Conformal supergravity in three dimensions: new off-shell formulation

Abstract

We propose a new off-shell formulation for N -extended conformal supergravity in three spacetime dimensions. Our construction is based on the gauging of the N -extended superconformal algebra in superspace. Covariant constraints are imposed such that the algebra of covariant derivatives is given in terms of a single curvature superfield which turns out to be the super Cotton tensor. An immediate corollary of this construction is that the curved superspace is conformally flat if and only if the super Cotton tensor vanishes. Upon degauging of certain local symmetries, our formulation is shown to reduce to the conventional one with the local structure group SL(2, R) × SO(N ).1 In the N = 1 case, the superconformal tensor calculus was independently developed in [7]. Early superspace approaches to N = 1 and N = 2 supergravity theories were given in [8,9,10,11]. 2 We are grateful to Jim Gates for bringing Ref.[6] to our attention. 3 This construction is a natural generalization of Howe's superspace formulation for 4D Nextended conformal supergravity in four dimensions [13]. 4 The special cases of N = 8 and N = 16 conformal supergravity theories were independently worked out in [15,16] and [17] respectively. 5 If a prepotential formulation is available, the conformal supergravity action may be written as a superspace integral in terms of the prepotentials.

The off-shell c-map

“…In particular, the constraints for R(S) µν ip and R(K) µν a are identically satisfied for the composite gauge fields in (3.6). These results are generally in agreement with earlier off-shell results [15,16,[18][19][20] as well as with on-shell results [23,24].…”

“…We emphasize that the results we are about to describe are consistent with results previously reported in the literature, such as in [15][16][17][18]. In particular they fully reproduce the non-linear results that were obtained in [19,20].…”

“…1 Some of the final results of the dimensional reduction procedure can be compared to existing results in the literature on N = 4 (conformal) supergravity theories in three dimensions (see e.g. [15][16][17][18][19][20] where further references can be found). We will discuss the details in due course.…”

“…It should be of interest to evaluate the full 3D superconformal invariant, either from the off-shell dimensional reduction or directly in three space-time dimensions. The latter can be done by utilizing the off-shell multiplet calculus obtained in [19,20].…”

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

“…Fortunately, the constraints on Ξ α i allow for a supergravity analogue. In order to lift Ξ α i to supergravity, it is advantageous to employ the superspace formulation for N = 2 conformal supergravity developed by Butter 11 [33] and further elaborated 11 Conformal superspace was originally constructed for 4D N = 1 supergravity [73] and then extended to the 4D N = 2 [33] case, 3D N -extended conformal supergravity [74], 5D conformal supergravity [75] and recently to the 6D N = (1, 0) case [76].…”