Schreiber Prequantum field theory and the Green-Schwarz WZW terms

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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

• Structure Theory for Higher WZW Terms

• super Cartan geometry

For more background and survey see

• Higher Prequantum Geometry

For full details see

• differential cohomology in a cohesive topos

Motivation

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 $n$-algebra cohomology, including strings ending on Dp-branes and M2-branes ending on M5-branes.

These lectures explain

• I) super $L_\infty$-algebras and their $L_\infty$-cocyles and extensions;

• II) how to integrate these to globally defined higher WZW terms for super p-branes in ordinary differential cohomology;

• III) and how to obtain conserved brane charges from these as higher Noether charges in prequantum field theory.

Finally in

• IV) we unfold the brane bouquet and compute that M2/M5-brane charges are rationally and unstably in degree-4 cohomotopy, controled by the quaternionic Hopf fibration.

This completes the lectures, but after that in an

• invited research talk we discuss how stably the M2/M5-brane charges exist in ADE-equivariant stable cohomotopy.

Lecture session I

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

Lecture notes at:

• super Cartan geometry

See also the beginning of

• Prequantum field theories from Shifted symplectic structures

Lecture session II

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

Lecture notes at:

• geometry of physics – WZW terms

• Structure Theory for Higher WZW Terms

Based on joint work with Domenico Fiorenza.

Lecture session III

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

Lecture notes at:

• Prequantum field theory (pdf, talk slides)

  (Based on joint work with Igor Khavkine.)

• Lie n-algebras of BPS charges

  (Based on joint work with Hisham Sati)

Lecture session IV

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

Lecture notes at

• The brane bouquet

• The WZW term of the M5-brane

Based on joint work with Domenico Fiorenza and Hisham Sati

Invited talk

• Urs Schreiber,

  Generalized cohomology of M2/M5-branes

  talk at

  main workshop of Higher Structures in String Theory and Quantum Field Theory, ESI Program, Vienna, Dec 7 - 11, 2015

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 $G$-equivariant stable cohomotopy, for $G$ 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.

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