Contact Geometry in Superconductors and New Massive Gravity

TL;DR

This paper explores the relationship between contact geometry in 3D manifolds and solutions to various gravity theories, including New Massive Gravity, revealing geometric conditions that lead to vacuum and charged solutions.

Contribution

It establishes a link between contact geometry conditions and solutions to 3D massive gravity theories, including new insights into charged solutions via coupling with gauge fields.

Findings

Contact geometry conditions are equivalent to London's equation in 2+1 electromagnetism.

Certain contact manifolds are shown to be vacuum solutions of New Massive Gravity.

Coupling gravity with gauge theories yields charged solutions from geometric structures.

Abstract

The defining property of every three-dimensional $\varepsilon$-contact manifold is shown to be equivalent to requiring the fulfillment of London's equation in 2+1 electromagnetism. To illustrate this point, we show that every such manifold that is also K-contact and $\eta$-Einstein is a vacuum solution to the most general quadratic-curvature gravity action, in particular of New Massive Gravity. As an example we analyse $S^3$ equipped with a contact structure together with an associated metric tensor such that the canonical generators of the contact distribution are null. The resulting Lorentzian metric is shown to be a vacuum solution of three-dimensional massive gravity. Moreover, by coupling the New Massive Gravity action to Maxwell-Chern-Simons we obtain a class of charged solutions stemming directly from the para-contact metric structure. Finally, we repeat the exercise for the Abelian Higgs theory.