String hypothesis for gl(n|m) spin chains: a particle/hole democracy
Pennsylvania State University
Mon, Nov. 15th 2010, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
Two important problems about integrable spin chains is what field theories can be obtained from them in a continuous limit and how to describe such theories in a finite volume. A possible first step for studying these problems is to formulate a string hypothesis for the solutions of the Bethe Ansatz equations and to approximate the Bethe Ansatz equations with linear integral equations. In this talk we show that the integral equations can be rewritten in a remarkably symmetrical way that treats equivalently the density of string configurations and the density of holes for string configurations.
The symmetrical integral equations are suitable for any kind of particle/hole transformations and therefore for construction of the field theories that may be obtained in the continuous limit of spin chains. Also, the symmetrical integral equations immediately suggest the structure of the Y-system which is defined in a general situation on a T-hook domain.
The discussion is valid for arbitrary choice of a Kac-Dynkin diagram of the gl(n $\vert$ m) symmetry algebra and for spin chains with all cites being in the same representation of the so called rectangular type. One can construct a bijection between possible string configurations and rectangular representations. The origin for this bijection is not clear.