Extremal black hole flows and duality invariants

Abstract

Extremal black holes in Maxwell‐Einstein supergravities with scalar fields can be obtained as solutions of first order flow equations even when they are not supersymmetric, provided one identifies a suitable superpotential W entering the effective potential, and driving the evolution of the scalar fields. We briefly review the main ideas for the construction of W in terms of duality invariants of N = 2 special geometry and we comment on its properties in wider contexts.

Multi-centered first order formalism

“…of the constraint I of (5.7) can be shown to be consistent with the first order equations and, as a consequence, the duality invariant i 3 [24,40] vanishes. In this sense, the constraint (5.11) can be regarded as the generalization, within the BPS multi-centered flow embedded into the branch I of the almost BPS class, of the constraint i 3 = 0.…”

“…based on the first order reformulation of the scalar equations of motion, was introduced in [18], and then developed in various works [19][20][21][22][23][24]. The possibility to switch from second order to first order differential equations of motion -without doubling their number -has an applicative relevance.…”

“…In terms of D-charges (3.23), (3.22) can be rewritten as 24) or equivalently, inverting in terms of the D-charges: …”

Duality covariant non-BPS first order systems

“…The simplest generalisation is the class of extremal black holes, which are still characterised by a vanishing Hawking temperature, but do not preserve any supersymmetry. The corresponding static solutions are known to be described by first order equations as well [10][11][12][13][14][15][16][17][18][19][20][21], although the latter are then not a direct consequence of supersymmetry.…”

“…In contrast, we will focus on the under-rotating (or ergo-free) black holes, which then admit a flat three-dimensional base, and include the static extremal black holes [22][23][24]. Single centre under-rotating non-BPS black holes have been studied throughout the last decade or so, from various aspects and using various techniques, see for example [10,12,15,[17][18][19][25][26][27][28] and references therein for some JHEP09 (2012)100 developments. Using the seed solution of [11,13,29] combined with a general duality transformation as explained in [30], one can construct any desired solution, but a manifestly duality covariant formulation was lacking.…”