Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BF Description of Higher-Dimensional Gravity Theories

L. Freidel, K. Krasnov, R. Puzio

TL;DR

The paper shows that gravity in any dimension can be reformulated as a BF theory with quadratic, non-derivative constraints on the B field, enforcing B to arise from a frame field and reproducing the Palatini action. It then develops a spin foam quantization for these higher-dimensional constrained BF theories, demonstrating that simple representations and simple intertwiners exist across dimensions and can be constructed via sphere-integral intertwiners, generalizing the 4D Barrett-Crane/Baez–Reisenberger framework. A central result is that simple SO(D) representations are labeled by a single integer N and realized by harmonic polynomials on S^{D-1}, with intertwiners solving quantum intersection constraints. The discussion highlights both the structural similarities to the 4D case and key differences in higher dimensions, including the absence of a topological sector and the redundancy of constraints, while proposing a universal generating functional framework for gravity as a deformation of BF theory.

Abstract

It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higher-dimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the ``spin foam'' quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions.

BF Description of Higher-Dimensional Gravity Theories

TL;DR

The paper shows that gravity in any dimension can be reformulated as a BF theory with quadratic, non-derivative constraints on the B field, enforcing B to arise from a frame field and reproducing the Palatini action. It then develops a spin foam quantization for these higher-dimensional constrained BF theories, demonstrating that simple representations and simple intertwiners exist across dimensions and can be constructed via sphere-integral intertwiners, generalizing the 4D Barrett-Crane/Baez–Reisenberger framework. A central result is that simple SO(D) representations are labeled by a single integer N and realized by harmonic polynomials on S^{D-1}, with intertwiners solving quantum intersection constraints. The discussion highlights both the structural similarities to the 4D case and key differences in higher dimensions, including the absence of a topological sector and the redundancy of constraints, while proposing a universal generating functional framework for gravity as a deformation of BF theory.

Abstract

It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higher-dimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the ``spin foam'' quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions.

Paper Structure

This paper contains 12 sections, 4 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Classical theory
  3. The action
  4. Gauge transformations
  5. Quantum models
  6. Spin foam quantization
  7. Simple representations
  8. Simple spin networks
  9. Discussion
  10. Pluecker relations
  11. Some facts about SO(N) and its representation theory
  12. Harmonic polynomial representations of SO(N)

Key Result

Theorem 1

In dimension $D > 4$ a generic $B$ field satisfies the constraints (constr) if and only if it comes from a frame field. In other words, a non-degenerate $B$ satisfies the constraints (constr) if and only if there exist $e^{\mu}_{i}$ such that: where $|e|$ is the absolute value of the determinant of the matrix $e^\mu_i$.

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3