publication . Preprint . 2006

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

Chervov, A.; Talalaev, D.;
Open Access English
  • Published: 19 Apr 2006
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(...
free text keywords: High Energy Physics - Theory, Mathematics - Quantum Algebra, Mathematics - Rings and Algebras
Related Organizations
Download from
34 references, page 1 of 3

[Talalaev04-N1] D. Talalaev Quantization of the Gaudin system, hep-th/0404153 Functional Analysis and Its application Vol. 40 No. 1 pp.86-91 (2006)

[ChervovTalalaev04-N2] D. Talalaev, A. Chervov Universal G-oper and Gaudin eigenproblem, hep-th/0409007

[GWHMFF-N7] R. Goodman, N. Wallach, Higher-order Sugawara operators for affine Lie algebras, Trans. Amer. Math. Soc. 315 (1989) 1-55. [OpenAIRE]

T. Hayashi, Sugawara operators and Kac-Kazhdan conjecture, Invent. Math. 94

F. Malikov On quantum motion integrals , Russ. Math. Surv. 43, No.4, 217-218

(1988); translation from Usp. Mat. Nauk 43, No.4(262), 209-210 (1988)

B. Feigin, E. Frenkel, Int. J. Mod. Phys. A7, Suppl. 1A 1992, 197-215.

[BabelonTalon02-N8] O. Babelon, M. Talon, Riemann surfaces, separation of variables and classical and quantum integrability, hep-th/0209071.

[EnriquezRubtsov01-N9] B. Enriquez, V. Rubtsov, Commuting families in skew fields and quantization of Beauville's fibration, e-print math.AG/0112276

[Birkhoff-N10] G. D. Birkhoff, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. of Amer. Acad. of Arts and Sciences 49, no. 9 (Oct. 1913), 521-568.

Math. Soc. 12, no. 2 (Apr. 1911), 243-284.

[ChervovTalalaev06-03-N13] A. Chervov, D. Talalaev, KZ equation, G-opers, quantum Drinfeld-Sokolov reduction and Cayley-Hamilton identity, hep-th/0607250

and curves. Adv. Math. 38, 267-317 (1980).

[RaisTauvel92-N23] Rais, Mustapha; Tauvel, Patrice Indice et polynomes invariants pour certaines algebres de Lie. (Index and invariant polynomials of certain Lie algebras) (French) J. Reine Angew. Math. 425, 123-140 (1992).

[Molev97-N24] A. Molev, Casimir elements for certain polynomial current Lie algebras “Group 21, Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras,” Vol. 1, (H.-D. Doebner, W. Scherer, P. Nattermann, Eds). World Scientific, Singapore, 1997, 172-176.

34 references, page 1 of 3
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue