Exact self dual vortices in $d=3$ Majumdar-Papapetrou dilaton gravity

Abstract

We show that self-dual Nielsen Olesen (NO) vortices in $3$ dimensions give rise to a class of exact solutions when coupled to Einstein Maxwell Dilaton gravity obeying the Majumdar-Papapetrou(MP) relation between gravitational and Maxwell couplings , provided certain Chern-Simons type interactions are present. The metric may be solved for explicitly in terms of the NO vortex function and is asymptotic to Euclidean space with signature (-1,-1,-1). The MP electric field is long range but, strictly speaking, the charge of the vortices is zero since the field dies off as $O(1/r (ln r)^2)$. The total ADM energy integral of such vortices is {\it{zero}}. These peculiarities are due to the nature of the two dimensional Greens function.