A Higher Stacky Perspective on Chern–Simons Theory

Abstract

The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. In the second part we provide a brief review and outlook of the higher prequantum field theory of which this is a first example. This includes a higher geometric description of supersymmetric Chern-Simons theory, Wilson loops and other defects, generalized geometry, higher Spin structures, anomaly cancellation, and various other aspects of quantum field theory.

Higher Structures in Algebraic Quantum Field Theory

“…As a last remark, we would like to add that moduli stacks of solutions to, e.g., the non‐Abelian Yang‐Mills equation or the Chern‐Simons equation can also be constructed from such a perspective. See [] for the details on Yang‐Mills theory and [] for Chern‐Simons theory.…”

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

“…One can also add source terms and also impose self-duality by hand, in which case the action (1.1) might be referred to as a pseudo-action (see [1]). Recent accounts of higher abelian gauge theory in this context, via differential cohomology, are given in [15,16,17,36].…”

“…This allows the partition function to be defined as a section of a line bundle over the intermediate Jacobian, and requires a quadratic refinement [39]. Discussions on extension to (higher) differential cohomology and stacks are given in [15,16,17]. The formulation that we propose via noncommutative geometry does not suffer from such an immediate problem, because ultimately H 2p+1 ∧ θ H 2p+1 = 0.…”

Lie n-algebras of BPS charges

“…This is in the general spirit of refining field theories to what are called "extended" or "multi-tiered" field theories; we refer the reader to the introductions of [17,20] for more on this general background. The urge to refine Lie 1-algebras of infinitesimal symmetries to L ∞ -algebras becomes more pronounced as one aims to pass from infinitesimal to finite symmetries.…”

“…The reader may find exposition and introduction for such objects in the context of field theory in [20], but this object is easily described and understood already by its defining universal property. That is, being a smooth universal moduli stack for higher connections means precisely that it is a generalized smooth space of sorts, with the following characterizing properties: for X any smooth manifold, then JHEP03 (2017)087 3. smooth homotopies-of-homotopies correspond to gauge-of-gauge transformations, and so forth, up to p-fold homotopies-of-homotopies.…”