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 Sd geometries admit such generalised parallelisations with an SO(d+1) frame algebra. Thus we show that the remarkable consistent truncations on S3, S4, S5 and S7 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 S5.

Spheres, Generalised Parallelisability and Consistent Truncations

We give a general review of extended supergravities and their gauging using the duality-covariant embedding tensor formalism. Although the focus is on four-dimensional theories, an overview of the gauging procedure and the related tensor hierarchy in the higher-dimensional models is given. The relation of gauged supergravities to flux compactifications is discussed and examples are worked out in detail.

Gauged Supergravities.

We use exceptional field theory as a tool to work out the full nonlinear reduction ansatz for the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ compactification of IIB supergravity and its noncompact counterparts in which the sphere ${S}^{5}$ is replaced by the inhomogeneous hyperboloidal space ${H}^{p,q}$. The resulting theories are the maximal 5D supergravities with gauge groups $\mathrm{SO}(p,q)$. They are consistent truncations in the sense that every solution of 5D supergravity lifts to a solution of IIB supergravity. In particular, every stationary point and every holographic renormalization group flow of the scalar potentials for the compact and noncompact 5D gaugings directly lift to solutions of IIB supergravity.

Consistent Type IIB Reductions to Maximal 5D Supergravity

The framework of exceptional field theory is extended by introducing consistent deformations of its generalised Lie derivative. For the first time, massive type IIA super-gravity is reproduced geometrically as a solution of the section constraint. This provides a unified description of all ten- and eleven-dimensional maximal supergravities. The actionof the E7(7) deformed theory is constructed, and reduces to those of exceptional field theory and gauged maximal supergravity in respective limits. The relation of this new framework to other approaches for generating the Romans mass non-geometrically is discussed.

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The exceptional story of massive IIA supergravity

Massive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the D = 4 fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an SO(1,3)×SL(7)–covariant way. Then, we employ an N = 8 SL(7)–covariant restriction of the D = 4 tensor hierarchy in order to find the full embedding. The redundant D = 4 degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present N = 8 truncation of massive IIA on S 6 and the N = 2 truncation ′

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Consistent N = 8 truncation of massive IIA on S 6

All N=4 conformal supergravities in four space-time dimensions are constructed. These are the only N=4 supergravity theories whose actions are invariant under off-shell supersymmetry. They are encoded in terms of a holomorphic function that is homogeneous of zeroth degree in scalar fields that parametrize an SU(1,1)/U(1) coset space. When this function equals a constant the Lagrangian is invariant under continuous SU(1,1) transformations. The construction of these higher-derivative invariants also opens the door to various applications for nonconformal theories.

Construction of all N=4 conformal supergravities.

IIB supergravity is reformulated with a manifest local USp(8) invariance that makes the embedding of five-dimensional maximal supergravities transparent. In this formulation the ten-dimensional theory exhibits all the 27 one-form fields and 22 of the 27 two-form fields that are required by the vector-tensor hierarchy of the five-dimensional theory. The missing 5 two-form fields must transform in the same representation as a descendant of the ten-dimensional ‘dual graviton’. The invariant E6(6) symmetric tensor that appears in the vector-tensor hierarchy is reproduced. Generalized vielbeine are derived from the supersymmetry transformations of the vector fields, as well as consistent expressions for the USp(8) covariant fermion fields. Implications are discussed for the consistency of the truncation of IIB supergravity compactified on the five-sphere to maximal gauged supergravity in five space-time dimensions with an SO(6) gauge group.

https://link.springer.com/content/pdf/10.1007%2FJHEP04%282015%29094.pdf

IIB supergravity and the E-6(6) covariant vector-tensor hierarchy

IIB supergravity is reformulated with a manifest local USp(8) invariance that makes the embedding of five-dimensional maximal supergravities transparent. In this formulation the ten-dimensional theory exhibits all the 27 one-form fields and 22 of the 27 two-form fields that are required by the vector-tensor hierarchy of the five-dimensional theory. The missing 5 two-form fields must transform in the same representation as a descendant of the ten-dimensional `dual graviton'. The invariant E6(6) symmetric tensor that appears in the vector-tensor hierarchy is reproduced. Generalized vielbeine are derived from the supersymmetry transformations of the vector fields, as well as consistent expressions for the USp(8) covariant fermion fields. Implications are discussed for the consistency of the truncation of IIB supergravity compactified on the five-sphere to maximal gauged supergravity in five space-time dimensions with an SO(6) gauge group.

/pdf/iib-supergravity-and-the-e6-6-covariant-vector-tensor-2lks9v2b7f.pdf

IIB Supergravity and the E6(6) covariant vector-tensor hierarchy

An extended field theory is presented that captures the full SL(2) × O(6, 6 + n) duality group of four-dimensional half-maximal supergravities. The theory has section constraints whose two inequivalent solutions correspond to minimal D = 10 supergravity and chiral half-maximal D = 6 supergravity, respectively coupled to vector and tensor multiplets. The relation with O(6, 6 + n) (heterotic) double field theory is thoroughly discussed. Non-Abelian interactions as well as background fluxes are captured by a deformation of the generalised diffeomorphisms. Finally, making use of the SL(2) duality structure, it is shown how to generate gaugings with non-trivial de Roo-Wagemans angles via generalised Scherk-Schwarz ansatze. Such gaugings allow for moduli stabilisation including the SL(2) dilaton.

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Franz Ciceri

Papers

The exceptional story of massive IIA supergravity

Construction of all N=4 conformal supergravities.

IIB supergravity and the E-6(6) covariant vector-tensor hierarchy

IIB Supergravity and the E6(6) covariant vector-tensor hierarchy

Double Field Theory at SL(2) angles