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

Abstract

We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ 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 $\mathfrak{gl}_{M|N}$ Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.