We develop geometric superspace settings to construct arbitrary higher derivative couplings (including R^n terms) in three-dimensional supergravity theories with N=1,2,3 by realising them as conformal supergravity coupled to certain compensators. For
all known off-shell supergravity formulations, we construct supersymmetric invariants with up to and including four derivatives. As a warming-up exercise, we first give a new and completely geometric derivation of such invariants in N=1 supergravity. Upon reduction to components, they agree with those given in arXiv:0907.4658 and arXiv:1005.3952. We then carry out a similar construction in the case of N=2 supergravity for which there exist two minimal formulations that differ by the choice of compensating multiplet: (i) a chiral scalar multipet; (ii) a vector multiplet. For these formulations all four derivative invariants are constructed in completely general and gauge independent form. For a general supergravity model (in the N=1 and minimal N=2 cases) with curvature-squared and lower order terms, we derive the superfield equations of motion, linearise them about maximally supersymmetric backgrounds and obtain restrictions on the parameters that lead to models for massive supergravity. We use the non-minimal formulation for N = 2 supergravity (which corresponds to a complex linear compensator) to construct a novel consistent theory of massive supergravity. In the case of N = 3 supergravity, we employ the off-shell formulation with a vector multiplet as compensator to construct for the first time various higher derivative invariants. These invariants may be used to derive models for N = 3 massive supergravity. As a bi-product of our analysis, we also present superfield equations for massive higher spin multiplets in (1,0), (1,1) and (2,0) anti-de Sitter superspaces.
We develop a formalism to construct supersymmetric backgrounds within the superspace formulation for five-dimensional (5D) conformal supergravity given in arXiv:0802.3953. Our approach is applicable to any off-shell formulation for 5D minimal Poincar
e and anti-de Sitter supergravity theories realized as the Weyl multiplet coupled with two compensators. For those superspace backgrounds which obey the equations of motion for (gauged) supergravity, we naturally reproduce the supersymmetric solutions constructed a decade ago by Gauntlett et al. For certain supersymmetric backgrounds with eight supercharges, we construct a large family of off-shell supersymmetric sigma models such that the superfield Lagrangian is given in terms of the Kahler potential of a real analytic Kahler manifold.
For all types of N=4 anti-de Sitter (AdS) supersymmetry in three dimensions, we construct manifestly supersymmetric actions for Abelian vector multiplets and explain how to extend the construction to the non-Abelian case. Manifestly N=4 supersymmetri
c Yang-Mills (SYM) actions are explicitly given in the cases of (2,2) and critical (4,0) AdS supersymmetries. The N=4 vector multiplets and the corresponding actions are then reduced to (2,0) AdS superspace, in which only N=2 supersymmetry is manifest. Using the off-shell structure of the N=4 vector multiplets, we provide complete N=4 SYM actions in (2,0) AdS superspace for all types of N=4 AdS supersymmetry. In the case of (4,0) AdS supersymmetry, which admits a Euclidean counterpart, the resulting N=2 action contains a Chern-Simons term proportional to q/r, where r is the radius of AdS_3 and q is the R-charge of a chiral scalar superfield. The R-charge is a linear inhomogeneous function of X, an expectation value of the N=4 Cotton superfield. Thus our results explain the mysterious structure of N=4 supersymmetric Yang-Mills theories on S^3 discovered in arXiv:1401.7952. In the case of (3,1) AdS supersymmetry, which has no Euclidean counterpart, the SYM action contains both a Chern-Simons term and a chiral mass-like term. In the case of (2,2) AdS supersymmetry, which admits a Euclidean counterpart, the SYM action has no Chern-Simons and chiral mass-like terms.
For general off-shell N=2 supergravity-matter systems in three spacetime dimensions, a formalism is developed to reduce the corresponding actions from superspace to components. The component actions are explicitly computed in the cases of Type I and
Type II minimal supergravity formulations. We describe the models for topologically massive supergravity which correspond to all the known off-shell formulations for three-dimensional N=2 supergravity. We also present a universal setting to construct supersymmetric backgrounds associated with these off-shell supergravities.
We introduce N-extended (p,q) AdS superspaces in three space-time dimensions, with p+q=N and p>=q, and analyse their geometry. We show that all (p,q) AdS superspaces with X^{IJKL}=0 are conformally flat. Nonlinear sigma-models with (p,q) AdS supersym
metry exist for p+q<=4 (for N>4 the target space geometries are highly restricted). Here we concentrate on studying off-shell N=3 supersymmetric sigma-models in AdS_3. For each of the cases (3,0) and (2,1), we give three different realisations of the supersymmetric action. We show that (3,0) AdS supersymmetry requires the sigma-model to be superconformal, and hence the corresponding target space is a hyperkahler cone. In the case of (2,1) AdS supersymmetry, the sigma-model target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations of the two-sphere of complex structures.
There exist two superspace approaches to describe N=2 supersymmetric nonlinear sigma models in four-dimensional anti-de Sitter (AdS_4) space: (i) in terms of N=1 AdS chiral superfields, as developed in arXiv:1105.3111 and arXiv:1108.5290; and (ii) in
terms of N=2 polar supermultiplets using the AdS projective-superspace techniques developed in arXiv:0807.3368. The virtue of the approach (i) is that it makes manifest the geometric properties of the N=2 supersymmetric sigma-models in AdS_4. The target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations on the two-sphere of complex structures. The power of the approach (ii) is that it allows us, in principle, to generate hyperkahler metrics as well as to address the problem of deformations of such metrics. Here we show how to relate the formulation (ii) to (i) by integrating out an infinite number of N=1 AdS auxiliary superfields and performing a superfield duality transformation. We also develop a novel description of the most general N=2 supersymmetric nonlinear sigma-model in AdS_4 in terms of chiral superfields on three-dimensional N=2 flat superspace without central charge. This superspace naturally originates from a conformally flat realization for the four-dimensional N=2 AdS superspace that makes use of Poincare coordinates for AdS_4. This novel formulation allows us to uncover several interesting geometric results.
We develop the superspace geometry of N-extended conformal supergravity in three space-time dimensions. General off-shell supergravity-matter couplings are constructed in the cases N=1,2,3,4.
We develop superspace techniques to construct general off-shell N=1,2,3,4 superconformal sigma-models in three space-time dimensions. The most general N=3 and N=4 superconformal sigma-models are constructed in terms of N=2 chiral superfields. Several
superspace proofs of the folklore statement that N=3 supersymmetry implies N=4 are presented both in the on-shell and off-shell settings. We also elaborate on (super)twistor realisations for (super)manifolds on which the three-dimensional N-extended superconformal groups act transitively and which include Minkowski space as a subspace.
We study dualities in off-shell 4D N = 2 supersymmetric sigma-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any s
uperconformal sigma-model is superconformal. Since N = 2 superconformal sigma-models (for which target spaces are hyperkahler cones) formulated in terms of polar multiplets are naturally associated with Kahler cones (which are target spaces for N = 1 superconformal sigma-models), polar-polar duality generates a transformation between different Kahler cones. In the non-superconformal case, we study implications of polar-polar duality for the sigma-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkahler potential and its dual. As an application of polar-polar duality, we study self-dual models.
Within the superspace formulation for four-dimensional N = 2 matter-coupled supergravity developed in arXiv:0805.4683, we elaborate two approaches to reduce the superfield action to components. One of them is based on the principle of projective inva
riance which is a purely N = 2 concept having no analogue in simple supergravity. In this approach, the component reduction of the action is performed without imposing any Wess-Zumino gauge condition, that is by keeping intact all the gauge symmetries of the superfield action, including the super-Weyl invariance. As a simple application, the c-map is derived for the first time from superfield supergravity. Our second approach to component reduction is based on the method of normal coordinates around a submanifold in a curved superspace, which we develop in detail. We derive differential equations which are obeyed by the vielbein and the connection in normal coordinates, and which can be used to reconstruct these objects, in principle in closed form. A separate equation is found for the super-determinant of the vielbein, which allows one to reconstruct it without a detailed knowledge of the vielbein. This approach is applicable to any supergravity theory in any number of space-time dimensions. As a simple application of this construction, we reduce an integral over the curved N = 2 superspace to that over the chiral subspace of the full superspace. We also give a new representation for the curved projective-superspace action principle as a chiral integral.
