Skip to main content
SpringerLink
Log in

Quiver varieties and the quantum Knizhnik–Zamolodchikov equation

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians and how these elements satisfy the rational level-one quantum Knizhnik–Zamolodchikov equation in some cases.

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. V. Ginzburg, M. Kapranov, and E. Vasserot, “Elliptic algebras and equivariant elliptic cohomology,” arXiv:qalg/9505012v1 (1995).

    Google Scholar 

  2. V. Ginzburg, M. Kapranov, and E. Vasserot, Adv. Math., 214, 40–77 (2007); arXiv:math/0503224v3 (2005).

    Article  MathSciNet  Google Scholar 

  3. P. Di Francesco and P. Zinn-Justin, J. Phys. A, 38, L815–L822 (2005); arXiv:math-ph/0508059v3 (2005).

    Article  MATH  Google Scholar 

  4. D. Maulik and A. Okounkov, “Quantum groups and quantum cohomology,” arXiv:1211.1287v1 [math.AG] (2012).

    Google Scholar 

  5. R. Rimányi, V. Tarasov, A. Varchenko, and P. Zinn-Justin, J. Geom. Phys., 62, 2188–2207 (2012); arXiv: 1110.2187v2 [math-ph] (2011).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. H. Nakajima, Duke Math. J., 76, 365–416 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Nakajima, Duke Math. J., 91, 515–560 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Nakajima, J. Amer. Math. Soc., 14, 145–238 (2001); arXiv:math/9912158v1 (1999).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. H. Nakajima, Invent. Math., 146, 399–449 (2001); arXiv:math/0103008v2 (2001).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. V. Ginzburg, “Lectures on Nakajima’s quiver varieties,” in: Geometric Methods in Representation Theory I (Séminairés et Congrès, Vol. 24, M. Brion, ed.), Société Mathématique de France, Paris (2012), pp. 145–219; arXiv:0905.0686v2 [math.RT] (2009).

    MathSciNet  Google Scholar 

  11. I. Mirkovic and M. Vybornov, “Quiver varieties and Beilinson–Drinfeld Grassmannians of type A,” arXiv: 0712.4160v2 [math.AG] (2007).

    Google Scholar 

  12. N. Spaltenstein, Classes unipotentes et sous-groupes de Borel (Lect. Notes Math., Vol. 946), Springer, Berlin (1982).

  13. S. Khoroshkin, “Central extension of the Yangian double,” in: Algèbre non commutative, groupes quantiques et invariants (Séminairés et Congrès, Vol. 2, J. Alev and G. Cauchon, eds.), Société Mathématique de France, Paris (1997), pp. 119–135; arXiv:q-alg/9602031v1 (1996).

    MathSciNet  Google Scholar 

  14. S. Khoroshkin, D. Lebedev, and S. Pakuliak, Phys. Lett. A, 222, 381–392 (1996); arXiv: q-alg/9602030v3 (1996).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. S. Khoroshkin, D. Lebedev, and S. Pakuliak, Lett. Math. Phys., 41, 31–47 (1997); arXiv:q-alg/9605039v1 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Nakayashiki, Commun. Math. Phys., 212, 29–61 (2000); arXiv:math/9902139v1 (1999).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. V. Tarasov and A. Varchenko, Invent. Math., 128, 501–588 (1997).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. I. Frenkel and N. Reshetikhin, Commun. Math. Phys., 146, 1–60 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. M. Idzumi, T. Tokihiro, K. Iohara, M. Jimbo, T. Miwa, and T. Nakashima, Internat. J. Mod. Phys. A, 8, 1479–1511 (1993); arXiv:hep-th/9208066v1 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  20. A. Ponsaing and P. Zinn-Justin, “Type Ĉ brauer loop schemes and loop model with boundaries,” arXiv: 1410.0262v2 [math-ph] (2014).

    Google Scholar 

  21. I. Cherednik, Commun. Math. Phys., 150, 109–136 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. D. Grayson and M. Stillman, “Macaulay2,” A software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/ (2015).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Énergies, CNRS UMR, Université Pierre et Marie Curie (Paris 6), Paris, France

    P. Zinn-Justin

  2. Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

    P. Zinn-Justin

Authors
  1. P. Zinn-Justin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Zinn-Justin.

Additional information

This research was supported by the ERC (Grant No. 278124 “LIC”) and ARC (Grant No. DP140102201).

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 3, pp. 438–459, December, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zinn-Justin, P. Quiver varieties and the quantum Knizhnik–Zamolodchikov equation. Theor Math Phys 185, 1741–1758 (2015). https://doi.org/10.1007/s11232-015-0376-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-015-0376-x

Keywords

Search

Navigation