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.
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.
A new class of N = 2 locally supersymmetric higher-derivative invariants is constructed based on logarithms of conformal primary chiral superfields. They characteristically involve a coupling to R µν 2 − 1 3 R 2 , which equals the non-conformal part of the GaussBonnet term. Upon combining one such invariant with the known supersymmetric version of the square of the Weyl tensor one obtains the supersymmetric extension of the Gauss-Bonnet term. The construction is carried out in the context of both conformal superspace and the superconformal multiplet calculus. The new class of supersymmetric invariants resolves two open questions. The first concerns the proper identification of the 4D supersymmetric invariants that arise from dimensional reduction of the 5D mixed gauge-gravitational Chern-Simons term. The second is why the pure Gauss-Bonnet term without supersymmetric completion has reproduced the correct result in calculations of the BPS black hole entropy in certain models.
We develop a new off-shell formulation for six-dimensional conformal supergravity obtained by gauging the 6D N = (1, 0) superconformal algebra in superspace. This formulation is employed to construct two invariants for 6D N = (1, 0) conformal supergravity, which contain C 3 and C C terms at the component level. Using a conformal supercurrent analysis, we prove that these exhaust all such invariants in minimal conformal supergravity. Finally, we show how to construct the supersymmetric F F invariant in curved superspace.
Using the off-shell formulation for N -extended conformal supergravity in three dimensions that has recently been presented in arXiv:1305.3132, we construct superspace actions for conformal supergravity theories with N < 6. For each of the cases considered, we work out the complete component action as well as the gauge transformation laws of the fields belonging to the Weyl supermultiplet. The N = 1 and N = 2 component actions derived coincide with those proposed by van Nieuwenhuizen and Roček in the mid-1980s. The offshell N = 3, N = 4 and N = 5 supergravity actions are new results. Upon elimination of the auxiliary fields, these actions reduce to those constructed by Roček in 1989 (and also by Gates andNishino in 1993).
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