Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets

Journal of High Energy Physics, Apr 2016

We consider N = 2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that imply the second-order equations of motion. The first-order flow turns out to be driven by Hamilton’s characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the flow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS first-order equations is obtained that corresponds to a different squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes.

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Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets

HJE 4 gauged supergravity with hypermultiplets Dietmar Klemm 0 1 3 Nicolo Petri 0 1 3 Marco Rabbiosi 0 1 3 0 Via Celoria 16, I-20133 Milano , Italy 1 Dipartimento di Fisica, Universita di Milano , and INFN 2 Sezione di Milano 3 = @ F . In terms of the sections v the 4 Notice that @ ( We consider N = 2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the elds, a one-dimensional e ective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to rst-order ow equations, that imply the second-order equations of motion. The rst-order ow turns out to be driven by Hamilton's characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the ow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS rst-order equations is obtained that corresponds to a di erent squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes. Black Holes; Black Holes in String Theory; Supergravity Models; Superstring Vacua Matter-coupled N = 2, D = 4 gauged supergravity Hamilton-Jacobi, ow equations and magnetic gaugings E ective action and Hamiltonian Flow equations with electric gaugings Magnetic gaugings and symplectic covariance Non-BPS ow equations Attractors 4.1 Attractor equations and near-horizon limit Examples of solutions Test for the BPS ow Symplectic rotation of the electromagnetic frame Some di erent gaugings 1 Introduction 2 3 4 5 3.1 3.2 3.3 3.4 5.1 5.2 5.3 6 Final remarks 1 Introduction Black holes in gauged supergravity theories provide an important testground to address fundamental questions of gravity, both at the classical and quantum level. Among these are for instance the problems of black hole microstates, the nal state of black hole evolution, uniqueness- or no hair theorems, to mention only a few of them. In gauged supergravity, the solutions often have AdS asymptotics, and one can then try to study these issues guided by the AdS/CFT correspondence. A nice example for this is the recent microscopic entropy calculation [1] for the black hole solutions to N = 2, D = 4 Fayet-Iliopoulos gauged supergravity constructed in [2]. These preserve two real supercharges, and are dual to a topologically twisted ABJM theory, whose partition function can be computed exactly using supersymmetric localization techniques. This partition function can also be interpreted as the Witten index of the superconformal quantum mechanics resulting from dimensionally reducing the ABJM theory on a two-sphere. To the best of our knowledge, the results of [1] represent the rst exact black hole microstate counting that uses AdS/CFT and that does not involve an AdS3 factor1 with a corresponding two-dimensional CFT, whose asymptotic level density is evaluated with the Cardy formula. 1Or geometries related to AdS3, like those appearing in the Kerr/CFT correspondence [3]. { 1 { number of recent developments in high energy- and especially in condensed matter physics, since they provide the dual description of certain condensed matter systems at nite temperature, cf. [4] for a review. In particular, models that contain Einstein gravity coupled to U( 1 ) gauge elds2 and neutral scalars have been instrumental to study transitions from Fermi-liquid to non-Fermi-liquid behaviour, cf. [5, 6] and references therein. In AdS/condensed matter applications one is often interested in including a charged scalar operator in the dynamics, e.g. in the holographic modeling of strongly coupled superconductors [7]. This is dual to a charged scalar eld in the bulk, that typically appears in supergravity coupled to gauged hypermultiplets. These theories are thus particularly appealing in an AdS/cond-mat context, and it would be nice to dispose of analytic black hole solutions to gauged supergravity with hyperscalars turned on. Up to now, the only known such solution in four dimensions was constructed recently in [8],3 by using the results of [12], where all supersymmetric backgrounds of N = 2, D = 4 gauged supergravity coupled to both vector- and hypermultiplets were classi ed. Such BPS solutions typically satisfy rst-order equations that arise from vanishing fermion variations, and that are much easier to solve than the full second-order equations of motion. In our paper we shall derive such a set of rst-order equations for static and spherically (or hyperbolically) symme (...truncated)


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Dietmar Klemm, Nicolò Petri, Marco Rabbiosi. Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets, Journal of High Energy Physics, 2016, pp. 8, Volume 2016, Issue 4, DOI: 10.1007/JHEP04(2016)008