Prequantum field theory and the Green-Schwarz WZW terms

Urs Schreiber,

Prequantum field theory and the Green-Schwarz WZW terms

lecture series and talk at

Higher Structures in String Theory and Quantum Field Theory,

ESI Vienna, Nov 30 - Dec 4 (2015)

Lecture notes are at

For more background and survey see

For full details see



The central open question in string theory is the identification of the non-perturbative theory. One aspect of this is the identification of the correct cohomology theory in which M-brane charges are cocycles.

In perturbative string theory the F1/Dp-brane charges are in Z/2-equivariant twisted differential K-theory on orientifolds (see there for details).

Non-perturbatively, this must be the limiting case of a richer cohomology theory. For instance in F-theory there is S-duality mixing the NS 3-form with the RR 3-form, which however are not mixable as components of twisted differential K-theory (one is part of the twist, the other is part of the K-theory itself). Potentially a modular equivariant elliptic cohomology is needed, which at the nodal curve-degeneration point of the F-theory elliptic fibration reduces to K-theory.

One way to make progress on this question is to first carefully consider the constraints from supersymmetry on brane charges in rational cohomology (de Rham cohomology) and then apply higher Lie integration to produce non-rational information from this.

The old brane scan classifies the rational charges of those branes with no “tensor multiplet” fields on their worldvolume, via cocycles in super Lie algebra cohomology. This is completed by the “brane bouquet” which classifies the rational charges of all branes in super Lie nn-algebra cohomology, including strings ending on Dp-branes and M2-branes ending on M5-branes.

These lectures explain

Finally in

This completes the lectures, but after that in an

Lecture session I

Topics: Super L L_\infty-algebroids, cocycles, and L L_\infty-extensions.

Lecture notes at:

See also the beginning of

Lecture session II

Topics: Lie integration of super L L_\infty-algebras to smooth higher groups and of L L_\infty-cocycles to higher WZW terms

Lecture notes at:

Based on joint work with Domenico Fiorenza.

Lecture session III

Topics: Prequantum field theory, L L_\infty-algebras of conserved Noether currents, BPS brane charges

Lecture notes at:

Lecture session IV

Topics: The brane bouquet, definite globalization of WZW terms, M2/M5-brane charges

Lecture notes at

Based on joint work with Domenico Fiorenza and Hisham Sati

Invited talk

Abstract. While it has become well-known that the charges of F1/Dp-branes in type II string theory need to be refined from de Rham cohomology to certain twisted generalized differential cohomology theories, it is an open problem to determine the generalized cohomology theory for M2-brane/M5-branes in 11 dimensions. I discuss how a careful re-analysis of the old brane scan (arXiv:1308.5264 , arXiv:1506.07557, joint with Fiorenza and Sati) shows that rationally and unstably, the M2/M5 brane charge is in degree-4 cohomotopy. While this does not integrate to the generalized cohomology theory called stable cohomotopy, it does integrate to GG-equivariant stable cohomotopy, for GG a non-cyclic finite group of ADE type. On general grounds, such an equivariant cohomology theory needs to be evaluated on manifolds with ADE orbifold singularities, and picks up contributions from the orbifold fixed points. Both of these statements are key in the hypothesized but open problem of gauge enhancement in M/F-theory.

Last revised on December 11, 2015 at 01:39:34. See the history of this page for a list of all contributions to it.