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Virasoro constraints in Drinfeld–Sokolov hierarchies

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Abstract

We describe a geometric theory of Virasoro constraints in generalized Drinfeld–Sokolov hierarchies. Solutions of Drinfeld–Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of a Higgs field near infinity. String solutions for these hierarchies are defined as points having a big stabilizer under a certain Lie algebra action. We characterize principal bundles coming from string solutions as those possessing connections compatible with the Higgs field near infinity. We show that tau-functions of string solutions satisfy second-order differential equations generalizing the Virasoro constraints of 2d quantum gravity.

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References

  1. Adler, M., van Moerbeke, P.: A matrix integral solution to two-dimensional \(W_p\)-gravity. Comm. Math. Phys. 147, 25–56 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bakalov, B., Milanov, T.: \(\cal W\)-constraints for the total descendant potential of a simple singularity. Compos. Math. 149, 840–888 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beĭlinson, A., Feigin, B., Mazur, B.: Notes on Conformal Field Theory (unpublished)

  4. Ben-Zvi, D., Frenkel, E.: Spectral curves, opers and integrable systems. Publ. Math. IHES 94, 87–159 (2001). arXiv:math/9902068

    Article  MATH  MathSciNet  Google Scholar 

  5. Ben-Zvi, D., Frenkel, E.: Geometric realization of the Segal-Sugawara construction, topology, geometry and quantum field theory. Lond. Math. Soc. Lect. Note Ser. 308, 46–97 (2004). arXiv:math/0301206

    MathSciNet  Google Scholar 

  6. Dijkgraaf, R., Verlinde, H., Verlinde, E.: Loop equations and virasoro constraints in non-perturbative 2-D quantum gravity. Nucl. Phys. B 348, 435–456 (1991)

    Article  MathSciNet  Google Scholar 

  7. Donagi, R., Gaitsgory, D.: The gerbe of Higgs bundles. Transform. Groups 7, 109–153 (2002). arXiv:math/0005132

    Article  MATH  MathSciNet  Google Scholar 

  8. Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korteweg-de Vries type (Russian), Itogi Nauki i Tekhniki, Current problems in mathematics, vol. 24, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., pp. 81–180 (1984)

  9. Gelfand, I., Dikii, L.: Asymptotic behaviour of the resolvent of Sturm–Liouville equations and the algebra of the Korteweg–de Vries equations. Russ. Math. Surveys 30(5), 77–113 (1975)

    Article  MathSciNet  Google Scholar 

  10. Kac, V., Peterson, D.: 112 constructions of the basic representation of the loop group of \(E_8\). In: Proceedings of the Conference Anomalies, Geometry, Topology (Argonne, 1985), World Scientific, pp. 276–298 (1985)

  11. Krichever, I.: Vector bundles and Lax equations on algebraic curves. Comm. Math. Phys. 229, 229–269 (2002). arXiv:hep-th/0108110

    Article  MATH  MathSciNet  Google Scholar 

  12. Kac, V., Schwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 257, 329–334 (1991)

    Article  MathSciNet  Google Scholar 

  13. Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent. Math. 89, 395–423 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Laszlo, Y.: Hitchin’s and WZW connections are the same. J. Differ. Geom. 49, 547–576 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Moore, G.: Geometry of the string equations. Comm. Math. Phys. 133, 261–304 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mulase, M.: Algebraic theory of the KP equations, perspectives in mathematical physics. Conf. Proc. Lect. Notes Math. Phys. III, 151–217 (1994)

    MathSciNet  Google Scholar 

  17. Plaza Martín, F.J.: Algebro-geometric Solutions of the String Equation. preprint. arXiv:1110.0729

  18. Schwarz, A.: On solutions to the string equation. Mod. Phys. Lett. A 6, 2713–2726 (1991). arXiv:hep-th/9109015

    Article  MATH  Google Scholar 

  19. Sorger, C.: Lectures on moduli of principal \(G\)-bundles over algebraic curves, moduli spaces in algebraic geometry. ICTP Lect. Note Ser. 1, 3–57 (2000)

    MathSciNet  Google Scholar 

  20. Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 61, 5–65 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Teleman, C.: Borel-Weil-Bott theory on the moduli stack of \(G\)-bundles over a curve. Invent. Math. 134, 1–57 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wu, C-Z.: Tau Functions and Virasoro Symmetries for Drinfeld–Sokolov Hierarchies. (preprint), arXiv:1203.5750

  23. Zhu, X.: Affine Demazure modules and \(T\)-fixed point subschemes in the affine Grassmannian. Adv. Math. 221, 570–600 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

First and foremost, the author would like to thank his advisor, David Ben-Zvi, without whose guidance and careful explanations this work would not be completed. The author also thanks Kevin Costello for useful discussions and the referee for pointing out a mistake in the definition of algebro-geometric solutions.

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Correspondence to Pavel Safronov.

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Safronov, P. Virasoro constraints in Drinfeld–Sokolov hierarchies. Sel. Math. New Ser. 22, 27–54 (2016). https://doi.org/10.1007/s00029-015-0182-1

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