We construct the scalar potential for the exceptional field theory based on the affine symmetry group E 9. The fields appearing in this potential live formally on an infinitedimensional extended spacetime and transform under E 9 generalised diffeomorphisms. In addition to the scalar fields expected from D = 2 maximal supergravity, the invariance of the potential requires the introduction of new constrained scalar fields. Other essential ingredients in the construction include the Virasoro algebra and indecomposable representations of E 9. Upon solving the section constraint, the potential reproduces the dynamics of either eleven-dimensional or type IIB supergravity in the presence of two isometries.

The framework of exceptional field theory is extended by introducing consistent deformations of its generalised Lie derivative. For the first time, massive type IIA supergravity is reproduced geometrically as a solution of the section constraint. This provides a unified description of all ten-and eleven-dimensional maximal supergravities. The action of the E 7(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.

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 non-conformal theories.

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 E 6(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.

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 ansätze. Such gaugings allow for moduli stabilisation including the SL(2) dilaton.

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends $$ {\mathfrak{e}}_9 $$ e 9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

The most general class of 4D N = 4 conformal supergravity actions depends on a holomorphic function of the scalar fields that parametrize an SU(1, 1)/U(1) coset space. The bosonic sector of these actions was presented in a letter [1]. Here we provide the complete actions to all orders in the fermion fields. They rely upon a new N = 4 density formula, which permits a direct but involved construction. This density formula also recovers the on-shell action for vector multiplets coupled to conformal supergravity. Applications of these results in the context of Poincaré supergravity are briefly discussed.1 Even the 4D N = 3 Weyl multiplet was terra incognita until recently [7,8], when its transformation laws were first written down, and the action remains unstudied. In three dimensions, the conformal supergravity actions are Lorentz-Chern-Simons and their complete form for N ≤ 6 were constructed in [9][10][11]. The half-maximal N = 8 multiplet possesses at most a pseudo-action [12]. In six dimensions, there are three conformal gravity actions, but (1, 0) supersymmetry selects out two [13,14]; the unique (2, 0) supersymmetric combination was partly constructed in [14] by lifting from (1, 0). In five dimensions, there is no pure conformal supergravity action although the Weyl multiplet exists for N = 1.

Based on the known non-linear transformation rules of the Weyl multiplet fields, the action of N = 4 conformal supergravity is constructed up to terms quadratic in the fermion fields. The bosonic sector corrects a recent result in the literature.