Browsing by Author "Tarasov, Vitaly"

Now showing 1 - 10 of 23
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    Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models
    (2013-11-06) Bothner, Thomas Joachim; Its, Alexander R.; Bleher, Pavel, 1947-; Tarasov, Vitaly; Eremenko, Alexandre; Mukhin, Evgeny
    We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.
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    Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations
    (IOP, 2016-04) Isaev, A. P.; Kirillov, A. N.; Tarasov, Vitaly; Department of Mathematical Sciences, School of Science
    Commutative sets of Jucys–Murphy elements for affine braid groups of ${A}^{(1)},{B}^{(1)},{C}^{(1)},{D}^{(1)}$ types were defined. Construction of R-matrix representations of the affine braid group of type ${C}^{(1)}$ and its distinguished commutative subgroup generated by the ${C}^{(1)}$-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the ${C}^{(1)}$-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements ${Y}^{\prime }{\rm{s}}$ in the double affine Hecke algebra of type A.
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    Certain Aspects of Quantum and Classical Integrable Systems
    (2022-08) Kosmakov, Maksim; Tarasov, Vitaly; Its, Alexander; Mukhin, Evgeny; Ramras, Daniel
    We derive new combinatorail formulas for vector-valued weight functions for the evolution modules over the Yangians Y (gl_n). We obtain them using the Nested Algebraic Bethe ansatz method. We also describe the asymptotic behavior of the radial solutions of the negative tt* equation via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method.
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    Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions
    (Springer, 2014-08) Rimányi, R.; Tarasov, Vitaly; Varchenko, A.; Mathematical Sciences, School of Science
    We consider the conormal bundle of a Schubert variety SI in the cotangent bundle T∗Gr of the Grassmannian Gr of k-planes in Cn. This conormal bundle has a fundamental class κI in the equivariant cohomology H∗T(T∗Gr). Here T=(C∗)n×C∗. The torus (C∗)n acts on T∗Gr in the standard way and the last factor C∗ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum YI of the Yangian Y(gl2) weight functions (WJ)J. We describe a relation of YI with the double Schur polynomial [SI]. A modified version of the κI classes, named κ′I, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold T∗Gr. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on Gr. The classes (κ′I)I form a basis in the suitably localized equivariant cohomology H∗T(T∗Gr). This basis depends on the choice of the coordinate flag in Cn. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.
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    Connection Problem for Painlevé Tau Functions
    (2019-08) Prokhorov, Andrei; Its, Alexander; Bleher, Pavel; Eremenko, Alexandre; Tarasov, Vitaly
    We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.
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    Duality for Knizhnik-Zamolodchikov and Dynamical Operators
    (SIGMA, 2020) Tarasov, Vitaly; Uvarov, Filipp; Mathematical Sciences, School of Science
    We consider the Knizhnik-Zamolodchikov and dynamical operators, both differential and difference, in the context of the (glk,gln)-duality for the space of polynomials in kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and dynamical operators naturally exchange under the duality.
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    Duality of Gaudin models
    (2020-08) Uvarov, Filipp; Tarasov, Vitaly; Mukhin, Evgeny; Its, Alexander; Ramras, Daniel
    We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1},\dots ,z_{k})$ and $\bar{\alpha}=(\alpha_{1},\dots ,\alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$. One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians. We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.
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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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    Fuchsian Equations with Three Non-Apparent Singularities
    (National Academy of Science of Ukraine, 2018) Eremenko, Alexandre; Tarasov, Vitaly; Mathematical Sciences, School of Science
    We show that for every second order Fuchsian linear differential equation E with n singularities of which n−3 are apparent there exists a hypergeometric equation H and a linear differential operator with polynomial coefficients which maps the space of solutions of H into the space of solutions of E. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations E with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature 1 on the punctured sphere with conic singularities, all but three of them having integer angles.
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    Gaudin models associated to classical Lie algebras
    (2020-08) Lu, Kang; Mukhin, Evgeny; Its, Alexander; Roeder, Roland; Tarasov, Vitaly
    We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.