Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations

Abstract

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of $(\infty ,1)$-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to $L_{\infty }$-algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.

v2: 80 pages, typos fixed, published version

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• $D=6$, $\mathcal{N}=(2,0)$ and $\mathcal{N}=(4,0)$ theories
• Towards an M5-Brane Model I: A 6d Superconformal Field Theory
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• Towards an M5-Brane Model II: Metric String Structures
• Adjusted Parallel Transport for Higher Gauge Theories
• The Non-Abelian Self-Dual String and the (2,0)-Theory
• Topological invariant of 4-manifolds based on a 3-group
• Supersymmetric Yang-Mills Theory as Higher Chern-Simons Theory
• Algebraic formulation of higher gauge theory
• Higher Structures in M-Theory
• Higher Gauge Theory
• Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
• Symmetry breaking mechanisms of the 3BF action for the Standard Model coupled to gravity
• Chern correspondence for higher principal bundles
• Correspondence between 3BF and Einstein-Cartan formulations of quantum gravity
• Differentiating $L_{\infty }$ groupoids
• Explicit Non-Abelian Gerbes with Connections