Browsing by Subject "quantum differential equation"

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    Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian
    (Springer Nature, 2024) Tarasov, Vitaly; Varchenko , Alexander; Mathematical Sciences, School of Science
    We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters.
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    q-hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem
    (Elsevier, 2019-08) Tarasov, Vitaly; Varchenko, Alexander; Mathematical Sciences, School of Science
    We describe q-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle T ∗F of a partial flag variety F . These q-hypergeometric solutions manifest a Landau-Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for T ∗F says that the leading term of the asymptotics of the q-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of T ∗F multiplied by the exponentials of the equivariant first Chern classes of the associated vector bundles. That statement is analogous to the statement of the gamma conjecture by B.Dubrovin and by S.Galkin, V.Golyshev, and H. Iritani, see also the Gamma theorem for F in Appendix B.