Commuting families in skew fields and quantization of Beauville’s fibration. (English) Zbl 1052.53065

From the summary: The authors construct commutating families in fraction fields of symmetric power of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a \(K3\)-surface \(S\), they correspond to Lagrangian fibration introduced by A. Beauville [Symp. Math. 32, 25–31 (1991; Zbl 0827.58022)]. When \(S\) is the canonical cone of a curve \(C\), the authors construct commutating families of differential operators on symmetric powers of \(C\), quantizing the Beauville systems.


53D50 Geometric quantization
14H70 Relationships between algebraic curves and integrable systems
12E15 Skew fields, division rings
14J99 Surfaces and higher-dimensional varieties


Zbl 0827.58022
Full Text: DOI arXiv Euclid


[1] A. Beauville, “Systèmes hamiltoniens complètement intégrables associés aux surfaces \(K3\)” in Problems in the Theory of Surfaces and their Classification (Cortona, Italy, 1988) , Sympos. Math. 32 , Academic Press, London, 1991, 25–31. · Zbl 0827.58022
[2] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable systems and Hecke eigensheaves , preprint, 2000,
[3] F. Bottacin, Poisson structures on Hilbert schemes of points of a surface and integrable systems , Manuscripta Math. 97 (1998), 517–527. · Zbl 0945.53049
[4] P. B. Cohen, Yu. Manin, and D. Zagier, “Automorphic pseudodifferential operators” in Algebraic Aspects of Integrable Systems , Progr. Nonlinear Differential Equations Appl. 26 , Birkhäuser, Boston, 1997, 17–47. · Zbl 1055.11514
[5] R. Donagi, L. Ein, and R. Lazarsfeld, “Nilpotent cones and sheaves on \(K3\) surfaces” in Birational Algebraic Geometry (Baltimore, 1996) , Contemp. Math. 207 , Amer. Math. Soc., Providence, 1997, 51–61. · Zbl 0907.32004
[6] B. Enriquez and A. Odesskii, Quantization of canonical cones of algebraic curves , · Zbl 1052.14035
[7] B. Enriquez and V. Rubtsov, Quantizations of Hitchin and Beauville-Mukai integrable systems ,
[8] E. Frenkel, “Affine algebras, Langlands duality and Bethe ansatz” in XIth International Congress of Mathematical Physics (Paris, 1994) , ed. D. Iagolnitzer, International Press, Cambridge, Mass., 1995, 606–642. · Zbl 1052.17504
[9] A. Gorsky, N. Nekrasov, and V. Rubtsov, Hilbert schemes, separated variables and \(D\)-branes , Comm. Math. Phys. 222 (2001), 299–318. · Zbl 0985.81107
[10] C. Grunspan, Discrete quantum Drinfeld-Sokolov correspondence , Comm. Math. Phys. 226 (2002), 627–662. · Zbl 1041.17029
[11] N. Hitchin, Stable bundles and integrable systems , Duke Math. J. 54 (1987), 91–114. · Zbl 0627.14024
[12] A. Källström and B. D. Sleeman, Solvability of a linear operator system , J. Math. Anal. Appl. 55 (1976), 785–793. · Zbl 0335.47008
[13] M. Kapranov, Noncommutative geometry based on commutator expansions , · Zbl 0918.14001
[14] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface , Invent. Math. 77 (1984), 101–116. · Zbl 0565.14002
[15] E. K. Sklyanin, Separation of variables in the Gaudin model , J. Soviet Math. 47 (1989), 2473–2488. · Zbl 0692.35107
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