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

References in corpus (6)

• AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Lie Algebroid Yang Mills with Matter Fields
• 2d Gauge Theories and Generalized Geometry
• Dual linearized gravity in D=6 coupled to a purely spin-two field of mixed symmetry (2,2)
• Reducible second-class constraints of order L: An irreducible approach

Cited by in corpus (14)

• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• Non-abelian Gerbes and Enhanced Leibniz Algebras
• Topological Field Theories induced by twisted R-Poisson structure in any dimension
• Fluxes in Exceptional Field Theory and Threebrane Sigma-Models
• All maximal gauged supergravities with uplift
• Leibniz-Yang-Mills Gauge Theories and the 2-Higgs Mechanism
• Higher Gauge Theory
• Duality and higher Buscher rules in p-form gauge theory and linearized gravity
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• Basic curvature and the Atiyah cocycle in gauge theory
• Y-algebroids and $E_{7(7)}\times {\mathbb{R}}^{+}$-generalised geometry
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Supersymmetric Poisson and Poisson-supersymmetric sigma models
• Instances of Higher Geometry in Field Theory