General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical frameworkorFrom 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, an L ∞ -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.

Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions

“…The structure of the paper is as follows: In Section 2 we review briefly the considerations of [41]; we restrict to trivial bundles in particular, since one of the main questions in the present article is the identification of the underlying structural L ∞ algebra (but a generalization to nontrivial bundles is rather immediate following the steps as outlined in [28]). In Section 3 we likewise review the ingredients needed from [18]: the precise content of the gauge fields, a possible set of field strengths, the gauge transformations and, in particular, the eight structural equations, (3.1), that are imposed for defining the theory.…”

Curving Yang-Mills-Higgs gauge theories

“…It can disappear with deformation parameters as in (19) or remain also for R ∇ = 0: in that case, the condition (13) reduces to ginvariance of B, B ∈ Ω 2 (M, g) G , where Lie(G) = g. Even without deformation, such terms, with higher derivatives respecting the symmetries, can arise in the process of renormalization of sigma models like (2). In higher gauge theories, part of B can also become dynamical [6].…”

“…Here, we call A simply a gauge field and F its field strength (cf. also [6,11])-or "YM-connection" and "YM-curvature", respectively, so as to distinguish them well from the connection ∇ and its curvature R ∇ : together with the target Lie algebroid, ∇ is fixed for a given "curved YMH theory"-like the Lie algebra g in the standard situation. It remains to calculate the behavior of (11) with respect to the gauge transformations (10).…”

Higher gauge theories from Lie n-algebras and off-shell covariantization

“…The gauge fields are obtained by a pullback map a * and a degree preserving mapã * as in [22]. Given a map between graded manifolds, a : T [1]Σ → M n , the pullback of elements of C ∞ (M n ) by a * gives superfields.…”

“…The authors of [22] discuss transformations generated by terms corresponding to α (1) above, as degree preserving coordinate transformations. In this section, we consider canonical transformations of QP-manifolds on T * [n]N .…”