$ \mathcal{N} = 2 $ conformal superspace in four dimensions

Abstract

Abstract:We develop the geometry of four dimensional N = 2 superspace where the entire conformal algebra of SU(2, 2|2) is realized linearly in the structure group rather than just the SL(2, C) × U(2) R subgroup of Lorentz and R-symmetries, extending to N = 2 our prior result for N = 1 superspace. This formulation explicitly lifts to superspace the existing methods of the N = 2 superconformal tensor calculus; at the same time the geometry, when degauged to SL(2, C) × U(2) R , reproduces the existing formulation of N = 2 conformal supergravity constructed by Howe.

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

“…Appendix A describes the component content of the nilpotent chiral scalar superfield Φ. Appendix B gives a summary of the SU(2) superspace [32], while appendix C briefly introduces N = 2 conformal superspace [33]. Finally, appendix D discusses nilpotent N = 1 supergravity following and extending [21].…”

“…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.…”

“…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].…”

“…Here the ellipsis denotes terms involving supergravity fields and at most three spinor derivatives (see [33,77] for the complete expression). In the unitary gauge, it is easy to see that the component reduction of the action (4.14) is…”

“…This appendix contains a summary of the formulation for N = 2 conformal supergravity in conformal superspace [33] employed in section 4. We use the notations of [77] which are consistent with those of [53] and appendix B and review the results necessary for deriving results in section 4.…”