspired by the MacDowell-Mansouri formulation of four-dimensional General Relativity, we study a class of four-dimensional gauge-theoretic functionals obtained from the Pontryagin density of a G-connection by inserting, under the trace, a matrix that breaks the gauge group G to a subgroup H. Concretely, we study the model with the pair (G,H) given by (SU(3), U(2)). We show that the critical points of the resulting functional are constant scalar curvature almost-Kahler 4-manifolds. On compact 4-manifolds, a stronger conclusion holds under the additional assumption that the scalar curvature is non-negative and the first Chern class is such that an Einstein metric can exist. In this case, results in the literature imply that the critical points are Kahler-Einstein 4-manifolds.
Paper Structure (19 sections, 8 theorems, 113 equations)

This paper contains 19 sections, 8 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. SU(3) MDM functional
  3. $\mathfrak{su}(2)$ description of the $\mathfrak{su}(3)$ connection
  4. Pseudo-Pontryagin density
  5. The arising model
  6. Field equations directly
  7. More general functionals
  8. Almost-Hermitian interpretation
  9. Metric and almost complex structure
  10. More general field equations
  11. Interpretation of the Cartan-flat geometries
  12. Functional for almost-Hermitian structures
  13. Consequences of the $\omega$ and $g$ compatibility
  14. Euler-Lagrange equations
  15. Consequences of the field equations
  16. ...and 4 more sections

Key Result

Proposition 1

The most general MDM functional for a pair $(w,\Psi)$ of an ${\rm U}(2)$ connection $w$ and a ${\rm U}(2)$ frame $\Psi$ is given by with real parameters $\lambda,\mu,\nu$ satisfying Here $A,a$ are the ${\rm SU}(2)$ and ${\rm U}(1)$ parts of $w$ as in (A-a), and $F=dA+A\wedge A$ is the curvature of the ${\rm SU}(2)$ connection $A$.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem
  • Theorem
  • Corollary
  • Corollary
  • Theorem