On the Gaudin and Xxx Models Associated to Lie Superalgebras

Chenliang Huang, Purdue University

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. We establish a duality of the non-periodic Gaudin model associated with superalgebra glm:n and the non-periodic Gaudin model associated with algebra glk . The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m + n) × (m + n) matrix in the case of glm:n and of a column determinant of a k × k matrix in the case of glk . We obtain our results by proving Capelli type identities for both cases and comparing the results. We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(glm:n ). To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains

Degree

Ph.D.

Advisors

Mukhin, Purdue University.

Subject Area

Mathematics

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