A Riemann-Hilbert approach to rotating attractors

Abstract

We construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable.

New gravitational solutions via a Riemann-Hilbert approach

“…As a starting point we pick the solution that describes an AdS 2 × S 2 space-time, supported by electric/magnetic charges (Q, P ) and a constant dilaton scalar field. The associated monodromy matrix was given in [4], and its entries contain double and simples poles at ω = 0. We then perform a two-parameter deformation of this monodromy matrix.…”

Geroch group description of bubbling geometries

“…We expect the techniques of [13,14] to be applicable for their explicit factorisation. It will also be interesting to relate monodromy matrices to rod-structure [41] in some precise way.…”

“…In order to reduce the solution to two dimensions, we need to take collinear centers; the matrix S( x) then only depends on two coordinates (r, θ). A recipe for obtaining the Geroch group matrix M(w) for a solution with known S( x) was given in [1,12]; see also [13]. To use this recipe we first need to change coordinates of the base space from (r, θ, φ) to the Weyl canonical coordinates (ρ, z, φ).…”

“…In the first approach [9][10][11][12], the authors have focused on monodromy matrices with simple poles with suitable rank residues. In the second approach [13,14], the authors have converted the matrix valued factorisation problem into a vectorial RiemannHilbert problem and solved it using complex analysis. Several examples have been worked out in both these approaches.…”

“…The indices I, J run from 1 to 3, and g IJ = ∂ I ∂J K with the potential 12) and where 13) where the split complex symmetric matrix N ΛΣ is constructed from the prepotential as…”