On supergravity and projective superspace: Dual formulations

Abstract

The superspace formulation for four-dimensional N = 2 matter-coupled supergravity recently developed in [1] makes use of a new type of conformal compensator with infinitely many off-shell degrees of freedom: the so-called covariant weight-one polar hypermultiplet. In the present note we prove the duality of this formulation to the known minimal (40 + 40) off-shell realization for N = 2 Poincaré supergravity involving the improved tensor compensator. Within the latter formulation, we present new off-shell matter couplings realized in terms of covariant weight-zero polar hypermultiplets. We also elaborate upon the projective superspace description of vector multiplets in N = 2 conformal supergravity. An alternative superspace representation for locally supersymmetric chiral actions is given. We present a model for massive improved tensor multiplet with both "electric" and "magnetic" types of mass terms.

Goldstino superfields in N = 2 $$ \mathcal{N}=2 $$ supergravity

“…In this section we make use of the superspace formulation for N = 2 conformal supergravity, which was developed in [53] and employed in [53,54] to construct general off-shell supergravity-matter couplings. 6 A brief summary of the corresponding curved superspace…”

“…The three formulations prove to be equivalent, and they are also related to each other in the following sense: (i) SU(2) superspace is a gauged fixed version of U(2) superspace [59]; and (ii) U(2) superspace is a gauge fixed version of conformal superspace [33]. The most general off-shell N = 2 supergravity-matter couplings were constructed in SU(2) superspace [53,54], a few years before the conformal superspace was introduced. They can uniquely be lifted to U(2) superspace [59] and also to conformal superspace [60].…”

“…The equation (3.23) is similar to that in spontaneously broken N = 1 supergravity [21,23], see appendix D for a review of the construction of [21]. The supergravity theory (3.15) possesses a dual formulation in which the tensor multiplet compensator is dualised into a polar hypermultiplet compensator [54]. To obtain the dual formulation, the first step is to recast the chiral action (3.15) as a projective action.…”

“…The projective Lagrangian corresponding to (3.26) was given in [54]. It is 27) where V (v) is the tropical prepotential for the vector multiplet, and Υ (1) (v) is a weightone arctic multiplet (both Υ (1) and its smile-conjugateΥ (1) are pure gauge degrees of freedom).…”

“…Since the tensor multiplet compensator appears in the Lagrangian (3.27) only via the gauge invariant field strength, G (2) , the tensor multiplet can be dualised into an offshell polar hypermultiplet following the scheme described in [54]. The dual supergravity Lagrangian is…”