Flux algebra, Bianchi identities and Freed–Witten anomalies in F-theory compactifications

Abstract

We discuss the structure of 4D gauged supergravity algebras corresponding to globally nongeometric compactifications of F-theory, admitting a local geometric description in terms of 10D supergravity. By starting with the well known algebra of gauge generators associated to nongeometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2, Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much information about the higher dimensional theory. In particular, tadpole equations and Bianchi identities are obtainable as Jacobi identities of the algebra. When a sector of magnetized (p, q) 7-branes is included, certain closed axions are gauged by the U (1) transformations on the branes. We indicate how the diagonal gauge generators of the branes can be incorporated into the full algebra, and show that Freed-Witten constraints and tadpole cancellation conditions for (p, q) 7-branes can be described as Jacobi identities satisfied by the algebra mixing bulk and brane gauge generators.

How to halve maximal supergravity

“…Furthermore, the connection between this formalism and string theory realisations has been investigated both for N = 4 in e.g. [6][7][8] and for N = 8 in e.g. [9][10][11].…”

“…. , 12 respectively denote SL (2) and SO (6,6) vector indices of the global symmetry group G = SL(2) × SO (6,6). Besides the consistency conditions imposed by half-maximal supergravity, the f -tensor has been conjectured to also satisfy…”

“…3) 6) where 32 and 32' respectively denote left-and right-handed Majorana-Weyl (M-W) spinorial representations of SO (6,6) and similarly for the other spinorial irrep's. 2 The decomposition of the 56 in (2.3) amounts to the index splitting M = (α, M ) ⊕ µ, where α = ± is an electric-magnetic SL(2) index, M = 1, .…”

“…By virtue of the truncation to half-maximal supergravity, we expect the quadratic constraints solely to furnish the parity even irrep's appearing in the decompositions (2.4) and (2.6). These are 5,6 133 Z 2 −→ (1, 66) + (3, 1) , (2.26)…”

“…They happen to be odd under the Z2-projection performed in the truncation from maximal to half-maximal theories and hence they are projected out. 6 The first set of quadratic constraints is empty for vanishing ξαM , while the second is for vanishing fαMNP .…”