We present a generalised geometry framework for systematically constructing consistent truncations of ten-and eleven-dimensional supergravity preserving varying fractions of supersymmetry. Truncations arise when there is a reduced structure group G S of the exceptional generalised geometry, such that the intrinsic torsion is a G S -singlet. The matter content of the truncated theory follows from group-theoretical arguments, while the gauging is determined by the sub-algebra of generalised diffeomorphisms generated by the G S -singlet vectors. After discussing the general ideas across different spacetime dimensions and amounts of supersymmetry, we provide detailed formulae for truncations to gauged half-maximal supergravity in five dimensions. In particular, we establish an expression for the generalised metric on the exceptional tangent bundle, which determines the scalar truncation ansatz. As applications, we show that this formalism gives a simple derivation of a new consistent truncation of type IIB supergravity on β-deformed Lunin-Maldacena geometries, yielding half-maximal supergravity coupled to two vector multiplets, and of the truncation of eleven-dimensional supergravity on Maldacena-Núñez geometries, given by S 4 twisted over a Riemann surface, which leads to half-maximal supergravity coupled to three vector multiplets. Conclusions 47A Type IIB E 6(6) generalised geometry 50 B Generalised vectors in angular coordinates on M 6 53 -1 -To give the ansatz for the two-forms one has to compute the tensors J A in the bundle N ≃ det T * M ⊗ E * . As for the Sasaki-Einstein truncation, these are obtained acting on the dual generalised vectors K * with the internal volume, as in (3.16),fl +r vol 4 −n C fl , J 5 = 1 √ 2 −r + β vol 4 −n vol 4 −r C fl , J 6 = 1 √ 2 n − β C fl +r vol 4 +n C fl , J 7 = 1 √ 2 r + β vol 4 −n vol 4 +r C fl , (4.74)

We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal $$ \mathcal{N} $$ N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5×wM solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the $$ \mathcal{N} $$ N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.

Using exceptional generalised geometry, we classify which five-dimensional $$ \mathcal{N} $$ N = 2 gauged supergravities can arise as a consistent truncation of 10-/11-dimensional supergravity. Exceptional generalised geometry turns the classification into an algebraic problem of finding subgroups GS ⊂ USp(8) ⊂ E6(6) that preserve exactly two spinors. Moreover, the intrinsic torsion of the GS structure must contain only constant singlets under GS, and these, in turn, determine the gauging of the five-dimensional theory. The resulting five-dimensional theories are strongly constrained: their scalar manifolds are necessarily symmetric spaces and only a small number of matter multiplets can be kept, which we completely enumerate. We also determine the largest reductive and compact gaugings that can arise from consistent truncations.