Prescribing topological defects for the coupled Einstein and Abelian Higgs equations

Abstract

We construct multi-string solutions of the coupled Einstein and Abelian Higgs equations so that the spacetime is uniform along the time axis and a vertical direction and nontrivial geometry is coded on a Riemann surface M. We concentrate on the critical BogomoΓnyi phase. When M is compact, the Abelian Higgs model is defined by a complex line bundle L over M. We prove that, due to the coupling of the Einstein equations, the Euler characteristic of M and the first Chern number of the line bundle L identified as the total string number impose an exact obstruction to the existence of a string solution. Such an obstruction leads to some interesting implications. We then study the existence of multi-string solutions which can realize a prescribed string distribution. We show that there are such solutions when the local string winding numbers do not exceed half of the total string number. When M is noncompact and globally conformal to a plane, we show that the energy scale of symmetry breaking plays a crucial role and there are finite-energy radially symmetric string solutions realizing a given string number if and only if the symmetry breaking scale is sufficiently small but nonvanishing. Finally, we obtain finite-energy multistring solutions with an arbitrary string distribution and associated local winding numbers. These solutions are not radially symmetric and are regular everywhere and topologically nontrivial so that both the energy of the matter-gauge sector and the energy of the gravitational sector viewed as the total Gauss curvature of M are quantized.

Geometry, Topology, and Gravitation Synthesized by Cosmic Strings

“…Our results [80,81] concerning the existence of cosmic string solutions of the system (3.3) can be stated as follows.…”

“…We choose two well known examples to illustrate such a picture. The first example is the classical Abelian Higgs model [38,70] also known as the Ginzburg-Landau theory for superconductivity [20,28,32,48,78,79], which is a simplest YangMills theory model, and the cosmic strings [80,81,82] are simply gravitating superconducting vortices [77]. The second example is the gauged σ-model [66,67] which allows the coexistence of magnetic vortices and antivortices and may be used to obtain cosmic strings with opposite string charges [84,85,88].…”

“…In Section 3, we consider cosmic strings which are in fact vortices which give rise to gravitation through introducing nontrivial geometry and topology. After stating the main existence theorem [80,81] concerning cosmic strings in the Abelian Higgs model within a noncompact setting, we illustrate the results using the explicit construction in [27] for solutions of the (bare) σ-model. We then state our existence theorem for cosmic string solutions of the Abelian Higgs model within a compact setting [81,82].…”

“…After stating the main existence theorem [80,81] concerning cosmic strings in the Abelian Higgs model within a noncompact setting, we illustrate the results using the explicit construction in [27] for solutions of the (bare) σ-model. We then state our existence theorem for cosmic string solutions of the Abelian Higgs model within a compact setting [81,82]. We also present a discussion comparing our structure with that in more familiar problems in three dimensions, the Positive Energy Theorem [59,62,63,64,65,76] and the Penrose Conjecture [60] in general relativity.…”

“…We state our results [81,82] as follows. 3) over a compact surface S is that S is topologically a 2-sphere, S 2 , and that the Newton constant G satisfies the quantization condition (3.23).…”