Supersymmetric higher derivative couplings and their applications

Abstract

After an elementary introduction on black hole physics, supersymmetry and effective theories, which motivates the importance of higher derivative couplings in supergravity, we give an exhaustive treatment of the covariant phase space formalism with many examples explicitly worked out. This sets the stage for the two distinct analyses that follow. The first analysis relies on the superconformal formalism of supergravity and its multiplet structure. We explicitly construct a new class of N=2 higher derivative invariants in four dimensional supergravity, based on logarithms of conformal primary chiral superfields. This class of locally supersymmetric invariants can be combined with the supersymmetrization of the square of the Weyl tensor to obtain the supersymmetric extension of the full Gauss-Bonnet term. The construction is carried out in the context of both conformal superspace and the superconformal multiplet calculus. This allows for the resolution of two open problems. First, we confirm that this new class of higher derivative invariants coincides with a particular 4D supersymmetric invariant arising from dimensional reduction of the 5D mixed gauge-gravitational Chern-Simons term. Secondly, it becomes clear why, in certain models, the pure Gauss-Bonnet term without its supersymmetric completion has reproduced the correct result in calculations of the BPS black hole entropy. We subsequently derive the conditions for fully supersymmetric backgrounds of general N=2 D=4 superconformal theories of gravity, and a non-renormalization theorem for this new class of higher derivative couplings is presented. Since the theorem implies that the invariant and its first order variation must vanish in a fully supersymmetric background, the macroscopic entropy of supersymmetric black hole solutions of N=2 D=4 supergravity is not modified by this class of higher derivative quantum corrections.. This confirms the results, obtained more than a decade ago, of the macroscopic entropy of supersymmetric black holes by de Wit et al. which numerically coincides with the microscopic results worked out by Witten, Strominger and Maldacena. The second analysis aims at studying the fate in higher derivative gravity of flat directions, i.e. scalar fields which are not fixed at the horizon by the attractor mechanism. We present two explicit examples of higher derivative supergravity theories in five and ten dimensions. In 5D, we consider as a background the 1/2 BPS supersymmetric spinning black hole solution in asymptotically AdS spacetime found by Gutowski and Reall. We find in this case that the two-derivative flat directions are not lifted after the addition of supersymmetric higher derivative terms, although the results can be extended to non-supersymmetric deformations of the two-derivative theory. This suggests that, as long as the two-derivative background solution preserves some supersymmetries, then flat directions remain flat even when quantum corrections are considered. We then analyzed type IIB theory in ten dimensions and consider the rotating D3-brane solution as background. Since this solution is not supersymmetric we find, as expected, that the dilaton gets fixed upon including the cubic α′corrections to type IIB action.