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Quantization of strings and branes coupled to BF theory

John C. Baez, Alejandro Perez

TL;DR

The paper extends BF theory by coupling it to extended objects: strings in 3+1 dimensions and general $(d-3)$-branes in $d$ dimensions, yielding flat connections away from the brane with conical singularities along the brane. A canonical analysis motivates a quantization scheme in which the auxiliary Hilbert space is a tensor product of a BF sector and a brane sector, leading to membrane/string spin networks that implement the Gauss constraint and, upon imposing the curvature constraint, give a spin-foam–type physical inner product. The construction generalizes to arbitrary dimensions and compact gauge groups, with potential connections to gravity formulations like MacDowell–Mansouri and Plebanski, and it suggests rich brane statistics described by loop braid groups in the nonperturbative regime. The framework provides a concrete path to incorporating matter in topological BF theories and offers a foundation for exploring higher-dimensional quantum gravity models with extended defects.

Abstract

BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in three dimensions generalizes in a direct way to BF theory in d dimensions coupled to (d-3)-branes. In the resulting model, the connection is flat except along the membrane world-sheet, where it has a conical singularity whose strength is proportional to the membrane tension. As a step towards canonically quantizing these models, we show that a basis of kinematical states is given by `membrane spin networks', which are spin networks equipped with extra data where their edges end on a brane.

Quantization of strings and branes coupled to BF theory

TL;DR

The paper extends BF theory by coupling it to extended objects: strings in 3+1 dimensions and general -branes in dimensions, yielding flat connections away from the brane with conical singularities along the brane. A canonical analysis motivates a quantization scheme in which the auxiliary Hilbert space is a tensor product of a BF sector and a brane sector, leading to membrane/string spin networks that implement the Gauss constraint and, upon imposing the curvature constraint, give a spin-foam–type physical inner product. The construction generalizes to arbitrary dimensions and compact gauge groups, with potential connections to gravity formulations like MacDowell–Mansouri and Plebanski, and it suggests rich brane statistics described by loop braid groups in the nonperturbative regime. The framework provides a concrete path to incorporating matter in topological BF theories and offers a foundation for exploring higher-dimensional quantum gravity models with extended defects.

Abstract

BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in three dimensions generalizes in a direct way to BF theory in d dimensions coupled to (d-3)-branes. In the resulting model, the connection is flat except along the membrane world-sheet, where it has a conical singularity whose strength is proportional to the membrane tension. As a step towards canonically quantizing these models, we show that a basis of kinematical states is given by `membrane spin networks', which are spin networks equipped with extra data where their edges end on a brane.

Paper Structure

This paper contains 9 sections, 56 equations, 1 figure.

Table of Contents

  1. The canonical analysis for $d=4$
  2. Quantization
  3. The $BF$ auxiliary Hilbert space
  4. The string auxiliary Hilbert space
  5. The kinematical Hilbert space
  6. The physical Hilbert space
  7. The general case: membranes coupled to $BF$ theory
  8. Conclusions
  9. Acknowledgements

Figures (1)

  • Figure 1: A typical string spin network. The Gauss law implies that if a single spin network edge $e$ ends at some point $x$ on the string, the representation $\rho_e$ is evaluated on the product of the associated holonomy $h_e(A)$ and the value of the $\lambda$ field at $x$.