# Topological Quantum Field Theory Seminar

### Structure Theory for Higher WZW Terms IV

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

1. Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
2. Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)