Abstract
Inspired by 5d supersymmetric Yang–Mills theories placed on the compact space \({\mathbb{S}^5}\), we propose an intriguing algebraic construction for the q-Virasoro algebra. We show that, when multiple q-Virasoro “chiral” sectors have to be fused together, a natural \({\mathrm{SL}(3,\mathbb{Z})}\) structure arises. This construction, which we call the modular triple, is consistent with the observed triple factorization properties of supersymmetric partition functions derived from localization arguments. We also give a 2d CFT-like construction of the modular triple, and conjecture for the first time a (non-local) Lagrangian formulation for a q-Virasoro model, resembling ordinary Liouville theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Drummond J.M., Henn J.M., Plefka J.: Yangian symmetry of scattering amplitudes in N = 4 super Yang–Mills theory. JHEP 05, 046 (2009) arXiv:0902.2987 [hep-th]
Gaiotto D.: N = 2 dualities. JHEP 08, 034 (2012) arXiv:0904.2715 [hep-th]
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]
Beisert N. et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3–32 (2012) arXiv:1012.3982 [hep-th]
Pestun V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129 (2012) arXiv:0712.2824 [hep-th]
Nekrasov N.: BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters. JHEP 03, 181 (2016) arXiv:1512.05388 [hep-th]
Nekrasov, N., Pestun, V.: Seiberg–Witten geometry of four dimensional N = 2 quiver gauge theories arXiv:1211.2240 [hep-th]
Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories arXiv:1312.6689 [hep-th]
Lossev, A., Nekrasov, N., Shatashvili, S.L.: Testing Seiberg–Witten solution. In: Strings, Branes and Dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26–June 14, 1997, pp. 359–372 arXiv:hep-th/9801061 [hep-th] (1997)
Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000) arXiv:hep-th/9712241 [hep-th]
Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B 534, 549–611 (1998) arXiv:hep-th/9711108 [hep-th]
Moore G.W., Nekrasov N., Shatashvili S.: D particle bound states and generalized instantons. Commun. Math. Phys. 209, 77–95 (2000) arXiv:hep-th/9803265 [hep-th]
Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003) arXiv:hep-th/0206161 [hep-th]
Nekrasov N., Okounkov A.: Seiberg–Witten theory and random partitions. Prog. Math. 244, 525–596 (2006) arXiv:hep-th/0306238 [hep-th]
Awata H., Yamada Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 01, 125 (2010) arXiv:0910.4431 [hep-th]
Awata H., Yamada Y.: Five-dimensional AGT relation and the deformed beta-ensemble. Prog. Theor. Phys. 124, 227–262 (2010) arXiv:1004.5122 [hep-th]
Awata, H., Feigin, B., Hoshino, A., Kanai, M., Shiraishi, J., Yanagida, S.: Notes on Ding–Iohara algebra and AGT conjecture arXiv:1106.4088 [math-ph]
Mironov A., Morozov A., Shakirov S., Smirnov A.: Proving AGT conjecture as HS duality: extension to five dimensions. Nucl. Phys. B 855, 128–151 (2012) arXiv:1105.0948 [hep-th]
Carlsson, E., Nekrasov, N., Okounkov, A.: Five dimensional gauge theories and vertex operators arXiv:1308.2465 [math.RT]
Aganagic, M., Haouzi, N., Kozcaz, C., Shakirov, S.: Gauge/Liouville triality arXiv:1309.1687 [hep-th]
Aganagic, M., Haouzi, N., Shakirov, S.: A n-Triality arXiv:1403.3657 [hep-th]
Aganagic, M., Shakirov, S.: Gauge/Vortex duality and AGT. In: Teschner, J. (ed.) New Dualities of Supersymmetric Gauge Theories, pp. 419–448 arXiv:1412.7132 [hep-th] (2016)
Kimura, T., Pestun, V.: Quiver W-algebras arXiv:1512.08533 [hep-th]
Mironov, A., Morozov, A., Zenkevich, Y.: Ding–Iohara–Miki symmetry of network matrix models arXiv:1603.05467 [hep-th]
Bourgine J.-E., Fukuda M., Matsuo Y., Zhang H., Zhu R.-D.: Coherent states in quantum \({\mathcal{W}_{1+\infty}}\) algebra and qq-character for 5d Super Yang–Mills. PTEP 2016(12), 123B05 (2016) arXiv:1606.08020 [hep-th]
Bourgine, J.-E., Fukuda, M., Harada, K., Matsuo, Y., Zhu, R.-D.: (p,q)-Webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters arXiv:1703.10759 [hep-th]
Nieri F., Pasquetti S., Passerini F.: 3d and 5d gauge theory partition functions as q-deformed CFT correlators. Lett. Math. Phys. 105(1), 109–148 (2015) arXiv:1303.2626 [hep-th]
Nieri F., Pasquetti S., Passerini F., Torrielli A.: 5D partition functions, q-Virasoro systems and integrable spin-chains. JHEP 12, 040 (2014) arXiv:1312.1294 [hep-th]
Bao L., Mitev V., Pomoni E., Taki M., Yagi F.: Non-Lagrangian theories from Brane junctions. JHEP 01, 175 (2014) arXiv:1310.3841 [hep-th]
Mitev V., Pomoni E.: Toda 3-point functions from topological strings. JHEP 06, 049 (2015) arXiv:1409.6313 [hep-th]
Cordova, C., Jafferis, D.L.: Toda theory from six dimensions arXiv:1605.03997 [hep-th]
Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and triality) arXiv:1506.04183 [hep-th]
Lockhart, G., Vafa, C.: Superconformal partition functions and non-perturbative topological strings arXiv:1210.5909 [hep-th]
Källén J., Zabzine M.: Twisted supersymmetric 5D Yang–Mills theory and contact geometry. JHEP 05, 125 (2012) arXiv:1202.1956 [hep-th]
Källén J., Qiu J., Zabzine M.: The perturbative partition function of supersymmetric 5D Yang–Mills theory with matter on the five-sphere. JHEP 08, 157 (2012) arXiv:1206.6008 [hep-th]
Qiu J., Zabzine M.: Factorization of 5D super Yang–Mills theory on Y p,q spaces. Phys. Rev. D 89(6), 065040 (2014) arXiv:1312.3475 [hep-th]
Kim, H.-C., Kim, J., Kim, S.: Instantons on the 5-sphere and M5-branes arXiv:1211.0144 [hep-th]
Kim H.-C., Kim S.: M5-Branes from gauge theories on the 5-sphere. JHEP 05, 144 (2013) arXiv:1206.6339 [hep-th]
Qiu J., Tizzano L., Winding J., Zabzine M.: Gluing Nekrasov partition functions. Commun. Math. Phys. 337(2), 785–816 (2015) arXiv:1403.2945 [hep-th]
Pan Y.: 5d Higgs branch localization, Seiberg–Witten equations and contact geometry. JHEP 01, 145 (2015) arXiv:1406.5236 [hep-th]
Pasquetti S.: Factorisation of N = 2 theories on the squashed 3-sphere. JHEP 04, 120 (2012) arXiv:1111.6905 [hep-th]
Beem C., Dimofte T., Pasquetti S.: Holomorphic blocks in three dimensions. JHEP 12, 177 (2014) arXiv:1211.1986 [hep-th]
Bullimore M., Kim H.-C., Koroteev P.: Defects and quantum Seiberg–Witten geometry. JHEP 05, 095 (2015) arXiv:1412.6081 [hep-th]
Nedelin A., Nieri F., Zabzine M.: q-Virasoro modular double and 3d partition functions. Commun. Math. Phys. 353(3), 1059–1102 (2017) arXiv:1605.07029 [hep-th]
Faddeev L.D.: Modular double of quantum group. Math. Phys. Stud. 21, 149–156 (2000) arXiv:math/9912078 [math-qa]
Faddeev L.D.: Modular double of quantum group. Math. Phys. Stud. 21, 149 (1999) arXiv:math/9912078 [math-qa]
Ponsot, B., Teschner, J.: Liouville bootstrap via harmonic analysis on a noncompact quantum group arXiv:hep-th/9911110 [hep-th]
Ponsot B., Teschner J.: Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of \({U(q)(sl(2,R))}\). Commun. Math. Phys. 224, 613–655 (2001) arXiv:math/0007097 [math-qa]
Hatsuda Y., Marino M., Moriyama S., Okuyama K.: Non-perturbative effects and the refined topological string. JHEP 09, 168 (2014) arXiv:1306.1734 [hep-th]
Aganagic M., Klemm A., Marino M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005) arXiv:hep-th/0305132 [hep-th]
Iqbal A., Kozcaz C., Vafa C.: The refined topological vertex. JHEP 10, 069 (2009) arXiv:hep-th/0701156 [hep-th]
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]
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 [q-alg]
Awata H., Kubo H., Odake S., Shiraishi J.: Quantum W(N) algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401–416 (1996) arXiv:q-alg/9508011 [q-alg]
Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras arXiv:q-alg/9708006 [q-alg]
Aganagic, M., Frenkel, E., Okounkov, A.: Quantum q-Langlands correspondence arXiv:1701.03146 [hep-th]
Narukawa, A.: The modular properties and the integral representations of the multiple elliptic gamma functions arXiv:math/0306164 [math.QA].
Morozov A.: Integrability and matrix models. Phys. Usp. 37, 1–55 (1994) arXiv:hep-th/9303139 [hep-th]
Nieri F.: An elliptic Virasoro symmetry in 6d. Lett. Math. Phys. 107(11), 2147–2187 (2017) arXiv:1511.00574 [hep-th]
Iqbal, A., Kozcaz, C., Yau, S.-T.: Elliptic Virasoro conformal blocks arXiv:1511.00458 [hep-th]
Kimura, T., Pestun, V.: Quiver elliptic W-algebras arXiv:1608.04651 [hep-th]
Tan M.-C.: An M-theoretic derivation of a 5d and 6d AGT correspondence, and relativistic and elliptized integrable systems. JHEP 12, 031 (2013) arXiv:1309.4775 [hep-th]
Tan M.-C.: Higher AGT correspondences, W-algebras, and higher quantum geometric langlands duality from M-theory. Adv. Theor. Math. Phys. 22, 429–507 (2018) arXiv:1607.08330 [hep-th]
Lodin R., Nieri F., Zabzine M.: Elliptic modular double and 4d partition functions. J. Phys. A 51(4), 045402 (2018) arXiv:1703.04614 [hep-th]
Grosse-Knetter C.: Effective Lagrangians with higher derivatives and equations of motion. Phys. Rev. D 49, 6709–6719 (1994) arXiv:hep-ph/9306321 [hep-ph]
Nieri, F., Pan, Y., Zabzine, M.: Bootstrapping the S 5 partition function. In: 20th International Seminar on High Energy Physics (Quarks 2018) Valday (Russia) arXiv:1807.11900 [hep-th] (2018)
Pan, Y., Peelaers, W.: Intersecting surface defects and instanton partition functions arXiv:1612.04839 [hep-th]
Gomis, J., Le Floch, B., Pan, Y., Peelaers, W.: Intersecting surface defects and two-dimensional CFT arXiv:1610.03501 [hep-th]
Schmude J.: Localisation on Sasaki–Einstein manifolds from holomorphic functions on the cone. JHEP 01, 119 (2015) arXiv:1401.3266 [hep-th]
Iqbal A., Shabbir K.: Elliptic CY3folds and non-perturbative modular transformation. Eur. Phys. J. C 76(3), 148 (2016) arXiv:1510.03332 [hep-th]
Qiu J., Tizzano L., Winding J., Zabzine M.: Modular properties of full 5D SYM partition function. JHEP 03, 193 (2016) arXiv:1511.06304 [hep-th]
Benvenuti S., Bonelli G., Ronzani M., Tanzini A.: Symmetry enhancements via 5d instantons, \({ q\mathcal{W} }\)-algebrae and (1, 0) superconformal index. JHEP 09, 053 (2016) arXiv:1606.03036 [hep-th]
Witten, E.: Some comments on string dynamics arXiv:hep-th/9507121 [hep-th]
Seiberg N.: New theories in six-dimensions and matrix description of M theory on T 5 and \({T^5 / Z(2)}\). Phys. Lett. B 408, 98–104 (1997) arXiv:hep-th/9705221 [hep-th]
Losev A., Moore G.W., Shatashvili S.L.: M & m’s. Nucl. Phys. B 522, 105–124 (1998) arXiv:hep-th/9707250 [hep-th]
Acknowledgement
We thank Guglielmo Lockhart, Jian Qiu, Shamil Shakirov, and Jörg Teschner for valuable discussions. We also thank the Simons Center for Geometry and Physics (Stony Brook University) for hospitality during the Summer Workshop 2017, at which some of the research for this paper was performed. M.Z. would like to thank Chambó the French bulldog for the inspiration. The research of the authors is supported in part by Vetenskapsrådet under Grant #2014-5517, by the STINT Grant and by the Grant “Geometry and Physics” from the Knut and Alice Wallenberg foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Nekrasov
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
OpenAccess 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.
About this article
Cite this article
Nieri, F., Pan, Y. & Zabzine, M. q-Virasoro Modular Triple. Commun. Math. Phys. 366, 397–422 (2019). https://doi.org/10.1007/s00220-019-03371-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03371-1