Quiver W-algebras

Article

Abstract

For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds–Kac–Moody Lie algebras, their quantum affinizations and associated W-algebras.

Keywords

Supersymmetric gauge theories Conformal field theories W-algebras Quantum groups Quiver instanton 

Mathematics Subject Classification

81T60 81R10 14D21 81R50 

Notes

Acknowledgements

We thank for discussions and comments Alexey Sevastyanov, Edward Frenkel, Nikita Nekrasov and Samson Shatashvili. TK is grateful to Institut des Hautes Études Scientifiques for hospitality where a part of this work has been done. The work of TK was supported in part by Keio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (No. JP17K18090), the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (No. S1511006), JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855), and “Discrete Geometric Analysis for Materials Design” (No. JP17H06462). VP acknowledges Grant RFBR 15-01-04217 and RFBR 16-02-01021. The research of VP on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT Grant Agreement 677368).

References

  1. 1.
    Frenkel, E., Reshetikhin, N.: Deformations of  \({\cal{W}} \)-algebras associated to simple Lie algebras. Commun. Math. Phys. 197, 1–32 (1998). arXiv:q-alg/9708006 [math.QA]MathSciNetMATHGoogle Scholar
  2. 2.
    Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum Geometry and Guiver Gauge Theories. Commun. Math. Phys. 357, 519–567 (2018).  https://doi.org/10.1007/s00220-017-3071-y CrossRefGoogle Scholar
  3. 3.
    Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters. JHEP 1603, 181 (2016). arXiv:1512.05388 [hep-th]ADSCrossRefGoogle Scholar
  4. 4.
    Marshakov, A., Nekrasov, N.: Extended Seiberg-Witten theory and integrable hierarchy. JHEP 0701, 104 (2007). arXiv:hep-th/0612019 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996). arXiv:q-alg/9507034 ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and \({\mathscr {W}}\)-algebras. Commun. Math. Phys. 178, 237–264 (1996). arXiv:q-alg/9505025 ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Nekrasov, N., Pestun, V.: Seiberg-Witten geometry of four dimensional \(N=2\) quiver gauge theories. arXiv:1211.2240 [hep-th]
  8. 8.
    Feigin, B., Frenkel, E.: Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras. Int. J. Mod. Phys. A 7, 197–215 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Frenkel, E., Hernandez, D.: Langlands duality for finite-dimensional representations of quantum affine algebras. Lett. Math. Phys. 96, 217–261 (2011). arXiv:0902.0447 [math.QA]ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Beilinson, A., Drinfeld, V.: Quantization of Hitchins integrable system and Hecke eigensheaves. http://www.math.uchicago.edu/~mitya/langlands/QuantizationHitchin.pdf
  11. 11.
    Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric langlands program. Commun. Number Theor. Phys. 1, 1–236 (2007). arXiv:hep-th/0604151 [hep-th]MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092 (2010). arXiv:1002.0888 ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Frenkel, E.: Lectures on the Langlands program and conformal field theory. In: Frontiers in Number Theory, Physics, and Geometry II, pp. 387–533. Springer (2007). arXiv:hep-th/0512172 [hep-th]
  14. 14.
    Hurtubise, J.C., Markman, E.: Elliptic Sklyanin integrable systems for arbitrary reductive groups. Adv. Theor. Math. Phys. 6, 873–978 (2002). arXiv:math/0203031 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantum affine algebras and deformations of \(\cal{W}\)-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics, vol. 248 of Contemp. Math., pp. 163–205. Amer. Math. Soc. (1999). arXiv:math/9810055 [math.QA]
  16. 16.
    Frenkel, E., Hernandez, D.: Baxters relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015). arXiv:1308.3444 [math.QA]MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14, 145–238 (2001). arXiv:math/9912158 ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101, 583–591 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4, 365–421 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ginzburg, V., Vasserot, É.: Langlands reciprocity for affine quantum groups of type \(A_n\). Int. Math. Res. Not. 3, 67–85 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nakajima, H.: Quiver varieties and \(t\)-analogs of \(q\)-characters of quantum affine algebras. Ann. Math. 160, 1057–1097 (2004). arXiv:math/0105173 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Commun. Num. Theor. Phys. 5, 231–352 (2011). arXiv:1006.2706 [math.AG]MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hou, B.-Y., Yang, W.-L.: A \(\hbar \)-deformation of the \(W_N\) algebra and its vertex operators. J. Phys. A 30, 6131–6145 (1997). arXiv:hep-th/9701101 ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Iqbal, A., Kozçaz, C., Yau, S.-T.: Elliptic Virasoro conformal blocks. arXiv:1511.00458 [hep-th]
  25. 25.
    Nieri, F.: An elliptic Virasoro symmetry in 6d. Lett. Math. Phys. 107, 2147–2187 (2017). arXiv:1511.00574 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Witten, E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B 500, 3–42 (1997). arXiv:hep-th/9703166 ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wyllard, N.: \(A_{N-1}\) conformal Toda field theory correlation functions from conformal \({\cal{N}} = 2\) \(SU(N)\) quiver gauge theories. JHEP 0911, 002 (2009). arXiv:0907.2189 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Braverman, A., Finkelberg, M., Nakajima, H.: Instanton moduli spaces and \(\mathscr {W}\)-algebras. arXiv:1406.2381 [math.QA]
  30. 30.
    Iqbal, A., Kozçaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). arXiv:hep-th/0701156 ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Awata, H., Kanno, H.: Refined BPS state counting from Nekrasov’s formula and Macdonald functions. Int. J. Mod. Phys. A 24, 2253–2306 (2009). arXiv:0805.0191 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Nekrasov, N., Okounkov, A.: Membranes and sheaves. arXiv:1404.2323 [math.AG]
  33. 33.
    Bao, L., Pomoni, E., Taki, M., Yagi, F.: M5-branes, toric diagrams and gauge theory duality. JHEP 1204, 105 (2012). arXiv:1112.5228 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and triality). arXiv:1506.04183 [hep-th]
  35. 35.
    Katz, S., Mayr, P., Vafa, C.: Mirror symmetry and exact solution of 4-D \(N=2\) gauge theories: 1. Adv. Theor. Math. Phys. 1, 53–114 (1998). arXiv:hep-th/9706110 MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Eynard, B.: All orders asymptotic expansion of large partitions. J. Stat. Mech. 0807, P07023 (2008). arXiv:0804.0381 [math-ph]MathSciNetGoogle Scholar
  37. 37.
    Sułkowski, P.: Matrix models for \(\beta \)-ensembles from Nekrasov partition functions. JHEP 1004, 063 (2010). arXiv:0912.5476 ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Nedelin, A., Zabzine, M.: \(q\)-Virasoro constraints in matrix models. JHEP 1703, 98 (2017). arXiv:1511.03471 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Mironov, A., Morozov, A., Zenkevich, Y.: On elementary proof of AGT relations from six dimensions. Phys. Lett. B 756, 208–211 (2016). arXiv:1512.06701 [hep-th]ADSCrossRefGoogle Scholar
  40. 40.
    Kostov, I.K.: Gauge invariant matrix model for the A-D-E closed strings. Phys. Lett. B 297, 74–81 (1992). arXiv:hep-th/9208053 ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Bourgine, J.-E., Mastuo, Y., Zhang, H.: Holomorphic field realization of SH\(^c\) and quantum geometry of quiver gauge theories. JHEP 1604, 167 (2016). arXiv:1512.02492 [hep-th]ADSGoogle Scholar
  42. 42.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76, 365–416 (1994)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ginzburg, V., Kapranov, M., Vasserot, É.: Langlands reciprocity for algebraic surfaces. Math. Res. Lett. 2, 147–160 (1995). arXiv:q-alg/9502013 MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316, 565–576 (2000). arXiv:math/9812016 [math.AG]MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of \({\mathbb{A}}^2\). Duke Math. J. 162, 279–366 (2013). arXiv:0905.2555 [math.QA]MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Varagnolo, M., Vasserot, E.: On the \(K\)-theory of the cyclic quiver variety. Int. Math. Res. Not. 1005–1028 (1999). arXiv:math/9902091
  47. 47.
    Schiffmann, O., Vasserot, E.: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on \({\mathbb{A}}^2\). Publ. Math. IHES 118, 213–342 (2013). arXiv:1202.2756 [math.QA]CrossRefMATHGoogle Scholar
  48. 48.
    Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. arXiv:1211.1287 [math.AG]
  49. 49.
    Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. arXiv:hep-th/9603167
  50. 50.
    Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161 MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Moore, G.W., Nekrasov, N., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000). arXiv:hep-th/9712241 ADSMathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces, vol. 18 of University Lecture Series. AMS (1999)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Keio UniversityTokyoJapan
  2. 2.IHESBures-sur-YvetteFrance

Personalised recommendations