In this paper we define the analogue of Calabi–Yau geometry for generic D=4, N=2 flux backgrounds in type II supergravity and M‐theory. We show that solutions of the Killing spinor equations are in one‐to‐one correspondence with integrable, globally defined structures in E7(7)×double-struckR+ generalised geometry. Such “exceptional Calabi–Yau” geometries are determined by two generalised objects that parametrise hyper‐ and vector‐multiplet degrees of freedom and generalise conventional complex, symplectic and hyper‐Kähler geometries. The integrability conditions for both hyper‐ and vector‐multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to D=5 and D=6 flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in Ofalse(d,dfalse)×double-struckR+ generalised geometry.

We analyse generic AdS flux backgrounds preserving eight supercharges in D = 4 and D = 5 dimensions using exceptional generalised geometry. We show that they are described by a pair of globally defined, generalised structures, identical to those that appear for flat flux backgrounds but with different integrability conditions. We give a number of explicit examples of such "exceptional Sasaki-Einstein" backgrounds in type IIB supergravity and M-theory. In particular, we give the complete analysis of the generic AdS 5 M-theory backgrounds. We also briefly discuss the structure of the moduli space of solutions. In all cases, one structure defines a "generalised Reeb vector" that generates a Killing symmetry of the background corresponding to the R-symmetry of the dual field theory, and in addition encodes the generic contact structures that appear in the D = 4 Mtheory and D = 5 type IIB cases. Finally, we investigate the relation between generalised structures and quantities in the dual field theory, showing that the central charge and Rcharge of BPS wrapped-brane states are both encoded by the generalised Reeb vector, as well as discussing how volume minimisation (the dual of a-and F -maximisation) is encoded.

We analyse the geometry of generic Minkowski $$ \mathcal{N} $$ N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

We consider finite deformations of the Hull-Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer-Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory. The supersymmetric locus of this action is described by an L 3 algebra. C The off-shell N = 1 parameter space and holomorphicity of Ω 41 C.1 SU(3) × SO(6) structures in the NS-NS sector 41 C.2 SU(3) × SO(6 + n) structures in heterotic supergravity 44 C.3 The off-shell hermitian structure on V 45 D Comments on D-terms 48 D.1 Massless deformations 48 D.2 Including bundle moduli 52 D.3 Polystable bundles 54 D.4 Full Maurer-Cartan equations 56 E Massless moduli 571 More precisely, it is the complex vector bundle V C (defined in appendix C.3) that is a holomorphic bundle. 2 The curvature R in the Bianchi identity is the curvature of a connection on T X, satisfying its own hermitian Yang-Mills conditions in order for the equations of motion to be fulfilled [33]. To O(α ′ ), this connection is ∇ − , given by taking the connection in (A.7) with the opposite sign for H.

We apply exceptional generalised geometry to the study of exactly marginal deformations of N = 1 SCFTs that are dual to generic AdS 5 flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries.Our analysis holds for any N = 2 AdS 5 flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.