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Categorical generalization of BF theory coupled to gravity

A. D. López-Hernández, Graciela Reyes-Ahumada, Javier Chagoya

TL;DR

The paper develops a pedagogical introduction to higher gauge theory based on 2-connections and then constructs a categorical generalization of BF theory, BFCG (aka 2BF). It couples gravity to BFCG through volume-form constructions, showing a bridge to unimodular gravity and extending BF sequestered gravity to generic crossed-module (2-group) settings, with abelian reductions recovering known results. It further extends the framework by introducing a C-based volume form and a kinetic term, revealing new couplings and potential implications for multi-metric gravity and cosmology. Overall, the work expands the landscape of gravity–gauge couplings through higher-categorical structures and lays groundwork for exploring non-abelian higher gauge effects in gravitational dynamics.

Abstract

We present a thorough introduction to the tools of category theory required for formulating gauge theories based on 2-connections. We provide a detailed construction of the categorical generalization of BF theory, dubbed BFCG, also known as 2BF. Similar to BF gravity, it is known that BFCG can be deformed to give general relativity. Here, we obtain an alternative relation between BFCG and gravity, which consists of coupling general relativity and BFCG by means of the volume form constructed out of the BFCG connections. The resulting theory, closely related to unimodular gravity, is a generalization of BF sequestered gravity not only in the sense that it adds new fields but also in that it allows for new choices for the volume form that is coupled to gravity. Furthermore, we show that BF sequestered gravity in the abelian case is recovered for a specific choice of the 2-group.

Categorical generalization of BF theory coupled to gravity

TL;DR

The paper develops a pedagogical introduction to higher gauge theory based on 2-connections and then constructs a categorical generalization of BF theory, BFCG (aka 2BF). It couples gravity to BFCG through volume-form constructions, showing a bridge to unimodular gravity and extending BF sequestered gravity to generic crossed-module (2-group) settings, with abelian reductions recovering known results. It further extends the framework by introducing a C-based volume form and a kinetic term, revealing new couplings and potential implications for multi-metric gravity and cosmology. Overall, the work expands the landscape of gravity–gauge couplings through higher-categorical structures and lays groundwork for exploring non-abelian higher gauge effects in gravitational dynamics.

Abstract

We present a thorough introduction to the tools of category theory required for formulating gauge theories based on 2-connections. We provide a detailed construction of the categorical generalization of BF theory, dubbed BFCG, also known as 2BF. Similar to BF gravity, it is known that BFCG can be deformed to give general relativity. Here, we obtain an alternative relation between BFCG and gravity, which consists of coupling general relativity and BFCG by means of the volume form constructed out of the BFCG connections. The resulting theory, closely related to unimodular gravity, is a generalization of BF sequestered gravity not only in the sense that it adds new fields but also in that it allows for new choices for the volume form that is coupled to gravity. Furthermore, we show that BF sequestered gravity in the abelian case is recovered for a specific choice of the 2-group.
Paper Structure (18 sections, 14 theorems, 157 equations)

This paper contains 18 sections, 14 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Principal bundles and connections
  4. Physical applications
  5. Categorical approach
  6. The connection functor
  7. Strict 2-categories and strict 2-functors
  8. Crossed modules and strict 2-groups
  9. 2-connections
  10. Physics from 2-connections
  11. BFCG
  12. BFCG sequestered gravity
  13. Coupling with the form C
  14. Couplings with kinetic term
  15. Conclusions
  16. ...and 3 more sections

Key Result

Lemma 7

Let $A\in \mathfrak{g}$. Then $X_{p}^{A}\in V_{p}P$ for any $p\in P$.

Theorems & Definitions (60)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • Theorem 9
  • ...and 50 more