Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory
Cosmic strings are static solutions to Einstein equations coupled to `matter' in which there is a trivial isometry along the time and one spatial direction. These solutions, therefore, have non-trivial dependence on the `transverse' coordinates, the geometry of which is changed by the gravitational back-reaction of the strings. In the present paper, `matter' is a linear sigma-model defined by a complex scalar field \(\varphi\) coupled to an abelian gauge field \(A\). It is shown that the Einstein equation for the coupled system reduces to a scalar equation determining the constant Gaussian curvature of the `transverse' surface \(S\) in terms of the energy density of the strings. The `matter' equations are shown to reduce to self-dual equations of Bogomolnyi type. A trivial solution to these are given by constant \(\varphi\), vanishing \(A\) and \(S\) a flat torus. Non-trivial solutions demand that \(S\) is topologically a 2-sphere, the explicit metric on \(S\) has not been determined. The scalar field \(\varphi\) is a meromorphic function on \(S\) -- it has \(N\) zeroes (strings) and \(P\) poles (anti-strings). However, solutions are possible only when \(N+P=1/4\pi G\), where \(G\) is the Newton's constant. These solutions are shown to give solutions on the plane \(\mathbb R^2\) under the condition \({1\over 8\pi G}<N+P<{1\over 4\pi G}\).
scientific article; zbMATH DE number 2158324
scientific article