Thursday, April 10, 2008

EGG 912: generalized G_2 structure

Warning: a somewhat technical post, not recommended to string theory outsiders

Pacheco and Waldram generalize the G_2 structure of 7-manifolds into the non-geometric (or generalized geometric) context with fluxes. Alternatively speaking, they explain the non-perturbative, M-theoretical counterpart of Hitchin's generalized geometry, namely "exceptional generalized geometry" (EGG).

If you wish, they identify a context in which the third non-trivial irrep of E_7 is relevant. Besides the singlet, E_7 has the fundamental 56 and the adjoint 133. Many people know these two irreps but have you ever heard what comes next? Well, it is 912. I didn't know this number before reading the article. ;-) This representation theory of E_{7(7)} is constructed in the appendix, using a SL(8,R) subgroup.

A quantity phi taking values in the 912-dimensional representation plays the same role as the "metric plus B-field" matrix of the size 2d x 2d transforming under O(d,d), but in the nonperturbative context. Amusingly enough, the relevant nonperturbative extension of Hitchin's generalized metric is a "spin 3" representation - given by a three-box Young diagram under Sp(56,R); the shape of the diagram is an upside-down-reflected "L".

As I mentioned at the beginning, you could also start with the G_2 structure (a special "spin 3" tensor under the spin(7) group, encoding the octonion multiplication table) that captures the preferred directions on a G_2 manifold, a seven-dimensional manifold that brings you from 11 to 4 dimensions and preserves the N=1 SUSY. And you could ask how to extend this structure to allow for non-trivial fluxes but to still preserve the four supercharges in four dimensions.

The three spin(7) indices in the tensor would be replaced by Sp(56) indices because the 7 directions in spin(7) - or their Kaluza-Klein momenta - must be supplemented with all the membrane, fivebrane, and KK monopole wrapping numbers, leading you to 7+21+21+7=56. When you impose all the legitimate linear symmetries and conditions on this rank-three tensor, you end up with the 912. This object knows about the SU(7) structure of the compactification much like the generalized 2d x 2d metric "G" knows about Hitchin's SU(3) x SU(3) structure of SUSY Calabi-Yau generalized compactifications.

I suppose that in acceptable compactifications, this 912-component object must be constrained by additional nonlinear constraints, much like its perturbative counterpart "G", the 2d x 2d matrix, is constrained by "eta^{-1} G eta^{-1} = G^{-1}" where "eta = ((0,1_d),(1_d,0))": what is exactly the constraint? Such a constraint should leave you with 70 components, parameterizing E_{7(7)}/SU(8), components that are constrained by extra differential equations, or do I misunderstand it?

I don't see a counting starting from the total number of moduli in the paper. It would be more enlightening for me. What causes additional confusion is that SU(n) groups appear both in the generic non-supersymmetric context as well as in the N=1 discussion (and in the type II simplifications, too) so it takes time to remember which of the spaces are relevant for the N=1, G_2 case only and which of them are generic.

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