Advertisement

Letters in Mathematical Physics

, Volume 108, Issue 6, pp 1383–1405 | Cite as

Quiver elliptic W-algebras

  • Taro Kimura
  • Vasily PestunEmail author
Article

Abstract

We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction (Kimura and Pestun in Quiver W-algebras, 2015. arXiv:1512.08533 [hep-th]) with six-dimensional gauge theory.

Keywords

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

Mathematics Subject Classification

81T60 81R10 14D21 81R50 

Notes

Acknowledgements

The work of T.K. 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.
    Kimura, T., Pestun, V.: Quiver W-algebras (2015). arXiv:1512.08533 [hep-th]
  2. 2.
    Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \(N=2\) supersymmetric QCD. Nucl. Phys. B431, 484–550 (1994). arXiv:hep-th/9408099 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang–Mills theory. Nucl. Phys. B426, 19–52 (1994). arXiv:hep-th/9407087 [hep-th]. [Erratum: Nucl. Phys. B430, 485 (1994)]
  4. 4.
    Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B355, 466–474 (1995). arXiv:hep-th/9505035 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Martinec, E.J., Warner, N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B459, 97–112 (1996). arXiv:hep-th/9509161 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Donagi, R., Witten, E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. B460, 299–334 (1996). arXiv:hep-th/9510101 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Seiberg, N., Witten, E.: Gauge dynamics and compactification to three-dimensions. In: The mathematical beauty of physics: a memorial volume for Claude Itzykson, vol. 24 of Advanced Series in Mathematical Physics, pp. 333–366. World Scientific (1997). arXiv:hep-th/9607163 [hep-th]
  8. 8.
    Gorsky, A., Marshakov, A., Mironov, A., Morozov, A.: \(\cal{N}=2\) supersymmetric QCD and integrable spin chains: Rational case \(N_F < 2 N_c\). Phys. Lett. B380, 75–80 (1996). arXiv:hep-th/9603140 [hep-th]ADSCrossRefGoogle Scholar
  9. 9.
    Nekrasov, N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B531, 323–344 (1998). arXiv:hep-th/9609219 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nekrasov, N., Pestun, V.: Seiberg–Witten geometry of four dimensional \(N=2\) quiver gauge theories (2012). arXiv:1211.2240 [hep-th]
  11. 11.
    Moore, G.W., Nekrasov, N., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000). arXiv:hep-th/9712241 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nekrasov, N., Shatashvili, S.: Bethe Ansatz and supersymmetric vacua. AIP Conf. Proc. 1134, 154–169 (2009)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. In: XVIth International Congress on Mathematical Physics, pp. 265–289 (2009). arXiv:0908.4052 [hep-th]
  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 Contemporary Mathematics, pp. 163–205. American Mathematical Society (1999). arXiv:math/9810055 [math.QA]
  16. 16.
    Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum Geometry and Quiver Gauge Theories. Commun. Math. Phys. 357, 519–567 (2018).  https://doi.org/10.1007/s00220-017-3071-y ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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]MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and \(qq\)-characters. JHEP 1603, 181 (2016). arXiv:1512.05388 [hep-th]ADSCrossRefGoogle Scholar
  19. 19.
    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]MathSciNetzbMATHGoogle Scholar
  20. 20.
    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 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kim, H.-C.: Line defects and 5d instanton partition functions. JHEP 1603, 199 (2016). arXiv:1601.06841 [hep-th]ADSCrossRefGoogle Scholar
  23. 23.
    Awata, H., Yamada, Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 1001, 125 (2010). arXiv:0910.4431 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005). arXiv:hep-th/0305132 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory, I. Compos. Math. 142, 1263–1285 (2006). arXiv:math/0312059 [math.AG]MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory, II. Compos. Math. 142, 1286–1304 (2006). arXiv:math/0406092 [math.AG]MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Iqbal, A., Kozçaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). arXiv:hep-th/0701156 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Mironov, A., Morozov, A., Runov, B., Zenkevich, Y., Zotov, A.: Spectral dualities in XXZ spin chains and five dimensional gauge theories. JHEP 1312, 034 (2013). arXiv:1307.1502 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Iqbal, A., Kozçaz, C., Yau, S.-T.: Elliptic Virasoro conformal blocks (2015). arXiv:1511.00458 [hep-th]
  32. 32.
    Nieri, F.: An elliptic Virasoro symmetry in 6d. Lett. Math. Phys. 107, 2147–2187 (2017). arXiv:1511.00574 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mironov, A., Morozov, A., Zenkevich, Y.: On elementary proof of AGT relations from six dimensions. Phys. Lett. B756, 208–211 (2016). arXiv:1512.06701 [hep-th]ADSCrossRefGoogle Scholar
  34. 34.
    Mironov, A., Morozov, A., Zenkevich, Y.: Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings. JHEP 1605, 121 (2016). arXiv:1603.00304 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Mironov, A., Morozov, A., Zenkevich, Y.: Ding-Iohara-Miki symmetry of network matrix models. Phys. Lett. B762, 196–208 (2016). arXiv:1603.05467 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Awata, H., Kanno, H., Matsumoto, T., Mironov, A., Morozov, A., Morozov, A., Ohkubo, Y., Zenkevich, Y.: Explicit examples of DIM constraints for network matrix models. JHEP 1607, 103 (2016). arXiv:1604.08366 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tan, M.-C.: An M-theoretic derivation of a 5d and 6d AGT correspondence, and relativistic and elliptized integrable systems. JHEP 1312, 031 (2013). arXiv:1309.4775 [hep-th]ADSCrossRefGoogle Scholar
  38. 38.
    Koroteev, P., Sciarappa, A.: Quantum hydrodynamics from large-\(n\) supersymmetric gauge theories. Lett. Math. Phys. (2017). arXiv:1510.00972 [hep-th]
  39. 39.
    Koroteev, P., Sciarappa, A.: On Elliptic Algebras and Large-\(n\) Supersymmetric Gauge Theories. J. Math. Phys. 57, 112302 (2016). arXiv:1601.08238 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Tan, M.-C.: Higher AGT correspondences, W-algebras, and higher quantum geometric langlands duality from M-theory (2016). arXiv:1607.08330 [hep-th]
  41. 41.
    Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). arXiv:hep-th/0310272 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Haghighat, B., Iqbal, A., Kozçaz, C., Lockhart, G., Vafa, C.: M-strings. Commun. Math. Phys. 334, 779–842 (2015). arXiv:1305.6322 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Benini, F., Eager, R., Hori, K., Tachikawa, Y.: Elliptic genera of 2d \({\cal{N}}\) = 2 gauge theories. Commun. Math. Phys. 333, 1241–1286 (2015). arXiv:1308.4896 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Aganagic, M., Okounkov, A.: Elliptic stable envelope (2016). arXiv:1604.00423 [math.AG]
  45. 45.
    Saito, Y.: Elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator. Pub. Res. Inst. Math. Sci. 50, 411–455 (2014). arXiv:1301.4912 [math.QA]MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Gadde, A., Gukov, S.: 2d index and surface operators. JHEP 1403, 080 (2014). arXiv:1305.0266 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Marshakov, A., Nekrasov, N.: Extended Seiberg–Witten theory and integrable hierarchy. JHEP 0701, 104 (2007). arXiv:hep-th/0612019 ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Clavelli, L., Shapiro, J.A.: Pomeron factorization in general dual models. Nucl. Phys. B57, 490–535 (1973)ADSCrossRefGoogle Scholar
  49. 49.
    Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and triality) (2015). arXiv:1506.04183 [hep-th]
  50. 50.
    Aganagic, M., Haouzi, N., Kozçaz, C., Shakirov, S.: Gauge/Liouville triality (2013). arXiv:1309.1687 [hep-th]
  51. 51.
    Aganagic, M., Haouzi, N., Shakirov, S.: \(A_n\)-triality (2014). arXiv:1403.3657 [hep-th]
  52. 52.
    Feigin, B., Frenkel, E.: Quantum \(\mathscr {W}\)-algebras and elliptic algebras. Commun. Math. Phys. 178, 653–678 (1996). arXiv:q-alg/9508009 [q-alg]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Farghly, R.M., Konno, H., Oshima, K.: Elliptic algebra \(U_{q,p}(\widehat{\mathfrak{g}})\) and quantum \(Z\)-algebras. Algebras Represent. Theory 18, 103–135 (2015). arXiv:1404.1738 [math.QA]MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    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
  55. 55.
    Bourgine, J.-E., Fukuda, M., Matsuo, Y., Zhang, H., Zhu, R.-D.: Coherent states in quantum \(\cal{W}_{1+\infty }\) algebra and qq-character for 5d super Yang-Mills. PTEP 2016, 123B05 (2016). arXiv:1606.08020 [hep-th]zbMATHGoogle Scholar
  56. 56.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  57. 57.
    Yamada, Y.: Introduction to Conformal Field Theory. Baifukan, Tokyo (2006). (in Japanese)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