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
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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)