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

Abstract

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.

On the on-shell: the action of AdS4 black holes

“…Since our line of reasoning did not require the explicit solution beyond the AdS 4 boundary conditions, it should be straightforward to generalize these arguments to theories with charged hypermultiplets where solutions are harder to find [40][41][42][43][44][45] but there are more avenues for embedding such theories into string or M-theory. It would also seem likely that our methods will generalize to asymptotically AdS solutions in other dimensions.…”

Black string first order flow in N = 2, d = 5 abelian gauged supergravity

“…eqs. (3.43) in [17]. Integrating T = 2 3 U and plugging this into the remaining equations of (5.16), one gets…”

“…e.g. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] for an (incomplete) list of references. In particular, ref.…”

“…A solution of (3.4) permits to write the action as a sum of squares and to derive a set of first-order flow equations by setting the squares to zero. 5 Guided by the four-dimensional case [9,13,17], the ansatz for the simplest non-trivial solution is…”

“…Similar to the case of N = 2, d = 4 abelian gauged supergravity [14,17], one introduces a 'field rotation matrix'…”