4D 𝒩 = 2 supergravity and projective superspace

Abstract

This paper presents a projective superspace formulation for 4D N = 2 mattercoupled supergravity. We first describe a variant superspace realization for the N = 2 Weyl multiplet. It differs from that proposed by Howe in 1982 by the choice of the structure group SO(3, 1) × SU(2) versus SO(3, 1) × U(2) , which implies that the super-Weyl transformations are generated by a covariantly chiral parameter instead of a real unconstrained one. We introduce various off-shell supermultiplets which are curved superspace analogues of the superconformal projective multiplets in global supersymmetry and which describe matter fields coupled to supergravity. A manifestly locally supersymmetric and super-Weyl invariant action principle is given. Off-shell locally supersymmetric nonlinear sigma models are presented in this new superspace.

On supergravity and projective superspace: Dual formulations

“…Recently, we have developed the superspace formulation for four-dimensional N = 2 matter-coupled supergravity [1], extending the earlier construction for 5D N = 1 supergravity [2,3]. From the purely geometrical point of view, this approach makes use of Grimm's curved superspace geometry [4], which is perfectly suitable to describe N = 2 conformal supergravity and has a simple relation to Howe's superspace formulation [5].…”

“…From the purely geometrical point of view, this approach makes use of Grimm's curved superspace geometry [4], which is perfectly suitable to describe N = 2 conformal supergravity and has a simple relation to Howe's superspace formulation [5]. Kinematically, matter fields in [1] are described in terms of covariant projective supermultiplets which are curved-space versions of the superconformal projective multiplets [6] living in rigid projective superspace [7,8]. In addition to the local N = 2 superspace coordinates z M = (x m , θ µ i ,θ iμ ), where m = 0, 1, · · · , 3, µ = 1, 2,μ = 1, 2 and i = 1, 2, such a supermultiplet, Q (n) (z, u + ), depends on auxiliary isotwistor variables u + i ∈ C 2 \ {0}, with respect to which Q (n) is holomorphic and homogeneous, Q (n) (c u + ) = c n Q (n) (u + ), on an open domain of C 2 \ {0} (the integer parameter n is called the weight of Q (n) ).…”

“…In the approach of [1], the dynamics of supergravity-matter systems is described by a locally supersymmetric action of the form:…”

“…the first superconformal compensator), such that the body of W (z) is everywhere non-vanishing, S ++ (z, u + ) = S ij (z)u + i u + j and S ++ (z, u + ) =S ij (z)u + i u + j are special dimension-1 components of the supertorsion; see [1] for more detail. The action (1.2) can be shown to be invariant under the supergravity gauge transformations, and is also manifestly super-Weyl invariant.…”

“…In [1], we presented a family of supergravity-matter systems in which the matter hypermultiplets are described by covariant weight-zero polar multiplets 2 , and the second superconformal compensator is identified with a covariant weight-one polar multiplet. This is a new type of supergravity compensator, although it is related to the q + -hypermultiplet compensator which emerges within the harmonic superspace approach [13,11] to 4D N = 2 supergravity, e.g., in the sense of [14].…”