Gravitating vortices, cosmic strings, and the Kähl er–Yang–Mills equations

Title: Gravitating vortices, cosmic strings, and the Kähler–Yang–Mills equations.

Author(s): Luis Álvarez-Cónsul (ICMAT-CSIC), Mario García-Fernández (ICMAT-UAM), Óscar García-Prada (ICMAT-CSIC).

Source: Communications in Mathematical Physics, Volume 351, Issue 1, pp. 361–385.

Date of publication: April 2017

doi: 10.1007/s00220-016-2728-2

Abstract: When a body of water cools below freezing and nucleation starts, ice is created about randomly separated nucleation sites, with the water molecules arranged in highly ordered crystal lattices. As the frozen areas grow, adjacent ice crystals join up erratically. The boundaries where the crystal structures are disrupted are topological defects. In general, phase transitions in condensed matter associated with a loss of symmetry, such as the formation of crystals, create similar topological defects.

A combination of ideas from cosmology and the unification of forces in particle physics suggests a sequence of similar symmetry-breaking phase transitions in the very early universe, as it expanded and cooled down. Like the transitions in condensed matter, they may have led to the formation of defects, some of which may have survived up until the present. A special type of topological defects, known as cosmic strings, would be especially interesting from the perspective of cosmology and particle physics. They would be exceedingly narrow filaments of primordial material, left over from the early moments of the universe. Although not observed yet, their existence would explain some exotic astrophysical phenomena, such as an unexpected surplus of high-energy positrons observed flitting through space [1], or the formation of super-massive black holes at high redshifts [2]. They may be observable due to their gravitational effects on the cosmic microwave background or gravitational wave experiments [3].

The basic mathematical structure of a string is a complex scalar field — the Higgs field φ — that winds around the location of the string, where there is a concentration of energy density. The existence of cosmic strings is intimately related with the choice of potential for the Higgs field

V (φ) = λ(|φ|² − τ)² .

The minimum energy configuration has |φ|² = τ but the phase of φ is undetermined and labels the points on the manifold of vacua, which is a circle. The choice of ground state in the circle — known as spontaneous symmetry breaking — may vary in space-time and, by continuity, φ can be forced to leave the manifold of vacua producing surfaces where φ = 0 (the two-dimensional worldsheet of the string in space-time).

For a class of physical models, B. Linet, and A. Comtet and G. W. Gibbons, showed in the late 1980s that the theoretical existence of cosmic strings reduces to solving a system of partial differential equation on a Riemann surface Σ, known as the Einstein Bogomol’nyi equations. For non-compact Σ, the analysis of these equations carried out during the early 1990s, mainly by Y. Yang and J. Spruck, led to the construction of families of finiteenergy cosmic strings. For compact Σ, the Einstein–Bogomol’nyi equations were studied by Y. Yang on the 2-dimensional sphere (the only allowed topology), who proved existence of solutions under certain conditions on the relative position of the zeros of the Higgs field.

In a recent publication in Communications in Mathematical Physics, L. Álvarez-Cónsul (ICMAT-CSIC), M. Garcia-Fernandez (ICMAT- UAM) and O. García-Prada (ICMAT-CSIC) found interesting relations between the Einstein–Bogomol’nyi equations and several areas of modern geometry. A first link was obtained applying dimensional reduction methods to the Kähler–Yang– Mills equations on the product of the 2-dimensional sphere with a compact Riemann surface. The Kähler–Yang–Mills equations, introduced in a previous paper [4], emerge from a natural extension of two well-studied theories in geometry, respectively devoted to constant scalar curvature Kähler metrics and Hermitian–Yang–Mills connections. The equations obtained by dimensional reduction generalize the Einstein–Bogomol’nyi equations to a Riemann surface of any genus. Since they couple the vortices to a Riemannian metric on the surface, they were called the gravitating vortex equations.

A second link started with the observation that the set of sufficient conditions found by Y. Yang to solve the Einstein–Bogomol’nyi equations have a natural meaning in the Geometric Invariant Theory, as introduced by D. Mumford in the 1960s to construct quotients by group actions in algebraic geometry. These links inspired a conjecture that identifies the moduli spaces parametrizing solutions of the Einstein–Bogomol’nyi equations and the simplest quotients appearing in Mumford’s theory, namely the ones attached to the theory of binary quantics, a topic of the second half of the 19th century in which computational open problems still remain.

Elaborating on the above interplay, L. Álvarez-Cónsul, M. Garcia-Fernandez, O. García-Prada and V. P. Pingali (IISc–Bangalore) proved in a recent preprint [5] a converse to Yang’s Theorem, establishing in this way a precise correspondence with Geometric Invariant Theory for the Einstein–Bogomol’nyi equations. This provided evidence for the above conjecture about moduli and binary quantics, and furthermore settled in the affirmative a conjecture by Y. Yang about the non-existence of cosmic strings on the Riemann sphere superimposed at a single point [6]. They also showed existence and uniqueness of solutions of the gravitating vortex equations in genus g≥ 2 for an explicit region of coupling parameters. This is a satisfying result that can be compared with the hyperbolic case of the celebrated Uniformization Theorem of Klein, Poincaré, and Koebe from the late 19th and the early 20th centuries, and the closely related result that a compact Riemann surface of genus g≥ 2 admits a unique metric of constant curvature with fixed volume. The proof requires a combination of modern techniques of symplectic geometry and non-linear analysis on manifolds