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MP: Theoretische und Mathematische Grundlagen der Physik

MP 7: Integrability and Variational Methods

MP 7.1: Fachvortrag

Mittwoch, 9. März 2005, 12:05–12:25, TU MA141

Noncompact SL(2,R) spin chains — •Marc Kirch¹ and Alexander N. Manashov^2,3 — ¹Institut für theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum — ²Institut für theoretische Physik, Universität Regensburg, D-93040 Regensburg — ³Department of theoretical physics, St.-Petersburg State University, 199034, St.-Petersburg, Russia

We consider completely integrable spin chain models whose spin operators are the generators of unitary representations of the noncompact group SL(2,R). Within the framework of the Quantum Inverse Scattering Method we construct R-operators, being solutions to the Yang-Baxter equation. These act on the corresponding vector spaces on wich the representations are realized. Examine the possible combinations of representations, the solutions exhibit different properties. Using the method of the separated variables and the Baxter Q-operator technique we construct and solve a spin chain model realized on the principal continuous series representation of SL(2,R). The main motivation for the study of such objects comes from the remarkable fact that various dilatation operators, governing the scaling behaviour of some composite field operators in certain (SUSY) Yang-Mills theories, have been found to be integrable. In fact they are in one to one correspondence with the Hamiltonian of a completely integrable spin chain - among them those with a noncompact symmetry group.