Cetraro, June 11-76, 2007 Vortices in Chern-Simons theory Prof. Gabriella Tarantello, Università Tor Vergata, Roma, Italia Abstract We shall discuss some analytical aspects related to the study of self-dual vortex configurations in Chern-Simons gauge field theory. We start with the introduction of basic notions of gauge theory in order to formulate Chern-Simons models of interest in matter physics and superconductivity. For those model we derive the corresponding self-dual equations (cfr [D]). Then, we show how to adapt an approach introduced by Taubes (cfr [JT]) for the abelian Higgs model, in order to reduce the self-dual vortex equations to elliptic problems of Liouville type in presence of Dirac measures supported at the vortex points. An analytical discussion of such elliptic problems will be provided in relation to their "concentration-compactness-principles" as well as sharp Moser-Trudinger type inequalities. Most of the material discussed in the lectures is taken from [T]. References [JT] Jaffe A., Taubes C. "Vortices and monopoles.", Birkhauser, Boston (1980). [T] Tarantello G. Selfdual Gauge Field Vortices: an analytical approach. PNLDE 72, Birkhauser Boston, Inc. Boston MA 2007. |