Geodesic
Eigenvalues of Bethe vectors in the Gaudin model
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 3, pp. 369-394 Cet article a éte moissonné depuis la source Math-Net.Ru

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According to the Feigin–Frenkel–Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical $\mathcal{W}$-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of $q$-characters and classical $\mathcal W$-algebras.
Keywords: Gaudin Hamiltonian, Bethe vector, $q$-character, classical $\mathcal{W}$-algebra. 
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A. I. Molev; E. E. Mukhin. Eigenvalues of Bethe vectors in the Gaudin model. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 3, pp. 369-394. http://geodesic.mathdoc.fr/item/TMF_2017_192_3_a1/

[1] B. Feigin, E. Frenkel, N. Reshetikhin, “Gaudin model, Bethe ansatz and critical level”, Commun. Math. Phys., 166:1 (1994), 27–62 | DOI | MR | Zbl

[2] B. Feigin, E. Frenkel, L. Rybnikov, “Opers with irregular singularity and spectra of the shift of argument subalgebra”, Duke Math. J., 155:2 (2010), 337–363 | DOI | MR | Zbl

[3] B. Feigin, E. Frenkel, V. Toledano Laredo, “Gaudin models with irregular singularities”, Adv. Math., 223:3 (2010), 873–948 | DOI | MR | Zbl

[4] L. G. Rybnikov, “Metod sdviga invariantov i model Godena”, Funkts. analiz i ego pril., 40:3 (2006), 30–43 | DOI | DOI | MR | Zbl

[5] B. Feigin, E. Frenkel, “Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras”, Internat. J. Modern Phys. A, 7, suppl. 1A (1992), 197–215 | DOI | MR | Zbl

[6] E. Frenkel, Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, 103, Cambridge Univ. Press, Cambridge, 2007 | MR | Zbl

[7] A. Chervov, D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, arXiv: hep-th/0604128

[8] A. I. Molev, “Feigin–Frenkel center in types $B$, $C$ and $D$”, Invent. Math., 191:1 (2013), 1–34 | DOI | MR | Zbl

[9] A. V. Chervov, A. I. Molev, “On higher-order Sugawara operators”, Int. Math. Res. Notices, 2009:9 (2009), 1612–1635 | DOI | MR | Zbl

[10] A. I. Molev, E. Ragoucy, “The MacMahon master theorem for right quantum superalgebras and higher sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$”, Mosc. Math. J., 14:1 (2014), 83–119 | MR | Zbl

[11] A. I. Molev, E. E. Mukhin, “Yangian characters and classical $\mathcal W$-algebras”, Conformal Field Theory, Automorphic Forms and Related Topics (Heidelberg, Germany, September 19–23, 2011), Contributions in Mathematical and Computational Sciences, 8, eds. W. Kohnen, R. Weissauer, Springer, Berlin, 2014, 287–334 | MR | Zbl

[12] E. Mukhin, V. Tarasov, A. Varchenko, “Bethe eigenvectors of higher transfer matrices”, J. Stat. Mech. Theory Exp., 2006:8 (2006), P08002, 44 pp. | DOI | MR

[13] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl | Zbl

[14] V. V. Schechtman, A. N. Varchenko, “Arrangements of hyperplanes and Lie algebra homology”, Invent. Math., 106:1 (1991), 139–194 | DOI | MR | Zbl

[15] H. M. Babujian, R. Flume, “Off-shell Bethe ansatz equation for Gaudin magnets and solutions of Knizhnik–Zamolodchikov equations”, Modern Phys. Lett. A, 9:22 (1994), 2029–2039 | DOI | MR | Zbl

[16] E. Mukhin, A. Varchenko, “Norm of a Bethe vector and the Hessian of the master function”, Compos. Math., 141:4 (2005), 1012–1028 | DOI | MR | Zbl

[17] S. Garoufalidis, T. T. Q. Lê, D. Zeilberger, “The quantum MacMahon master theorem”, Proc. Nat. Acad. Sci. USA, 103:38 (2006), 13928–13931 | DOI | MR | Zbl

[18] A. Chervov, G. Falqui, V. Rubtsov, “Algebraic properties of Manin matrices 1”, Adv. Appl. Math., 43:3 (2009), 239–315 | DOI | MR | Zbl

[19] N. Rozhkovskaya, “A new formula for the Pfaffian-type Segal–Sugawara vector”, J. Lie Theory, 24:2 (2014), 529–543 | MR | Zbl

[20] E. Mukhin, A. Varchenko, “Critical points of master functions and flag varieties”, Commun. Contemp. Math., 6:1 (2004), 111–163 | DOI | MR | Zbl

[21] E. Frenkel, N. Reshetikhin, “The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal W$-algebras”, Recent Developments in Quantum Affine Algebras and Related Topics (North Carolina State University, Raleigh, NC, USA, May 21–24, 1998), Contemporary Mathematics, 248, eds. N. Jing, K. C. Misra, 1999, 163–205 | DOI | MR | Zbl

[22] E. Frenkel, E. Mukhin, “Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras”, Commun. Math. Phys., 216:1 (2001), 23–57 | DOI | MR | Zbl

[23] W. Nakai, T. Nakanishi, “Paths, tableaux and $q$-characters of quantum affine algebras: the $C_n$ case”, J. Phys. A: Math. Gen., 39:9 (2006), 2083–2115 | DOI | MR | Zbl

[24] A. I. Molev, E. Ragoucy, “Classical $\mathcal W$-algebras in types $A$, $B$, $C$, $D$ and $G$”, Commun. Math. Phys., 336:2 (2015), 1053–1084 | DOI | MR | Zbl

[25] D. Arnaudon, A. Molev, E. Ragoucy, “On the $R$-matrix realization of Yangians and their representations”, Ann. H. Poincaré, 7:7–8 (2006), 1269–1325 | DOI | MR | Zbl