Skip to main content
SpringerLink
Log in

Supersymmetric Bi-Hamiltonian Systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We construct super Hamiltonian integrable systems within the theory of supersymmetric Poisson vertex algebras (SUSY PVAs). We provide a powerful tool for the understanding of SUSY PVAs called the super master formula. We attach some Lie superalgebraic data to a generalized SUSY W-algebra and show that it is equipped with two compatible SUSY PVA brackets. We reformulate these brackets in terms of odd differential operators and obtain super bi-Hamiltonian hierarchies after performing a supersymmetric analog of the Drinfeld–Sokolov reduction on these operators. As an example, an integrable system is constructed from \({\mathfrak {g}}=\mathfrak {osp}(2|2)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bouwknegt, P., Schoutens, K.: W symmetry in conformal field theory. Phys. Rep. 223(4), 183–276 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  2. Burroughs, N.J., De Groot, M.F., Hollowood, T.J., Miramontes, J.L.: Generalized Drinfel’d–Sokolov hierarchies, II. The Hamiltonian structures. Commun. Math. Phys. 153, 187–215 (1993)

    Article  ADS  Google Scholar 

  3. Bakalov, B., Kac, V.G.: Field algebras. IMRN 3, 123–159 (2003)

    Article  MathSciNet  Google Scholar 

  4. Barakat, A., De Sole, A., Kac, V.: Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math. 4(2), 141–252 (2009)

    Article  MathSciNet  Google Scholar 

  5. Carpentier, S., De Sole, A., Kac, V., Valeri, D, Van de Leur, J.: \(p\)-reduced multicomponent KP hierarchy and classical \(W\)-algebras \({\cal{W}}(gl_N,p)\), submitted. arXiv:1909.03301 (2019)

  6. Delduc, F., Feher, L.: Gallot, nonstandard Drinfeld–Sokolov reduction. J. Phys. A 31(25), 5545–5563 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  7. De Groot, M.F., Hollowood, T.J., Miramontes, J.L.: Generalized Drinfel’d-Sokolov hierarchies. Commun. Math. Phys. 145, 57–84 (1992)

    Article  ADS  Google Scholar 

  8. De Sole, A., Kac, V.G.: Finite vs affine W-algebras. Jpn. J. Math. 1, 137–261 (2006)

    Article  MathSciNet  Google Scholar 

  9. De Sole, A., Kac, V.G., Valeri, D.: Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras. Commun. Math. Phys. 360(3), 851–918 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  10. De Sole, A., Kac, V.G., Valeri, D.: A new scheme of integrability for (bi)Hamiltonian PDE. Commun. Math. Phys. 347(2), 449–488 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  11. De Sole, A., Kac, V.G., Valeri, D.: Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras. Int. Math. Res. Not. 21, 11186–11235 (2015)

    Article  MathSciNet  Google Scholar 

  12. De Sole, A., Kac, V.G., Valeri, D.: Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras. Commun. Math. Phys. 323(2), 663–711 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  13. De Sole, A., Kac, V. G., Valeri, D.: Classical W-algebras and generalized Drinfeld–Sokolov hierarchies for minimal and short nilpotents, Commun. Math. Phys. 331(2), 623–676 (2014). Erratum in Commun. Math. Phys. 333(3), 1617–1619 (2015)

  14. De Sole, A., Kac, V.G., Valeri, D.: Classical W-algebras for \(gl_N\) and associated integrable Hamiltonian hierarchies. Commun. Math. Phys. 348(1), 265–319 (2016)

    Article  Google Scholar 

  15. De Sole, A., Kac, V.G., Valeri, D.: Double Poisson vertex algebras and non-commutative Hamiltonian equations. Adv. Math. 281, 1025–1099 (2015)

    Article  MathSciNet  Google Scholar 

  16. Delduc, F., Gallot, L.: Supersymmetric Drinfeld–Sokolov reduction. J. Math. Phys. 39(9), 4729–4745 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  17. Drinfel’d, V.G., Sokolov, V.V.: Lie algebras and equations of Korteseg–de Vries type. J. Sov. Math. 30, 1975–2036 (1984)

    Article  Google Scholar 

  18. Fehér, L., Harnad, J., Marshall, I.: Generalized Drinfel’d–Sokolov reductions and KdV type hierarchies. Commun. Math. Phys. 154(1), 181–214 (1993)

    Article  ADS  Google Scholar 

  19. Fehér, L., Marshall, I.: Extensions of the matrix Gelfand–Dickey hierarchy from generalized Drinfeld–Sokolov reduction. Commun. Math. Phys. 183(2), 423–461 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  20. Heluani, R., Kac, V.G.: Supersymmetric vertex algebras. Commun. Math. Phys. 271(1), 103–178 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  21. Inami, T., Kanno, H.: Generalized N = 2 super KdV hierarchies: Lie superalgebraic methods and scalar super Lax formalism. In nite analysis Part A, B (Kyoto, 1991), 419–447, Adv. Ser. Math. Phys. 16, World Sci. Publ., River Edge, NJ (1992)

  22. Inami, T., Kanno, H.: Lie superalgebraic approach to super Toda lattice and generalized super KdV equations. Commun. Math. Phys. 136, 519–542 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  23. Inami, T.: Super-W algebras and generalized super-KdV equations. Strings ’90 (College Station, TX, 1990), 321–334, World Sci. Publ., River Edge, NJ (1991). 81R10

  24. Inami, T., Kanno, H.: N = 2 super KdV and super sine-Gordon equations based on Lie super algebra \(A(1,1)^{(1)}\), Nucl. Phys. B 359(1): 201–217. 58F07 (17B67) (1991)

  25. Kac, V.: Vertex algebras for beginners, University Lecture Series, AMS, vol. 10, 2nd ed. 1996, AMS (1998)

  26. Kupershmidt, B.A.: A super Korteweg–de Vries equation: an integrable system. Phys. Lett. A 102(5–6), 213–215 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  27. Kulish, P., Zeitlin, A.: Super-KdV equation: classical solutions and quantization, PAMM \(\cdot \). Proc. Appl. Math. Mech. 4, 576–577 (2004)

    Article  Google Scholar 

  28. Liu, S.-Q., Wu, C.-Z., Zhang, Y.: On the Drinfeld–Sokolov hierarchies of D type. Int. Math. Res. Not. IMRN 8, 1952–1996 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Madsen, J.O., Ragoucy, E.: Quantum Hamiltonian reduction in superspace formalism. Nucl. Phys. B 429, 277–290 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  30. Manin, Y.I., Radul, A.O.: A supersymmetric extension of the Kadomtsev–Petviashvili hierarchy. Commun. Math. Phys. 98, 65–77 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  31. McArthur, I.N.: On the integrability of the super-KdV equation. Commun. Math. Phys. 148, 177–188 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  32. Oevel, W., Popowicz, Z.: The bi-Hamiltonian structure of fully supersymmetric Korteweg–de Vries systems. Commun. Math. Phys. 139(3), 441–460 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  33. Pedroni, M.: Marco equivalence of the Drinfel’d–Sokolov reduction to a bi-Hamiltonian reduction. Lett. Math. Phys. 35(4), 291–302 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  34. Suh, U.R.: Classical affine W-superalgebras via generalized Drinfeld–Sokolov reductions and related integrable systems. Commun. Math. Phys. 358, 199–236 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  35. Suh, U.R.: Structures of (supersymmetric) classical W-algebras. J. Math. Phys. 61(11), 111701 (2020)

  36. Wu, C.-Z.: Tau functions and Virasoro symmetries for Drinfeld–Sokolov hierarchies. Adv. Math. 306, 603–652 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported by a Junior Fellow award from the Simons Foundation. He is extremely grateful to the Seoul National University for its hospitality during a short visit in the fall of 2019, where most of the work for this paper was accomplished. The second author was supported by the New Faculty Startup Fund from Seoul National University and the NRF Grant # NRF-2019R1F1A1059363.

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University in the City of New York, Broadway and 116th, New York, NY, 10027, USA

    Sylvain Carpentier

  2. Department of Mathematical Sciences and Research institute of Mathematics, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul, 08826, Korea

    Uhi Rinn Suh

Authors
  1. Sylvain Carpentier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uhi Rinn Suh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uhi Rinn Suh.

Additional information

Communicated by Y. Kawahigashi

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carpentier, S., Suh, U.R. Supersymmetric Bi-Hamiltonian Systems. Commun. Math. Phys. 382, 317–350 (2021). https://doi.org/10.1007/s00220-021-03974-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-03974-7

Search

Navigation