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Extended matter coupled to BF theory

Winston J. Fairbairn, Alejandro Perez

TL;DR

The paper addresses how to couple extended matter, in the form of strings, to BF theory in four dimensions and develops a background-independent quantum description. It shows that classical solutions include flat-background string propagation and that the full set of solutions corresponds to distributional GR with cosmic strings; in the quantum theory, a regulated projection onto physical states yields transition amplitudes dual to 3D gravity coupled to particles, and regulator moves demonstrate topological invariance. The key contributions are a canonical quantization with string spin networks, a carefully regularized physical inner product, and a robustness result: amplitudes are independent of the regulator and depend only on topology. The work advances the understanding of topological matter-gravity couplings, offers a route to compute amplitudes in higher-dimensional topological theories, and suggests extensions to Yang-Mills fields on the world-sheet.

Abstract

Recently, a topological field theory of membrane-matter coupled to BF theory in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss various aspects of the four-dimensional theory. Firstly, we study classical solutions leading to an interpretation of the theory in terms of strings propagating on a flat spacetime. We also show that the general classical solutions of the theory are in one-to-one correspondence with solutions of Einstein's equations in the presence of distributional matter (cosmic strings). Secondly, we quantize the theory and present, in particular, a prescription to regularize the physical inner product of the canonical theory. We show how the resulting transition amplitudes are dual to evaluations of Feynman diagrams coupled to three-dimensional quantum gravity. Finally, we remove the regulator by proving the topological invariance of the transition amplitudes.

Extended matter coupled to BF theory

TL;DR

The paper addresses how to couple extended matter, in the form of strings, to BF theory in four dimensions and develops a background-independent quantum description. It shows that classical solutions include flat-background string propagation and that the full set of solutions corresponds to distributional GR with cosmic strings; in the quantum theory, a regulated projection onto physical states yields transition amplitudes dual to 3D gravity coupled to particles, and regulator moves demonstrate topological invariance. The key contributions are a canonical quantization with string spin networks, a carefully regularized physical inner product, and a robustness result: amplitudes are independent of the regulator and depend only on topology. The work advances the understanding of topological matter-gravity couplings, offers a route to compute amplitudes in higher-dimensional topological theories, and suggests extensions to Yang-Mills fields on the world-sheet.

Abstract

Recently, a topological field theory of membrane-matter coupled to BF theory in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss various aspects of the four-dimensional theory. Firstly, we study classical solutions leading to an interpretation of the theory in terms of strings propagating on a flat spacetime. We also show that the general classical solutions of the theory are in one-to-one correspondence with solutions of Einstein's equations in the presence of distributional matter (cosmic strings). Secondly, we quantize the theory and present, in particular, a prescription to regularize the physical inner product of the canonical theory. We show how the resulting transition amplitudes are dual to evaluations of Feynman diagrams coupled to three-dimensional quantum gravity. Finally, we remove the regulator by proving the topological invariance of the transition amplitudes.

Paper Structure

This paper contains 28 sections, 3 theorems, 51 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Classical theory
  3. Action principle and classical symmetries
  4. Physical interpretation: the flat solution
  5. The point particle in $2+1$ dimensions
  6. The string in $3+1$ dimensions
  7. Geometrical interpretation: cosmic strings and topological defects
  8. Cosmic strings
  9. Many-strings-solution
  10. Quantum Theory
  11. Canonical setting
  12. Quantum kinematics : the Gauss law
  13. Auxiliary Hilbert space $\mathcal{H}$
  14. String spin network states
  15. Quantum dynamics: the curvature constraint
  16. ...and 13 more sections

Key Result

Theorem 1

(Invariance theorem) Let $\Psi_{\Gamma}$ denote a string spin network element of a given basis of $\mathcal{H}_{kin}$ defined with respect to the one-complex $\Gamma$. Choose a regulator $\mathcal{R}_{(\eta,\epsilon)} = (T_{\eta}, (\Delta_{\epsilon},\partial \Delta_{\epsilon}), T, \alpha, C)$ consis

Figures (7)

  • Figure 1: A typical string spin network (the string is represented by the bold line).
  • Figure 2: The $(4,1)$ and $(3,2)$ bistellar moves.
  • Figure 3: The $(3,1)$ and $(2,2)$ shellings, their dual moves and the associated boundary bistellars.
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Proposition 1
  • Corollary 1