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Universal G-oper and Gaudin eigenproblem | | A. Chervov
; D. Talalaev
; | | Date: |
1 Sep 2004 | | Subject: | High Energy Physics - Theory; Quantum Algebra | hep-th math.QA | | Abstract: | 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. Firstly we explore the quantum analog of the characteristic polynomial which is differential operator in variable z with the coefficients in U(gl(n))^{otimes N}. We will call it "universal G-oper". We find a slight different definition of "Det"(L(z)-partial_z) where L(z) is the quantum Lax operator for the Gaudin model such that 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 tensor product of the finite-dimensional representation of gl(n) and substituting them into the universal G-oper we obtain the scalar differential operator (scalar G-oper) which does not have monodromy. This result is basically due to Frenkel (q-alg/9506003), but his context is slightly different (there is no notion of universal G-oper in his works) and approach is not based on the explicit formulas. We strongly believe that out 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’s D-modules to the 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. | | Source: | arXiv, hep-th/0409007 | | Services: | Forum | Review | PDF | Favorites | |
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