Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym n (S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb n (S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism Hilb n (S) → Sym n (S) and yields a method of reconstructing the cup product for Hilb n (S) from the orbifold invariants of [Sym n (S)].
Let L be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show that the Gaudin Hamiltonians associated to G are diagonalizable with simple spectrum on the space spanned by singular vectors of any fixed weight in L. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the space spanned by singular vectors of any fixed weight in the tensor product of infinitedimensional unitarizable irreducible highest weight modules.
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