Prescribing Zeros and Poles on a Compact Riemann Surface for a Gravitationally Coupled Abelian Gauge Field Theory

In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results.The authors research was supported in part by NSF grants DMS–9972300 and DMS–9729992 through the Institute for Advanced Study.