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Explicit Non-Abelian Gerbes with Connections

Dominik Rist, Christian Saemann, Martin Wolf

TL;DR

The paper advances non-Abelian gerbes by introducing adjusted connections, removing the conventional fake-flatness constraint and enabling consistent higher parallel transport in physical settings such as supergravity. It develops an explicit cocycle framework via adjusted BRST 2-groupoids and adjusted crossed modules, including a detailed stackification to define adjusted differential cocycles and coboundaries. The authors then compute concrete examples, lifting spin structures on S^4 to string structures as string 2-bundles with adjusted connections, and outlining a string-based realization of non-Abelian self-dual strings through a Penrose–Ward type transform. The work also clarifies the relation between loop-space constructions and string 2-groups, and provides a comprehensive toolkit for applying higher gauge theory to geometric and physical problems, including twistor approaches. Overall, it delivers explicit, implementable cocycle data for adjusted higher bundles and demonstrates physically relevant constructions beyond fake-flat theories.

Abstract

We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on $S^4$, corresponding to an instanton-anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.

Explicit Non-Abelian Gerbes with Connections

TL;DR

The paper advances non-Abelian gerbes by introducing adjusted connections, removing the conventional fake-flatness constraint and enabling consistent higher parallel transport in physical settings such as supergravity. It develops an explicit cocycle framework via adjusted BRST 2-groupoids and adjusted crossed modules, including a detailed stackification to define adjusted differential cocycles and coboundaries. The authors then compute concrete examples, lifting spin structures on S^4 to string structures as string 2-bundles with adjusted connections, and outlining a string-based realization of non-Abelian self-dual strings through a Penrose–Ward type transform. The work also clarifies the relation between loop-space constructions and string 2-groups, and provides a comprehensive toolkit for applying higher gauge theory to geometric and physical problems, including twistor approaches. Overall, it delivers explicit, implementable cocycle data for adjusted higher bundles and demonstrates physically relevant constructions beyond fake-flat theories.

Abstract

We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on , corresponding to an instanton-anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.
Paper Structure (95 sections, 18 theorems, 303 equations)

This paper contains 95 sections, 18 theorems, 303 equations.

Table of Contents

  1. Introduction and results
  2. Principal 2-bundles with adjusted connections
  3. Crossed modules and their morphisms
  4. Crossed modules of Lie groups.
  5. Crossed modules of Lie algebras.
  6. Adjusted BRST 2-groupoid for local connections
  7. Generalities.
  8. Unadjusted BRST complex.
  9. Adjusted BRST complex and adjusted BRST Lie 2-algebroid.
  10. Adjusted BRST 2-groupoid.
  11. Adjusted crossed modules and their morphisms
  12. Adjusted crossed modules of Lie groups.
  13. Special adjustments.
  14. Decomposition induced by special adjustment.
  15. Example.
  16. ...and 80 more sections

Key Result

proposition 1

The unique such deformation are the adjusted (higher) gauge transformations given by where $\kappa$ is a bilinear function $\kappa:\frg\times \frg\rightarrow \frh$ (extended in the evident way to forms taking values in $\frg$ and $\frh$) that satisfies for all $V_{1,2}\in\frg$ and $W\in\frh$.

Theorems & Definitions (48)

  • remark 1
  • proposition 1
  • proof
  • corollary 1
  • lemma 1
  • proof
  • theorem 1
  • proof
  • definition 1
  • remark 2
  • ...and 38 more