Spheres, Generalised Parallelisability and Consistent Truncations

Abstract

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten-and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises as a generalised version of a conventional Scherk-Schwarz reduction with the frame algebra encoding the embedding tensor of the reduced theory. The key new result is that all round-sphere S d geometries admit such generalised parallelisations with an SO(d + 1) frame algebra. Thus we show that the remarkable consistent truncations on S 3 , S 4 , S 5 and S 7 are in fact simply generalised Scherk-Schwarz reductions. This description leads directly to the standard non-linear scalar-field ansatze and as an application we give the full scalar-field ansatz for the type IIB truncation on S 5 .

Duality twisted reductions of Double Field Theory of Type II strings

“…Therefore, the existence of this extra constraint can be seen as a sign that the reduction we have studied here should be related to duality twisted reductions of Exceptional Field Theory (EFT), which is a U-duality invariant extension of supergravity [56][57][58][59]. Indeed, the reduction of EFT on generalised parallelisable manifolds [60] (which corresponds to a reduction with a duality twisted anzats of the type we have considered here) gives rise to maximal gauged supergravity upon imposing a section constraint, which is the analogue of the strong constraint of DFT [61][62][63]. A flux formulation of (a particular type of) EFT is also available and geometric and non-geometric RR fluxes were studied also in this formulation [64].…”

The bosonic string on string-size tori from double field theory

“…the vielbein (4.23) gives the desired g × g algebra [9,[14][15][16]. We now show that this vielbein also satisfies the assumption (4), namely the generalized metric when restricted to the Cartan subsector reduces to that of the torus.…”

“…we use a different vielbein for the left and the right sectors, e,ē, giving rise to the same metric (see footnote 12)), namely [9] …”

“…The setup relevant to this paper is that of generalized Scherk-Schwarz reductions of double field theory [7,8] on generalized parallelizable manifolds [9], namely manifolds for which there is a globally defined generalized frame on the double space, such that the Cbracket algebra (the generalization of the Lie algebra that is needed in double field theory) on the frame gives rise to (generalized) structure constants, which are on the one hand trully constant, and on the other should satisfy Jacobi identities. As in standard ScherkSchwarz reductions [10], the only allowed dependence on the internal (double) coordinates is through this frame.…”

Generalised structures for N = 1 $$ \mathcal{N}=1 $$ AdS backgrounds

“…The singlet in the torsion that we identified above would not break this commutant group (since it is a singlet ofH 4 = USp(4)), reflecting that the AdS R-symmetry is the full USp(4) group for N = 2. The generalised parallelisation on AdS 7 × S 4 is presented in detail in [30], as an example of the generic appearance of this structure in maximally supersymmetric compactifications. …”