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Principal 3-Bundles with Adjusted Connections

Gianni Gagliardo, Christian Saemann, Roberto Tellez-Dominguez

TL;DR

We develop a framework for adjusted connections on principal $3$-bundles by deriving explicit adjustment data for $3$-term $L_infty$-algebras and formulating a local BRST description. The paper then introduces adjusted $2$-crossed modules of Lie groups and provides a complete global description of principal $3$-bundles with adjusted connections through differential cocycles, including the necessary cocycle, coboundary, and higher coboundary relations and their adjustment conditions. Applications span tensor hierarchies in $d=4$ gauged supergravity, twisted differential string structures, and categorified tori that may uplift T-duality to M-theory via $U$-duality, with the categorified torus serving as a potential tool for lifting dualities in higher dimensions. The central result is that adjusted data, encoded in maps $\kappa_{1,2,3,4}$, yield deformed curvatures $F,H,G$ with modified Bianchi identities and globular/gluing relations, providing a differential refinement of higher gauge theory for principal $3$-bundles.

Abstract

We explore the notion of adjusted connection for principal 3-bundles. We first derive the explicit form of an adjustment datum for 3-term $L_\infty$-algebras, which allows us to give a local description of such adjusted connections and their infinitesimal symmetries. We then introduce the notion of an adjusted 2-crossed module of Lie groups and provide the explicit global description of principal 3-bundles with adjusted connections in terms of differential cocycles. These connections appear in a number of context within high-energy physics, and we list local examples arising in gauged supergravity and a global example arising in various contexts in string/M-theory. Our primary motivation, however, stems from U-duality, and we define a notion of categorified torus that forms an adjusted 2-crossed module, which we hope to be useful in lifting T-duality to M-theory.

Principal 3-Bundles with Adjusted Connections

TL;DR

We develop a framework for adjusted connections on principal -bundles by deriving explicit adjustment data for -term -algebras and formulating a local BRST description. The paper then introduces adjusted -crossed modules of Lie groups and provides a complete global description of principal -bundles with adjusted connections through differential cocycles, including the necessary cocycle, coboundary, and higher coboundary relations and their adjustment conditions. Applications span tensor hierarchies in gauged supergravity, twisted differential string structures, and categorified tori that may uplift T-duality to M-theory via -duality, with the categorified torus serving as a potential tool for lifting dualities in higher dimensions. The central result is that adjusted data, encoded in maps , yield deformed curvatures with modified Bianchi identities and globular/gluing relations, providing a differential refinement of higher gauge theory for principal -bundles.

Abstract

We explore the notion of adjusted connection for principal 3-bundles. We first derive the explicit form of an adjustment datum for 3-term -algebras, which allows us to give a local description of such adjusted connections and their infinitesimal symmetries. We then introduce the notion of an adjusted 2-crossed module of Lie groups and provide the explicit global description of principal 3-bundles with adjusted connections in terms of differential cocycles. These connections appear in a number of context within high-energy physics, and we list local examples arising in gauged supergravity and a global example arising in various contexts in string/M-theory. Our primary motivation, however, stems from U-duality, and we define a notion of categorified torus that forms an adjusted 2-crossed module, which we hope to be useful in lifting T-duality to M-theory.
Paper Structure (21 sections, 8 theorems, 182 equations)

This paper contains 21 sections, 8 theorems, 182 equations.

Table of Contents

  1. Introduction and results
  2. Motivation.
  3. Key results.
  4. Applications.
  5. Local connections on principal 3-bundles
  6. Local L infinity-algebra-valued connection forms
  7. Adjusting L infinity-algebras
  8. Semi-strict Lie 3-algebras
  9. Global connections on principal 3-bundles
  10. Derivation of the unadjusted differential cohomology
  11. Unadjusted differential cocycles and coboundaries
  12. Derivation of the adjustment conditions
  13. Adjusted differential cocycles
  14. Examples
  15. Infinitesimal case: firm adjustments and d=4 gauged supergravity
  16. ...and 6 more sections

Key Result

Proposition 2.1

Consider an $L_\infty$-algebra $\frL$ together with an automorphism $\phi$ on its Weil algebra covering the identity map on its Chevalley--Eilenberg algebra as in (eq:respect_projection). The map $\phi$ defines an adjustment if and only if we have for some structure constants $m_{\alpha_1\ldots \alpha_i}$ and $n_{\alpha_0\alpha_1\ldots \alpha_i}$, where $n_{\alpha_0\alpha_1\ldots \alpha_i}$ vani

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Definition 2.2: Fischer:2024vak
  • Corollary 2.3
  • Remark 2.4
  • Definition 3.1: Adjustment
  • Definition 3.2: Adjusted 2-crossed module of Lie groups
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 11 more