Browsing by Subject "Braid group action"

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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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    Quantum Toroidal Superalgebras
    (2020-05) Pereira Bezerra, Luan; Mukhin, Evgeny; Ramras, Daniel; Roeder, Roland; Tarasov, Vitaly
    We introduce the quantum toroidal superalgebra E(m|n) associated with the Lie superalgebra gl(m|n) and initiate its study. For each choice of parity "s" of gl(m|n), a corresponding quantum toroidal superalgebra E(s) is defined. To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. The superalgebra E(s) contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra Uq sl̂(m|n) with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E(s), which exchanges the vertical and horizontal subalgebras. If m and n are different and "s" is standard, we give a construction of level 1 E(m|n)-modules through vertex operators. We also construct an evaluation map from E(m|n)(q1,q2,q3) to the quantum affine algebra Uq gl̂(m|n) at level c=q3^(m-n)/2.