New gravitational solutions via a Riemann-Hilbert approach

Abstract

We consider the Riemann-Hilbert factorization approach to solving the field equations of dimensionally reduced gravity theories. First we prove that functions belonging to a certain class possess a canonical factorization due to properties of the underlying spectral curve. Then we use this result, together with appropriate matricial decompositions, to study the canonical factorization of non-meromorphic monodromy matrices that describe deformations of seed monodromy matrices associated with known solutions. This results in new solutions, with unusual features, to the field equations.

Weyl metrics and Wiener-Hopf factorization

“…As a result one finds, in particular, that from rational n × n monodromy matrices M, whose canonical Wiener-Hopf factorization can be constructed explicitly and in a computationally simple manner, one obtains explicit exact solutions that would be very difficult to obtain through other approaches. This is the case of the novel solutions presented in [6], whose construction, based on the Riemann-Hilbert approach of [5], is hereby rigorously justified. Using our improved understanding of the role of the factorization contour, we return to one of the new solutions obtained in [6], which was restricted to a certain domain in space-time.…”

“…This is the case of the novel solutions presented in [6], whose construction, based on the Riemann-Hilbert approach of [5], is hereby rigorously justified. Using our improved understanding of the role of the factorization contour, we return to one of the new solutions obtained in [6], which was restricted to a certain domain in space-time. Here we complete its analysis, discuss its meaning and properties, and we also consider other ranges of parameters.…”

“…By restricting to the subspace of solutions that only depend on two of the D spacetime coordinates, various approaches to solving the field equations become available (see for instance [1] for a recent survey thereof). In this paper, we focus on the Riemann-Hilbert approach, which can be applied to gravitational theories that satisfy certain requirements, see for instance [2][3][4][5][6] and references therein. This approach is remarkable in that it allows us to obtain explicit solutions to the reduced field equations through the appropriate factorization of a matrix function which depends on a complex variable τ that does not appear in the original problem.…”

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 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.…”