General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identities to twisted Courant algebroids

Abstract

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into a so-called Q-structure or, equivalently, a Lie infinity algebroid. This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction. This article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree p, we pay particular attention to p=2, i.e. 1- and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

89 pages, 1 figure, to be published in the International Journal of Geometric Methods in Modern Physics

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Cited by in corpus (39)

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• Extended Riemannian Geometry I: Local Double Field Theory
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• Gauging without Initial Symmetry
• A new formulation of higher parallel transport in higher gauge theory
• Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations
• Towards an M5-Brane Model II: Metric String Structures
• Curving Yang-Mills-Higgs Gauge Theories
• The Pontryagin Class for Pre-Courant Algebroids
• Non-abelian Gerbes and Enhanced Leibniz Algebras
• $L_{\infty }$-Algebra Models and Higher Chern-Simons Theories
• Enhanced Leibniz Algebras: Structure Theorem and Induced Lie 2-Algebra
• Topological Field Theories induced by twisted R-Poisson structure in any dimension
• Fluxes in Exceptional Field Theory and Threebrane Sigma-Models
• The Non-Abelian Self-Dual String and the (2,0)-Theory
• Strings in Singular Space-Times and their Universal Gauge Theory
• A Lie based 4-dimensional higher Chern-Simons theory
• Courant sigma model and $L_{\infty }$-algebras
• Leibniz-Yang-Mills Gauge Theories and the 2-Higgs Mechanism
• AKSZ constructions for topological membranes on $G_2$-manifolds
• All maximal gauged supergravities with uplift
• Generalized Higher Gauge Theory
• On higher holonomy invariants in higher gauge theory I
• Algebraic formulation of higher gauge theory
• On higher holonomy invariants in higher gauge theory II
• Duality and higher Buscher rules in p-form gauge theory and linearized gravity
• Higher Gauge Theory
• Higher Gauge Theories from Lie n-algebras and Off-Shell Covariantization
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• Basic curvature and the Atiyah cocycle in gauge theory
• Representation of a gauge field via intrinsic "BRST" operator
• Omni $n$-Lie algebras and linearization of higher analogues of Courant algebroids
• Y-algebroids and $E_{7(7)}\times {\mathbb{R}}^{+}$-generalised geometry
• Twisted Weil Algebras for the String Lie 2-Algebra
• Off-Shell Covariantization of Algebroid Gauge Theories
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Exceptional algebroids and type IIA superstrings
• Supersymmetric Poisson and Poisson-supersymmetric sigma models
• Instances of Higher Geometry in Field Theory