Black holes in supergravity and integrability

Abstract

Stationary black holes of massless supergravity theories are described by certain geodesic curves on the target space that is obtained after dimensional reduction over time. When the target space is a symmetric coset space we make use of the grouptheoretical structure to prove that the second order geodesic equations are integrable in the sense of Liouville, by explicitly constructing the correct amount of Hamiltonians in involution. This implies that the Hamilton-Jacobi formalism can be applied, which proves that all such black hole solutions, including non-extremal solutions, possess a description in terms of a (fake) superpotential. Furthermore, we improve the existing integration method by the construction of a Lax integration algorithm that integrates the second order equations in one step instead of the usual two step procedure. We illustrate this technology with a specific example

On extremal limits and duality orbits of stationary black holes

“…The ADM-mass M ADM , NUT-charge n NUT , electric and magnetic charges Γ M = (p Λ , q Λ ), scalar charges Σ s and angular momentum M ϕ , associated with the solution are then obtained as components of Q and Q ψ [23,26,31]:…”

Einstein branes

“…This method is based on reducing flat p-branes over their worldvolume [38] (see also [30]), a technique inspired from the special case of black holes [39]. Since the worldvolume is flat, this does not generate a scalar potential, rather one just obtains a sigma model that is solvable and whose integrability can be understood in a formal way using group theory and the Hamilton-Jacobi formalism [40]. This works for a very large class of generalisations with much less worldvolume symmetries [38].…”

Nilpotent orbits in real symmetric pairs and stationary black holes

“…In it was found that a subset of all solutions to the 4‐dimensional theory, namely, the stationary (locally‐)asymptotically‐flat ones , actually feature a larger symmetry group G which is not manifest in four space‐time dimensions (), but rather in an effective Euclidean 3‐dimensional description which is formally obtained by compactifying the 4‐dimensional model along the time direction and dualising the vector fields into scalars. Stationary 4‐dimensional asymptotically‐flat black hole solutions can be conveniently arranged in orbits with respect to this larger symmetry group G , whose action has proven to be a valuable tool for their classification (see ). It also yields a “solution‐generating technique” (see ) for constructing new solutions from known ones (see ).…”