Browsing by Author "Varchenko, Alexander"

Now showing 1 - 4 of 4
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    Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space
    (Springer, 2021-06) Tarasov, Vitaly; Varchenko, Alexander; Mathematical Sciences, School of Science
    We consider the equivariant quantum differential equation for the projective space $$P^{n-1}$$and introduce a compatible system of difference equations. We prove an equivariant gamma theorem for $$P^{n-1}$$, which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $$P^{n-1}$$. We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of $$P^{n-1}$$and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-known results of Dubrovin and Guzzetti.
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    Frobenius-like structure in Gaudin model
    (World Scientific, 2022-06) Mukhin, Evgeny; Varchenko, Alexander; Mathematical Sciences, School of Science
    We introduce a Frobenius-like structure for the 𝔰𝔩2 Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.
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    Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety
    (Elsevier, 2023-02) Tarasov, Vitaly; Varchenko, Alexander; Mathematical Sciences, School of Science
    We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau–Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the K-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and qKZ difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit. Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety.
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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.