The solutions of $${\mathfrak {g}}{\mathfrak {l}}_{M|N}$$ Bethe ansatz equation and rational pseudodifferential operators

Abstract

We describe a reproduction procedure which, given a solution of the glM|N Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all glM|N Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.