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Register a new userComponent versus Superspace Approaches to D=4, N=1 Conformal Supergravity
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The superspace formulation of N=1 conformal supergravity in four dimensions is demonstrated to be equivalent to the conventional component field approach based on the superconformal tensor calculus. The detailed correspondence between two approaches is explicitly given for various quantities; superconformal gauge fields, curvatures and curvature constraints, general conformal multiplets and their transformation laws, and so on. In particular, we carefully analyze the curvature constraints leading to the superconformal algebra and also the superconformal gauge fixing leading to Poincare supergravity since they look rather different between two approaches.
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We formulate a unimodular N=1, d=4 supergravity theory off shell. We see that the infinitesimal Grassmann parameters defining the unimodular supergravity transformations are constrained and show that the conmutator of two infinitesinal unimodular supergravity transformations closes on transverse diffeomorphisms, Lorentz transformations and unimodular supergravity transformations. Along the way, we also show that the linearized theory is a supersymmetric theory of gravitons and gravitinos. We see that de Sitter and anti-de Sitter spacetimes are non-supersymmetric vacua of our unimodular supergravity theory.
The projective superspace formulation for four-dimensional N = 2 matter-coupled supergravity presented in arXiv:0805.4683 makes use of the variant superspace realization for the N = 2 Weyl multiplet in which the structure group is SL(2,C) x SU(2) and the super-Weyl transformations are generated by a covariantly chiral parameter. An extension to Howes realization of N = 2 conformal supergravity in which the tangent space group is SL(2,C) x U(2) and the super-Weyl transformations are generated by a real unconstrained parameter was briefly sketched. Here we give the explicit details of the extension.
We put forward a unimodular $N=1, d=4$ anti-de Sitter supergravity theory off shell. This theory, where the Cosmological Constant does not couple to gravity, has a unique maximally supersymmetric classical vacuum which is Anti-de Sitter spacetime with radius given by the equation of motion of the auxiliary scalar field, ie, $S=frac{3}{kappa L}$. However, we see that the non-supersymmetric classical vacua of the unimodular theory are Minkowski and de Sitter spacetimes as well as anti-de Sitter spacetime with radius $l eq L$.
Based on the known non-linear transformation rules of the Weyl multiplet fields, the action of $N=4$ conformal supergravity is constructed up to terms quadratic in the fermion fields. The bosonic sector corrects a recent result in the literature.
We review the question of quantum consistency of N=4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard (minimal) conformal supergravity where the complex scalar $phi$ is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to N=4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the non-minimal N=4 conformal supergravity with the action (recently constructed in arXiv:1609.09083) depending on an arbitrary holomorphic function $f(phi)$. The special case of the non-minimal conformal supergravity with $f= e^{2phi}$ appears in the twistor-string theory. We show that divergences and anomalies do not depend on the form of the function $f$ and thus can be cancelled just as in the minimal $f=1$ case by coupling the theory to four N=4 vector multiplets.
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