The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz. (English) Zbl 1213.14101

Mukhin, Evgeny; Tarasov, Vitaly; Varchenko, Alexander

Ann. Math. (2) 170, No. 2, 863-881 (2009).

Let \(V\) be an \(r+1\) dimensional complex linear subspace of \(\mathbb{C}[x]\). The Wronskian of \(V\) is defined as the Wronskian of an ordered basis of \(V\); hence it is defined up to multiplication by a non-zero scalar. The main result of the paper is the proof of the following theorem (earlier B. and M. Shapiro conjecture): If the roots of the Wronskian \(Wr\) of \(V\) are real, then \(V\subset \mathbb{R}[x] \subset \mathbb{C}[x]\). A continuity argument shows that it is enough to prove this theorem in the case when \(Wr\) has simple roots. The authors recall a Schubert calculus argument to find an upper bound for the number of subspaces \(V \subset C[x]\) with a given exponent at infinity (degrees of a basis) and \(Wr=\prod (x-z_i)\), where the \(z_i\)’s are distinct. In Sections 2 and 3 the Bethe ansatz construction for a Gaudin model is studied and used to construct the expected number of subspaces with given exponent and given \(Wr\). A thorough study of the Gaudin Hamiltonians shows that the constructed subspaces \(V\) are all real if the roots of \(Wr\) are all real. This proves the main theorem.
The authors also discuss some related topics. They show that their main theorem implies that if the ramified points of a map \(\phi: \mathbb{C}\mathbb{P}^1 \to \mathbb{C}\mathbb{P}^r\) are contained in a circle in \(\mathbb{C}\mathbb{P}^1\), then \(\phi\) of this circle is contained in an (appropriate) \(\mathbb{R}\mathbb{P}^r \subset \mathbb{C}\mathbb{P}^r\). They also prove a statement about how the transversality of intersections of Schubert cycles in Grassmannians is related to the simplicity of the spectrum of Gaudin Hamiltonians. In Appendix B they show that their results imply the reality of orbits of critical points of master functions for the root systems A, B, and C; and conjecture this statement in general.

Reviewer: Richárd Rimányi (Chapel Hill)

MSC:

14N15	Classical problems, Schubert calculus
17B80	Applications of Lie algebras and superalgebras to integrable systems
34M35	Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

real Schubert calculus; B. & M. Shapiro conjecture; Bethe ansatz; Gaudin model

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