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Black holes in supergravity and integrability. (English) Zbl 1291.81299

Summary: 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 group-theoretical 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.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83C57 Black holes
83E50 Supergravity
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
70H20 Hamilton-Jacobi equations in mechanics
22E70 Applications of Lie groups to the sciences; explicit representations
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
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