Supergravity and Exceptional Field Theory

Abstract

We review E 6(6) exceptional field theory with a particular emphasis on the embedding of type IIB supergravity, which is obtained by picking the GL(5) × SL(2) invariant solution of the section constraint. We work out the precise decomposition of the E 6(6) covariant fields on the one hand and the Kaluza-Klein-like decomposition of type IIB supergravity on the other. Matching the symmetries, this allows to establish the precise dictionary between both sets of fields. Finally, we establish on-shell equivalence. In particular, we show how the self-duality constraint for the fourform potential in type IIB is reconstructed from the duality relations in the off-shell formulation of the E 6(6) exceptional field theory.

Uplifting GPPZ: a ten-dimensional dual of $$ \mathcal{N}={1}^{\ast } $$

“…While this was expected to be true for a long time, and supported by preliminary evidence (see for example [7,8]), it was only recently that the consistency of this truncation was rigorously established in [9,10]. Moreover, [10][11][12] give explicit uplift formulae for all ten-dimensional fields in terms of the five-dimensional N = 8 supergravity fields, thus allowing for the explicit construction of…”

$$ \mathcal{N} $$ = 2 supersymmetric S-folds

“…Using the dictionary between the fields of type IIB supergravity and those of E 7(7) -EFT [52,53], together with the S-S ansatz (3.2) involving generalised twist parameters (3.5)-(3.9), one arrives at the final uplift formulae be extracted from the internal generalised metric M M N (x, y i ,ỹ) by performing the grouptheoretical decomposition that is relevant for the embedding of type IIB supergravity into E 7(7) -EFT:…”

“…We refer the reader to the original works [52,53] (and also [27,32]) for more details on the generalised S-S reductions of E 7(7) -EFT and their connection with the gauged maximal supergravities.…”

S-folds and (non-)supersymmetric Janus solutions

“…Equipped with the SL(8) twist matrix in (3.14) and the scaling function in (3.10) it then becomes straightforward (but tedious) to uplift the AdS 4 solutions of Section 2.3 to solutions of E 7(7) -ExFT via the SS ansatz (3.5). Once the explicit form of the external metric g µν (x µ , y i ,ỹ) and the generalised metric M M N (x µ , y i ,ỹ) is determined, one can make use of the dictionary between the fields of E 7(7) -ExFT and the ones of type IIB supergravity [48,49] to obtain an explicit type IIB background. To this end one must first perform a decomposition of the generalised metric M M N under the relevant GL(6) × SL(2) IIB ⊂ E 7 (7) embedding in (3.7) so that the dictionary reads…”