Abstract
In this note, we introduce the generalization of opers (superopers) for a certain class of superalgebras with a root system, which admits a basis of odd roots. We study in detail SPL 2-superopers and in particular derive the corresponding Bethe ansatz equations, which describe the spectrum of the \({\mathfrak{osp}(1|2)}\) Gaudin model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arvis J.F.: Classical dynamics of supersymmetric Liouville theory. Nucl. Phys. B 212, 151–172 (1983)
Beilinson, A., Drinfeld, V.: Opers. arXiv:math.AG/0501398
Beilinson A., Manin Y., Schechtman V.V.: Sheaves of the Virasoro and Neveu–Schwarz algebras. Lect. Notes Math. 1289, 52–66 (1987)
Berezin, F.A.: Introduction to Superanalysis. Springer, Berlin (1987)
Bergvelt, M.J., Rabin, J.M.: Supercurves, their Jacobians, and super KP equations, Duke Math. J. 98(1), 1–57 (1999)
Crane L., Rabin J.M.: Super Riemann surfaces: uniformization and Teichmuller theory. Commun. Math. Phys. 113, 601–623 (1988)
Delduc F., Ragoucy E., Sorba P.: Super-Toda theories and W-algebras from superspace Wess–Zumino–Witten models. Commun. Math. Phys. 146, 403–426 (1992)
Delduc, F., Gallot, A.: Supersymmetric Drinfeld–Sokolov reduction. arXiv:solv-int/9802013
Drinfeld V., Sokolov V.: Lie algebras and KdV type equations. J. Sov. Math. 30, 1975–2036 (1985)
Feigin B., Frenkel E., Reshetikhin N.: Gaudin model, Bethe ansatz and critical level. Commun. Math. Phys. 166, 27–62 (1994)
Frappat, L., Ragoucy, E., Sorba, P.: W-algebras and superalgebras from constrained WZW models: a group theeoretical classification. arXiv:hep-th/9207102
Frenkel, E.: Affine Algebras, Langlands Duality and Bethe ansatz. In: Proceedings of International Congress of Mathematical Physics, Paris, 1994, International Press, pp. 606–642
Frenkel, E.: Langlands correspondence for loop groups, CUP, 2007
Frenkel, E.: Opers on the projective line, flag manifolds and Bethe ansatz. arXiv:math/0308269
Heluani R.: SUSY vertex algebras and supercurves. Commun. Math. Phys. 275, 607–658 (2007)
Inami, T., Kanno, H.: Lie superalgebraic approach to super Toda lattice and generalized super KdV equations. Commun. Math. Phys. 136, 519–542 (1991)
Kulish P.P., Manojlovic N.: Bethe vectors of the osp(1|2) Gaudin model. Lett. Math. Phys. 55, 77–95 (2001)
Kulish P.P., Zeitlin A.M.: Group theoretical structure and inverse scattering method for super-KdV equation. J. Math. Sci. 125, 203–214 (2005)
Leites, D.A.: Theory of supermanifolds. In: Akad, K.F., Mauk, S.S.S.R. (eds.) Petrozavodsk (1983)
Manin Y.I.: Critical dimensions of the string theories and the dualizing sheaf on the moduli space of (super) curves. Funct. Anal. Appl. 20, 244–246 (1986)
Manin, Y.I.: Gauge Field Theory and Complex Geometry. Springer, Berlin (1997)
Manin, Y.I.: Topics in Noncommutative Geometry, Princeton University Press, USA (1991)
Manin, Y.I., Voronov, A.A.: Supercellular partitions of flag superspaces. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. VINITI Moscow 32, 27–70 (1988)
Penkov, I.B.: Borel–Weil–Bott theory for classical Lie supergroups, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. VINITI Moscow 32, 71–124 (1988)
Mathieu, P.: Super Miura transformations, super Schwarzian derivatives and super Hill Operators. In: Integrable and Superintegrable systems, World Scientific, pp. 352–388 (1991)
Rakowski M., Thompson G.: Connections on vector bundles over super Riemann surfaces. Phys. Lett. B 220, 557–561 (1989)
Vaintrob, A.Y.: Deformation of complex superspaces and coherent sheaves on them. J. Soviet Math. 51(1), 2140–2188 (1990)
Witten, E.: Khovanov homology and Gauge theory. arXiv:1108.3103
Witten, E.: Notes on super Riemann surfaces and their moduli. arXiv:1209.2459
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zeitlin, A.M. Superopers on Supercurves. Lett Math Phys 105, 149–167 (2015). https://doi.org/10.1007/s11005-014-0735-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-014-0735-9