Skip to main content
SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Selecta Mathematica
  3. Article

Quantum character varieties and braided module categories

  • Open Access
  • Published: 26 July 2018
  • volume 24, pages 4711–4748 (2018)
Download PDF

You have full access to this open access article

Selecta Mathematica Aims and scope Submit manuscript
Quantum character varieties and braided module categories
Download PDF
  • David Ben-Zvi1,
  • Adrien Brochier2 &
  • David Jordan3 
  • 745 Accesses

  • 24 Citations

  • Explore all metrics

  • Cite this article

Abstract

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants \(\int _S{\mathcal {A}}\) of a surface S, determined by the choice of a braided tensor category \({\mathcal {A}}\), and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for \({\mathcal {A}}\), and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided \({\mathcal {A}}\)-modules are objects of the torus category \(\int _{T^2}{\mathcal {A}}\). We initiate a theory of character sheaves for quantum groups by identifying the torus integral of \({\mathcal {A}}={\text {Rep}}_{q}G\) with the category \({\mathcal {D}}_q(G/G)\)-mod of equivariant quantum \({\mathcal {D}}\)-modules. When \(G=GL_n\), we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra \({\mathbb {SH}}_{q,t}\).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ayala, D., Francis, J.: Factorization homology of topological manifolds. J. Topol. 8(4), 1045–1084 (2015)

    Article  MathSciNet  Google Scholar 

  2. Ayala, D., Francis, J., Tanaka, H.: Factorization homology of stratified spaces. Selecta Mathematica 23(1), 293–362 (2017)

    Article  MathSciNet  Google Scholar 

  3. Alekseev, A., Malkin, A., Meinrenken, E.: Lie group valued moment maps. J. Differ. Geom. 48(3), 445–495 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bellamy, G., Ginzburg, V.: Hamiltonian reduction and nearby cycles for mirabolic D-modules. Adv. Math. 269, 71–161 (2015)

    Article  MathSciNet  Google Scholar 

  5. Brochier, A., Jordan, D.: Fourier transform for quantum \(D\)-modules via the punctured torus mapping class group. Quantum Topol. 8(2), 361–379 (2017)

    Article  MathSciNet  Google Scholar 

  6. Balagovic, M., Jordan, D.: The quantum Harish-Chandra isomorphism for \({GL}_2\). arXiv:1603.09218 (2016)

  7. Brochier, A.: A Kohno–Drinfeld theorem for the monodromy of cyclotomic KZ connections. Commun. Math. Phys. 311, 55–96 (2012). https://doi.org/10.1007/s00220-012-1424-0

    Article  MathSciNet  MATH  Google Scholar 

  8. Brochier, A.: Cyclotomic associators and finite type invariants for tangles in the solid torus. Algebr. Geom. Topol. 13, 3365–3409 (2013)

    Article  MathSciNet  Google Scholar 

  9. Bruguières, A., Virelizier, A.: The double of a Hopf monad. arXiv preprint arXiv:0812.2443 (2008)

  10. Ben-Zvi, D., Brochier, A., Jordan, D.: Integrating quantum groups over surfaces. J. Topology (2015) arXiv:1501.04652 (To appear)

  11. Ben-Zvi, D., Nadler, D.: The character theory of a complex group. arXiv preprint arXiv:0904.1247 (2009)

  12. Ben-Zvi, D., Nadler, D.: Loop spaces and representations. Duke Math. J. 162(9), 1587–1619 (2013)

    Article  MathSciNet  Google Scholar 

  13. Ben-Zvi, D., Nadler, D.: Betti geometric Langlands. In: Proceedings of Symposia in Pure Mathematics, vol. 97.2, pp. 3–41. AMS (2018)

  14. Cherednik, I.: DAHA and Verlinde algebras. In: Quantum Theory and Symmetries, pp. 53–64. World Scientific Publishing, Hackensack, NJ (2004)

  15. Cherednik, I.: Double Affine Hecke Algebras. London Mathematical Society Lecture Note Series, vol. 319. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  16. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  17. Donin, J., Kulish, P.P., Mudrov, A.I.: On a universal solution to the reflection equation. Lett. Math. Phys. 63(3), 179–194 (2003)

    Article  MathSciNet  Google Scholar 

  18. Donin, J., Mudrov, A.: Explicit equivariant quantization on coadjoint orbits of \({GL}(n, {\mathbb{C}})\). Lett. Math. Phys. 62(1), 17–32 (2002)

    Article  MathSciNet  Google Scholar 

  19. Donin, J., Mudrov, A.: Quantum coadjoint orbits of \({GL}(n)\) and generalized Verma modules. Lett. Math. Phys. 67(3), 167–184 (2004)

    Article  MathSciNet  Google Scholar 

  20. Donin, J., Mudrov, A.: Quantum groupoids and dynamical categories. J. Algebra 296(2), 348–384 (2006)

    Article  MathSciNet  Google Scholar 

  21. Douglas, C.L., Schommer-Pries, C., Snyder, N.: Dualizable tensor categories. arXiv:1312.7188 (2013)

  22. Etingof, P., Geer, N.: Monodromy of trigonometric KZ equations. Int. Math. Res. Not. IMRN 24, 15 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Enriquez, B.: Quasi-reflection algebras and cyclotomic associators. Sel. Math. New Ser. 13, 391–463 (2008). https://doi.org/10.1007/s00029-007-0048-2

    Article  MathSciNet  MATH  Google Scholar 

  24. Frohman, C., Gelca, R.: Skein modules and the noncommutative torus. Trans. Am. Math. Soc. 352(10), 4877–4888 (2000)

    Article  MathSciNet  Google Scholar 

  25. Finkelberg, M., Ginzburg, V.: On mirabolic D-modules. Int. Math. Res. Not. 2010(15), 2947–2986 (2010)

    Article  MathSciNet  Google Scholar 

  26. Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted \({K}\)-theory III. Ann. Math. 174(2), 947–1007 (2011)

    Article  MathSciNet  Google Scholar 

  27. Fehér, L., Klimčík, C.: Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction. Nucl. Phys. B 860(3), 464–515 (2012)

    Article  MathSciNet  Google Scholar 

  28. Fehér, L., Klimčík, C.: The Ruijsenaars Self-Duality Map as a Mapping Class Symplectomorphism, pp. 423–437. Springer, Tokyo (2013)

    MATH  Google Scholar 

  29. Fehér, L., Kluck, T.: New compact forms of the trigonometric Ruijsenaars–Schneider system. Nucl. Phys. B 882, 97–127 (2014)

    Article  MathSciNet  Google Scholar 

  30. Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and the \(r\)-matrix. In: Moscow Seminar in Mathematical Physics, vol. 191 of Am. Math. Soc. Transl. Ser. 2, pp. 67–86. Am. Math. Soc., Providence, RI (1999)

  31. Francis, J.: The tangent complex and Hochschild cohomology of \({E}_n\)-rings. Compos. Math. 149(3), 430–480 (2013)

    Article  MathSciNet  Google Scholar 

  32. Fuchs, J., Schaumann, G., Schweigert, S.: A trace for bimodule categories. arXiv:1412.6968 (2014)

  33. Gaitsgory, D.: Quantum Langlands correspondence. arXiv preprint arXiv:1601.05279 (2016)

  34. Gan, W.L., Ginzburg, V.: Almost-commuting variety, D-modules, and Cherednik algebras. Int. Math. Res. Pap. (2006)

  35. Ginot, G.: Notes on factorization algebras, factorization homology and applications. In: Mathematical aspects of quantum field theories, pp. 429–552. Springer (2015)

  36. Gorsky, A., Nekrasov, N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B 436(3), 582–608 (1995)

    Article  MathSciNet  Google Scholar 

  37. Gunningham, S.: A generalized Springer decomposition for D-modules on a reductive Lie algebra. arXiv preprint arXiv:1510.02452 (2015)

  38. Jordan, D.: Quantized multiplicative quiver varieties. Adv. Math. 250, 420–466 (2014)

    Article  MathSciNet  Google Scholar 

  39. Kolb, S., Stokman, J.: Reflection equation algebras, coideal subalgebras, and their centres. Sel. Math. 15(4), 621–664 (2009)

    Article  MathSciNet  Google Scholar 

  40. Lurie, J.: Higher algebra. http://www.math.harvard.edu/~lurie/

  41. Lurie, J.: On the classification of topological field theories. Curr. Dev. Math. 2008, 129–280 (2009)

    Article  MathSciNet  Google Scholar 

  42. Nevins, T.: Mirabolic Langlands duality and the quantum Calogero–Moser system. Transform. Groups 14(4), 931–983 (2009)

    Article  MathSciNet  Google Scholar 

  43. Oblomkov, A.: Double affine Hecke algebras and Calogero–Moser spaces. Represent. Theory Am. Math. Soc. 8(10), 243–266 (2004)

    Article  MathSciNet  Google Scholar 

  44. Ruijsenaars, S., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. 170(2), 370–405 (1986)

    Article  MathSciNet  Google Scholar 

  45. Voronov, A.A.: The Swiss-cheese operad. In: Homotopy invariant algebraic structures (Baltimore, MD, 1998), vol. 239 of Contemp. Math., pp. 365–373. Am. Math. Soc., Providence, RI (1999)

  46. Varagnolo, M., Vasserot, E.: Double affine Hecke algebras at roots of unity. Represent. Theory 14, 510–600 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas, Austin, TX, 78712-0257, USA

    David Ben-Zvi

  2. MPIM, Bonn, Germany

    Adrien Brochier

  3. School of Mathematics, University of Edinburgh, Edinburgh, UK

    David Jordan

Authors
  1. David Ben-Zvi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adrien Brochier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Jordan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Jordan.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben-Zvi, D., Brochier, A. & Jordan, D. Quantum character varieties and braided module categories. Sel. Math. New Ser. 24, 4711–4748 (2018). https://doi.org/10.1007/s00029-018-0426-y

Download citation

  • Published: 26 July 2018

  • Issue Date: November 2018

  • DOI: https://doi.org/10.1007/s00029-018-0426-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification

  • 17B37
  • 16T99

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature