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Generalized forms of types N = 1, 2 and higher gauge theory

Danhua Song, Mengyao Wu

TL;DR

The paper advances a unified framework for higher gauge theory by employing generalized differential calculus to encode higher connections, curvatures, and gauge transformations within generalized forms of type $N=1$ and $N=2$. It constructs complete calculi for higher algebra- and group-valued generalized forms, develops corresponding 2- and 3-gauge theories, and derives action principles for higher Chern–Simons and Yang–Mills theories, including explicit transformation laws and Bianchi identities. This approach yields a modular, recursive description that naturally extends ordinary gauge theory to higher structures, enabling systematic derivations of topological and Yang–Mills-type actions in higher settings. The results offer a versatile toolkit with potential applications in string theory, topological field theories, and quantum gravity, while suggesting avenues for extending the formalism to higher $N$, supersymmetry, and related physical contexts.

Abstract

We present a unified formulation for higher gauge theory using generalized forms, encompassing higher connections, curvatures, and gauge transformations. We begin by developing the calculus of generalized forms valued in higher algebras and groups, including the associated higher Maurer--Cartan forms and equations. Using generalized forms of types N = 1, 2, we then provide a complete description of higher gauge structures. Finally, we derive the action functionals for higher Chern--Simons and Yang--Mills theories as applications of the formalism.

Generalized forms of types N = 1, 2 and higher gauge theory

TL;DR

The paper advances a unified framework for higher gauge theory by employing generalized differential calculus to encode higher connections, curvatures, and gauge transformations within generalized forms of type and . It constructs complete calculi for higher algebra- and group-valued generalized forms, develops corresponding 2- and 3-gauge theories, and derives action principles for higher Chern–Simons and Yang–Mills theories, including explicit transformation laws and Bianchi identities. This approach yields a modular, recursive description that naturally extends ordinary gauge theory to higher structures, enabling systematic derivations of topological and Yang–Mills-type actions in higher settings. The results offer a versatile toolkit with potential applications in string theory, topological field theories, and quantum gravity, while suggesting avenues for extending the formalism to higher , supersymmetry, and related physical contexts.

Abstract

We present a unified formulation for higher gauge theory using generalized forms, encompassing higher connections, curvatures, and gauge transformations. We begin by developing the calculus of generalized forms valued in higher algebras and groups, including the associated higher Maurer--Cartan forms and equations. Using generalized forms of types N = 1, 2, we then provide a complete description of higher gauge structures. Finally, we derive the action functionals for higher Chern--Simons and Yang--Mills theories as applications of the formalism.
Paper Structure (26 sections, 11 theorems, 171 equations, 1 figure, 1 table)

This paper contains 26 sections, 11 theorems, 171 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Generalized forms and gauge theory
  3. Generalized forms of type $N$
  4. Basic properties and formalism
  5. Exterior derivatives and canonical basis
  6. Generalized gauge theory
  7. Generalized connections and curvatures
  8. Generalized gauge transformations
  9. Higher algebra- and group-valued generalized forms
  10. Higher algebra-valued generalized forms
  11. Higher group-valued generalized forms
  12. Type $N=1$: Lie 2-group-valued generalized 0-forms
  13. Type $N=2$: Lie 3-group-valued generalized 0-forms
  14. Higher gauge theories in generalized formulations
  15. Type $N=1$: 2-connections and 2-gauge transformations
  16. ...and 11 more sections

Key Result

Proposition 2.1

Within any locally contractible domain, there exists a canonical basis $\xi^i$ such that $\underline{d} \xi^i = 0$.

Figures (1)

  • Figure 1: Correspondence between generalized forms and (higher) gauge theories

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2: Exterior product
  • Definition 2.3: Symmetric inner product
  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1: Generalized forms of type $N=1$ valued in Lie 2-algebras
  • Definition 3.2: Generalized forms of type $N=2$ valued in Lie 3-algebras
  • Definition 3.3: Lie 2-group-valued generalized 0-forms
  • Theorem 3.1
  • proof
  • ...and 18 more