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D. Talalaev

Publications and source records attributed to D. Talalaev.

13 recordsLinked to original sources
arXiv2010

Quantum generic Toda system

The Toda chains take a particular place in the theory of integrable systems, in contrast with the linear group structure for the Gaudin model this system is related to the corresponding Borel group and mediately to the geometry of flag varieties. The main goal of this paper is to reconstruct a "spectral curve" in a wider context of the generic Toda system. This appears to be an efficient way to find its quantization which is obtained here by the technique of quantum characteristic polynomial for the Gaudin model and an appropriate AKS reduction. We discuss also some relations of this result with the recent consideration of the Drinfeld Zastava space, the monopole space and corresponding Borel Yangian symmetries.

math-ph↗
arXiv2008

Bethe ansatz and Isomonodromic deformations

We study symmetries of the Bethe equations for the Gaudin model appeared naturally in the framework of the geometric Langlands correspondence under the name of Hecke operators and under the name of Schlesinger transformations in the theory of isomonodromic deformations, and particularly in the theory of Painlev\'e transcendents.

math-ph↗
arXiv2006

Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the ``quantum spectral curve'' and argue that it takes the analogous structural and unifying role on the quantum level also. In the simplest, but essential case the ``quantum spectral curve'' is given by the formula "det"(L(z)-dz) [Talalaev04] (hep-th/0404153). As an easy application of our constructions we obtain the following: quite a universal receipt to define quantum commuting hamiltonians from the classical ones, in particular an explicit description of a maximal commutative subalgebra in U(gl(n)[t])/t^N and in U(\g[t^{-1}])\otimes U(t\g[t]); its relation with the center on the of the affine algebra; an explicit formula for the center generators and a conjecture on W-algebra generators; a receipt to obtain the q-deformation of these results; the simple and explicit construction of the Langlands correspondence; the relation between the ``quantum spectral curve'' and the Knizhnik-Zamolodchikov equation; new generalizations of the KZ-equation; the conjecture on rationality of the solutions of the KZ-equation for special values of level. In the simplest cases we observe the coincidence of the ``quantum spectral curve'' and the so-called Baxter equation. Connection with the KZ-equation offers a new powerful way to construct the Baxter's Q-operator.

hep-th↗
arXiv2006

KZ equation, G-opers, quantum Drinfeld-Sokolov reduction and quantum Cayley-Hamilton identity

The Lax operator of the Gaudin type models is a 1-form on the classical level. In virtue of the quantization scheme proposed in [Talalaev04] (hep-th/0404153) it is natural to treat the quantum Lax operator as a connection; this connection is a particular case of the Knizhnik-Zamolodchikov connection [ChervovTalalaev06] (hep-th/0604128). In this paper we find a gauge transformation which produces the "second normal form" or the "Drinfeld-Sokolov" form. Moreover the differential operator naturally corresponding to this form is given precisely by the quantum characteristic polynomial [Talalaev04] of the Lax operator (this operator is called the G-oper or Baxter equation). This observation allows to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ-equations has only meromorphic solutions. As a corollary we obtain the quantum Cayley-Hamilton identity for the Gaudin-type Lax operators (including the general gl(n)[t] case). The presented construction sheds a new light on a geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism.

hep-th↗
arXiv2005

On the Lie-formality of Poisson manifolds

Starting from the problem of describing cohomological invariants of Poisson manifolds we prove in a sense a ``no-go'' result: the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold is formal.

math.SG↗
arXiv2004

Universal G-oper and Gaudin eigenproblem

This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without monodromy in general gl(n) case modulo a hypothesys on the analytic properties of the solution of a KZ-type equation. Firstly we explore the quantum analog of the characteristic polynomial which is a differential operator in a variable $u$ with the coefficients in U(gl(n))^{\otimes N}. We will call it "universal G-oper". It is constructed by the formula "Det"(L(u)-\partial_u) where L(u) is the quantum Lax operator for the Gaudin model and "Det" is appropriate definition of the determinant. The coefficients of this differential operator are quantum Gaudin Hamiltonians obtained by one of the authors (D.T. hep-th/0404153). We establish the correspondence between eigenvalues and $G$-opers as follows: taking eigen-values of the Gaudin's hamiltonians on the joint eigen-vector in the tensor product of finite-dimensional representation of gl(n) and substituting them into the universal G-oper we obtain the scalar differential operator (scalar G-oper) which conjecturally does not have monodromy. We strongly believe that our quantization of the Gaudin model coincides with quantization obtained from the center of universal enveloping algebra on the critical level and that our scalar G-oper coincides with the G-oper obtained by the geometric Langlands correspondence, hence it provides very simple and explicit map (Langlands correspondence) from Hitchin D-modules to G-opers in the case of rational base curves. It seems to be easy to generalize the constructions to the case of other semisimple Lie algebras and models like XYZ.

hep-th↗
arXiv2004

Quantization of the Gaudin System

In this article we exploit the known commutative family in Y(gl(n)) - the Bethe subalgebra - and its special limit to construct quantization of the Gaudin integrable system. We give explicit expressions for quantum hamiltonians QI_k(u), k=1,..., n. At small order k=1,...,3 they coincide with the quasiclassic ones, even in the case k=4 we obtain quantum correction.

hep-th↗
arXiv2004

Rational Lax operators and their quantization

We investigate the construction of the quantum commuting hamiltonians for the Gaudin integrable model. We prove that [Tr L^k(z), Tr L^m(u) ]=0, for k,m < 4 . However this naive receipt of quantization of classically commuting hamiltonians fails in general, for example we prove that [Tr L^4(z), Tr L^2(u) ] \ne 0. We investigate in details the case of the one spin Gaudin model with the magnetic field also known as the model obtained by the "argument shift method". Mathematically speaking this method gives maximal Poisson commutative subalgebras in the symmetric algebra S(gl(N)). We show that such subalgebras can be lifted to U(gl(N)), simply considering Tr L(z)^k, k\le N for N<5. For N=6 this method fails: [Tr L_{MF}(z)^6, L_{MF}(u)^3]\ne 0 . All the proofs are based on the explicit calculations using r-matrix technique. We also propose the general receipt to find the commutation formula for powers of Lax operator. For small power exponents we find the complete commutation relations between powers of Lax operators.

hep-th↗
arXiv2003

Hitchin systems on singular curves II. Gluing subschemes

In this paper we continue our studies of Hitchin systems on singular curves (started in hep-th/0303069). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec $\{f \in \CC[z]: f(A(\ep))=f((B(\ep)); \ep^N=0 \}$, where $A(\ep), B(\ep)$ are arbitrary polynomials) . The most simple examples are the generalized cusp curves which are projectivizations of Spec $\{f \in \CC[z]: f'(0)=f''(0)=...f^{N-1}(0)=0 \}$). We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials $A(\ep), B(\ep)$ defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop $r$-matrix formalism for the functions on the truncated loop group $GL_n(\CC[z]), z^N=0$. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.

hep-th↗
arXiv2003

Riemann bilinear form and Poisson structure in Hitchin-type systems

In this paper we reinterpret the Poisson structure of the Hitchin-type system in cohomological terms. The principal ingredient of a new interpretation in the case of the Beauville system is the meromorphic cohomology of the spectral curve, and the main result is the identification of the Riemann bilinear form and the symplectic structure of the model. Eventual perspectives of this approach lie in the quantization domain.

hep-th↗
arXiv2003

Hitchin system on singular curves I

In this paper we study Hitchin system on singular curves. Some examples of such system were first considered by N. Nekrasov (hep-th/9503157), but our methods are different. We consider the curves which can be obtained from the projective line by gluing several points together or by taking cusp singularities. (More general cases of gluing subschemas will be considered in the next paper). It appears that on such curves all ingredients of Hitchin integrable system (moduli space of vector bundles, dualizing sheaf, Higgs field etc.) can be explicitly described, which may deserve independent interest. As a main result we find explicit formulas for the Hitchin hamiltonians. We also show how to obtain the Hitchin integrable system on such a curve as a hamiltonian reduction from a more simple system on some finite-dimensional space. In this paper we also work out the case of a degenerate curve of genus two and find the analogue of the Narasimhan-Ramanan parameterization of SL(2)-bundles. We describe the Hitchin system in such coordinates. As a demonstration of the efficiency of our approach we also rederive the rational and trigonometric Calogero systems from the Hitchin system on cusp and node with a marked point.

hep-th↗
arXiv2002

The elliptic Gaudin system with spin

The elliptic Gaudin model was obtained as the Hitchin system on an elliptic curve with two fixed points. In the present paper the algebraic-geometrical structure of the system with two fixed points is clarified. We identify this system with poles dynamics of the finite gap solutions of Davey-Stewartson equation. The solutions of this system in terms of theta-functions and the action-angle variables are constructed. We also discuss the geometry of its degenerations.

hep-th↗