### Urs Schreiber

Shared publicly -**Supergravity BPS Charges**, that's is the topic of the next session of the

*Higher Cartan Geometry Seminar*[1]. Lecture notes are at [2].

[1] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry

[2] http://ncatlab.org/nlab/show/geometry+of+physics+--+BPS+charges

Consider (X,g) a super-spacetime and ω a degree (p+2)-differential form on X which is a WZW curvature form definite on the super Lie algebra cocycle for Green-Schwarz super p-brane sigma model with target space X. Then the Polyvector extensions of the super Lie algebra of super-isometries of (X,g) by charges Z of Noether currents of the super p-brane sigma model are known as algebras of

*BPS charges*. (The spacetime (X,g,ω) is called a supergravity 1/k-BPS state if the dimension of the space of supercharges Q in the kernel of the above bracket is 1/k-th of that of super Minkowski spacetime).

This is well understood in the literature (Azcárraga-Gauntlett-Izquierdo-Townsend 89) for the case that X is locally modeled on an ordinary super Minkowski spacetime and that the p-brane species is in the old brane scan (e.g the type II superstrings, the heterotic superstring and the M2-brane, also e.g. the super 1-brane in 3d and the 3-brane in 6d, but not the D-branes and not the M5-brane).

For the full story of string theory this needs to be refined in three ways (see Fiorenza-Sati-Schreiber 13), and this has been left open in the literature, for previous lack of a higher differential geometry that could handle this:

1. For a genuine global definition of the Green-Schwarz super p-brane sigma model with target (X,ω), the WZW curvature form ω needs to be prequantized to a globally well-defined WZW term, a genuine cocycle in Deligne cohomology (a circle (p1+1)-bundle with connection).

(The need for this has broadly been ignored, one place where it is mentioned is (Witten 86, p. 17).)

2. For the inclusion of p2-brane charges of p2-branes on which p1-branes may end (for p1=1: type II strings ending on D-branes, for p1=2 and p2=5 M2-branes ending on M5-branes) then X is to be locally modeled on an extended super Minkowski spacetime, hence on a super orbispace, hence a curved spacetime now is an object in higher Cartan geometry and one needs to make sense of Noether currents there.

(Arguments in this direction for the D-branes have been given in (Hammer 97) and for the M5-brane in (Sorokin-Townsend 97).)

3. For inclusion of non-infinitesimal isometries one needs the global structure of the full supergroup of BPS charges, not just its super Lie algebra.

Here we discuss how to solve these problems in full generality. Specified to the situation in 11-dimensional supergravity with M2-branes and M5-branes we find that the BPS charges traditionally seen in the M-theory super Lie algebra as living in ordinary cohomology H5(X)⊕H2(X) receive corrections by d4-differentials of a Serre spectral sequence given by cup product with the class of the supergravity C-field. This is in higher analogy to how D-brane charges are well known (Maldacena-Moore-Seiberg 01) to be in ordinary cohomology only up to corrections of the d3-differential (and higher) in an Atiyah-Hirzebruch spectral sequence for twisted K-theory,given by cup product with the class of the B-field. This supports the conjecture (Sati 10) that M5-brane charge should really be in twisted elliptic cohomology, since this is what is canonically twisted by these degree-4 classes (Ando-Blumberg-Gepner 10). (Realizing this fully amounts to refining the M5-WZW term that we construct in ordinary differential cohomology below to ellitptic differential cohomology. Discussion of that refinement is beyond the scope of this page here.)

We also close a gap in (AGIT89): what is strictly derived there from the Noether theorem is extension of the supersymmetry algebra by differential forms, while the argument that it is only the de Rham cohomology class of these forms that matters relies on physics intuition. We find here that the Lie algebra of conserved currents extending the (super)isometry algebra is naturally not just a (super)Lie algebra but a (super)Lie (p+1)-algebra including higher order symmetries of Noether symmetries. It is by quotienting these out when restricting the current Lie n-algebra to its lowest Postnikov stage that current forms pass to their de Rham equivalence classes. Accordingly, the fully globalized current groups that we find are really (super)smooth n-groups. For instance the M-theory super Lie algebra is refines to a super Lie 6-group, where 6=5+1 is the dimension of the M5-brane worldvolume.

http://ncatlab.org/nlab/show/geometry+of+physics+--+BPS+charges

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