2014/07/28 by Melchior Grützmann, Melchior Grutzmann, Thomas Strobl · 1 citation
Mathematics · Physics and Astronomy · #Axiom #BRST quantization #Black Holes and Theoretical Physics #Differential geometry #Gauge (firearms) #Gauge theory #Homotopy and Cohomology in Algebraic Topology #Introduction to gauge theory #Nonlinear Waves and Solitons #Scalar (mathematics) #Supersymmetric gauge theory #Tower #Type (biology) #hep-th #math-ph #math.DG #math.MP
paper · pdf · doi:10.1142/s0219887815500097
89 pages, 1 figure, to be published in the International Journal of Geometric Methods in Modern Physics
arxiv created 2014/07/28 · openalex publication_date 2014/09/05 · arxiv updated 2015/06/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 what is called a 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 and is given by what is called the derived bracket construction. This paper 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 nonlinearly 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.