Abstract
We study partition functions of 3d \({\mathcal{N}=2}\) \({{\rm U}(N)}\) gauge theories on compact manifolds which are S 1 fibrations over S 2. We show that the partition functions are free field correlators of vertex operators and screening charges of the q-Virasoro modular double, which we define. The inclusion of supersymmetric Wilson loops in arbitrary representations allows us to show that the generating functions of Wilson loop vacuum expectation values satisfy two \({{\rm SL}(2,\mathbb{Z})}\)-related commuting sets of q-Virasoro constraints. We generalize our construction to 3d \({\mathcal{N}=2}\) unitary quiver gauge theories and as an example we give the free boson realization of the ABJ(M) model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Witten E.: Quantum field theory and the jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)
Aganagic M., Shakirov S.: Knot homology and refined Chern–Simons index. Commun. Math. Phys. 333, 187–228. (2015) arXiv:1105.5117
Aganagic M., Shakirov S.: Refined Chern–Simons theory and knot homology. Proc. Symp. Pure Math. 85, 3–32. (2012) arXiv:1202.2489
Aganagic, M., Shakirov, S.: Refined Chern–Simons theory and topological string. arXiv:1210.2733 (2012)
Witten E.: Chern–Simons gauge theory as a string theory. Prog. Math. 133, 637–678. (1995) arXiv:hep-th/9207094
Gopakumar R., Vafa C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443. (1999) arXiv:hep-th/9811131
Aganagic, M., Marino, M., Vafa, C.: All loop topological string amplitudes from Chern–Simons theory. Commun. Math. Phys. 247, 467–512. arXiv:hep-th/0206164 (2004)
Hanany A., Witten E.: Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics. Nucl. Phys. B 492, 152–190. (1997) arXiv:hep-th/9611230
Aharony O., Bergman O., Jafferis D.L., Maldacena J.: N = 6 superconformal chern-simons-matter theories, m2-branes and their gravity duals. JHEP 10, 091. (2008) arXiv:0806.1218
Aharony O., Bergman O., Jafferis D.L.: Fractional M2-branes. JHEP 11, 043. (2008) arXiv:0807.4924
Dimofte T., Gukov S., Hollands L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287. (2011) arXiv:1006.0977
Dimofte T., Gaiotto D., Gukov S.: Gauge theories labelled by three-manifolds. Commun. Math. Phys. 325, 367–419. (2014) arXiv:1108.4389
Dimofte T., Gaiotto D., Gukov S.: 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17, 975–1076 (2013) arXiv:1112.5179
Chung H.-J., Dimofte T., Gukov S., Sulkowski P.: 3d–3d correspondence revisited. JHEP. 04, 140. (2016) arXiv:1405.3663
Dimofte, T.: 3d Superconformal theories from three-manifolds. In: Teschner, J. (ed). New Dualities of Supersymmetric Gauge Theories, pp. 339–373. arXiv:1412.7129 (2016)
Witten E.: Quantization of Chern–Simons gauge theory with complex gauge group. Commun. Math. Phys. 137, 29–66 (1991)
Karch A.: Seiberg duality in three-dimensions. Phys. Lett. B 405, 79–84. (1997) arXiv:hep-th/9703172
Aharony O.: IR duality in d = 3 N=2 supersymmetric USp(2N(c)) and U(N(c)) gauge theories. Phys. Lett. B 404, 71–76. (1997) arXiv:hep-th/9703215
Niarchos V.: Seiberg duality in Chern–Simons theories with fundamental and adjoint matter. JHEP 11, 001. (2008) arXiv:0808.2771
Benini F., Closset C., Cremonesi S.: Comments on 3d Seiberg-like dualities. JHEP. 10, 075. (2011) arXiv:1108.5373
Aharony O., Razamat S.S., Seiberg N., Willett B.: 3d dualities from 4d dualities. JHEP. 07, 149. (2013) arXiv:1305.3924
Aharony O., Razamat S.S., Seiberg N., Willett B.: 3d dualities from 4d dualities for orthogonal groups. JHEP 08, 099. (2013) arXiv:1307.0511
Intriligator K.A., Seiberg N.: Mirror symmetry in three-dimensional gauge theories. Phys. Lett. B 387, 513–519. (1996) arXiv:hep-th/9607207
de Boer J., Hori K., Oz Y., Yin Z.: Branes and mirror symmetry in n = 2 supersymmetric gauge theories in three-dimensions. Nucl. Phys. B 502, 107–124. (1997) arXiv:hep-th/9702154
Aharony, O., Hanany, A., Intriligator, K.A., Seiberg, N., Strassler, M.J.: Aspects of n = 2 supersymmetric gauge theories in three-dimensions. Nucl. Phys. B 499, 67–99. arXiv:hep-th/9703110 (1997)
Gadde, A., Gukov, S., Putrov, P.: Fivebranes and 4-manifolds. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds). Arbeitstagung Bonn 2013, pp. 155–245. arXiv:1306.4320 (2016)
Gukov, S., Pei, D.: Equivariant Verlinde formula from fivebranes and vortices. arXiv:1501.01310 (2015)
Festuccia G., Seiberg N.: Rigid supersymmetric theories in curved superspace. JHEP 06, 114. (2011) arXiv:1105.0689
Dumitrescu T.T., Festuccia G., Seiberg N.: Exploring curved superspace. JHEP 08, 141. (2012) arXiv:1205.1115
Closset C., Dumitrescu T.T., Festuccia G., Komargodski Z.: Supersymmetric field theories on three-manifolds. JHEP 05, 017. (2013) arXiv:1212.3388
Closset C., Dumitrescu T.T., Festuccia G., Komargodski Z.: The geometry of supersymmetric partition functions. JHEP 01, 124. (2014) arXiv:1309.5876
Closset C., Dumitrescu T.T., Festuccia G., Komargodski Z.: From rigid supersymmetry to twisted holomorphic theories. Phys. Rev. D 90, 085006. (2014) arXiv:1407.2598
Imbimbo C., Rosa D.: Topological anomalies for Seifert 3-manifolds. JHEP 07, 068. (2015) arXiv:1411.6635
Pestun V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129. (2012) arXiv:0712.2824
Kapustin A., Willett B., Yaakov I.: Exact results for wilson loops in superconformal chern-simons theories with matter. JHEP 03, 089. (2010) arXiv:0909.4559
Hama N., Hosomichi K., Lee S.: SUSY gauge theories on squashed three-spheres. JHEP 05, 014. (2011) arXiv:1102.4716
Imamura Y., Yokoyama D.: N = 2 supersymmetric theories on squashed three-sphere. Phys. Rev. D 85, 025015. (2012) arXiv:1109.4734
Alday L.F., Martelli D., Richmond P., Sparks J.: Localization on three-manifolds. JHEP 10, 095 (2013) arXiv:1307.6848
Gang, D.: Chern–Simons theory on L(p,q) lens spaces and localization. arXiv:0912.4664 (2009)
Benini F., Nishioka T., Yamazaki M.: 4d Index to 3d Index and 2d TQFT. Phys. Rev. D 86, 065015. (2012) arXiv:1109.0283
Alday L.F., Fluder M., Sparks J.: The large N limit of M2-branes on lens spaces. JHEP 10, 057. (2012) arXiv:1204.1280
Imamura Y., Yokoyama D.: \({S^3/\mathbb{Z}_n}\) partition function and dualities. JHEP 11, 122. (2012) arXiv:1208.1404
Imamura Y., Matsuno H., Yokoyama D.: Factorization of the \({S^3/\mathbb{Z}_n}\) partition function. Phys. Rev. D 89, 085003. (2014) arXiv:1311.2371
Imamura Y., Yokoyama S.: Index for three dimensional superconformal field theories with general r-charge assignments. JHEP 04, 007. (2011) arXiv:1101.0557
Kapustin, A., Willett, B.: Generalized Superconformal Index for Three Dimensional Field Theories. arXiv:1106.2484 (2011)
Benini, F., Zaffaroni, A.: A topologically twisted index for three-dimensional supersymmetric theories. JHEP 07, 127. arXiv:1504.03698 (2015)
Benini, F., Zaffaroni, A.: Supersymmetric partition functions on Riemann surfaces. arXiv:1605.06120 (2016)
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. arXiv:q-alg/9507034 (1996)
Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197. arXiv:0906.3219 (2010)
Wyllard, N.: A(n-1) conformal toda field theory correlation functions from conformal n = 2 su(n) quiver gauge theories. JHEP 11, 002. arXiv:0907.2189 (2009)
Awata H., Yamada Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 01, 125. (2010) arXiv:0910.4431
Nieri F., Pasquetti S., Passerini F.: 3d and 5d gauge theory partition functions as q-deformed CFT correlators. Lett. Math. Phys. 105, 109–148. (2015) arXiv:1303.2626
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
Aganagic, M., Haouzi, N.: ADE Little String Theory on a Riemann Surface (and Triality). arXiv:1506.04183 (2015)
Aganagic M., Cheng M.C.N., Dijkgraaf R., Krefl D., Vafa C.: Quantum geometry of refined topological strings. JHEP. 11, 019. (2012) arXiv:1105.0630
Gadde A., Gukov S., Putrov P.: Walls, lines, and spectral dualities in 3d gauge theories. JHEP 05, 047. (2014) arXiv:1302.0015
Gaiotto D., Koroteev P.: On three dimensional quiver gauge theories and integrability. JHEP 05, 126. (2013) arXiv:1304.0779
Bullimore M., Kim H.-C., Koroteev P.: Defects and quantum Seiberg–Witten geometry. JHEP 05, 095. (2015) arXiv:1412.6081
Koroteev, P., Sciarappa, A.: Quantum Hydrodynamics from Large-n Supersymmetric Gauge Theories. arXiv:1510.00972 (2016)
Koroteev P., Sciarappa A.: On elliptic algebras and large-n supersymmetric gauge theories. J. Math. Phys. 57, 112302. (2016) arXiv:1601.08238
Pasquetti S.: Factorisation of N = 2 theories on the squashed 3-sphere. JHEP 04, 120. (2012) arXiv:1111.6905
Beem C., Dimofte T., Pasquetti S.: Holomorphic blocks in three dimensions. JHEP 12, 177. (2014) arXiv:1211.1986
Benini F., Peelaers W.: Higgs branch localization in three dimensions. JHEP 05, 030. (2014) arXiv:1312.6078
Fujitsuka M., Honda M., Yoshida Y.: Higgs branch localization of 3d N = 2 theories. PTEP 2014, 123B02. (2014) arXiv:1312.3627
Taki, M.: Holomorphic Blocks for 3d Non-abelian Partition Functions. arXiv:1303.5915 (2013)
Hwang C., Park J.: Factorization of the 3d superconformal index with an adjoint matter. JHEP 11, 028. (2015) arXiv:1506.03951
Nieri F., Pasquetti S.: Factorisation and holomorphic blocks in 4d. JHEP 11, 155. (2015) arXiv:1507.00261
Aganagic, M., Haouzi, N., Kozcaz, C., Shakirov, S.: Gauge/Liouville Triality. arXiv:1309.1687 (2013)
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 (2016)
Nekrasov N.: Bps/cft correspondence: non-perturbative Dyson-Schwinger equations and qq-characters. JHEP 03, 181. (2016) arXiv:1512.05388
Nekrasov, N., Pestun, V.: Seiberg–Witten Geometry of Four Dimensional N = 2 Quiver Gauge Theories. arXiv:1211.2240 (2012)
Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum Geometry and Quiver Gauge Theories. arXiv:1312.6689 (2013)
Kimura, T., Pestun, V.: Quiver W-algebras. arXiv:1512.08533 (2015)
Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864. (2003) arXiv:hep-th/0206161
Nekrasov N., Okounkov A.: Seiberg–Witten theory and random partitions. Prog. Math. 244, 525–596. (2006) arXiv:hep-th/0306238
Leung N.C., Vafa C.: Branes and toric geometry. Adv. Theor. Math. Phys. 2, 91–118. (1998) arXiv:hep-th/9711013
Kol B., Rahmfeld J.: BPS spectrum of five-dimensional field theories, (p,q) webs and curve counting. JHEP 08, 006. (1998) arXiv:hep-th/9801067
Aharony O., Hanany A.: Branes, superpotentials and superconformal fixed points. Nucl. Phys. B 504, 239–271. (1997) arXiv:hep-th/9704170
Aharony O., Hanany A., Kol B.: Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams. JHEP 01, 002. (1998) arXiv:hep-th/9710116
Iqbal A., Kozcaz C., Vafa C.: The Refined topological vertex. JHEP 10, 069. (2009) arXiv:hep-th/0701156
Awata H., Kanno H.: Instanton counting, Macdonald functions and the moduli space of D-branes. JHEP 05, 039. (2005) arXiv:hep-th/0502061
Taki M.: Refined topological vertex and instanton counting. JHEP 03, 048. (2008) arXiv:0710.1776
Mironov A., Morozov A., Zenkevich Y.: Ding–Iohara–Miki symmetry of network matrix models. Phys. Lett. B 762, 196–208. (2016) arXiv:1603.05467
Awata H., Kanno H., Matsumoto T., Mironov A., Morozov A., Morozov A. et al.: Explicit examples of DIM constraints for network matrix models. JHEP 07, 103. (2016) arXiv:1604.08366
Ding, J., Iohara, K.: Generalization and Deformation of Drinfeld Quantum Affine Algebras. arXiv:q-alg/9608002 (1996)
Miki, K.: A (q,\({\gamma}\)) analog of the W\({_{1+?}}\) algebra. J. Math. Phys. 48, 123520–123520 (2007)
Iqbal A., Kashani-Poor A.-K.: The vertex on a strip. Adv. Theor. Math. Phys. 10, 317–343. (2006) arXiv:hep-th/0410174
Awata H., Yamada Y.: Five-dimensional AGT Relation and the deformed beta-ensemble. Prog. Theor. Phys. 124, 227–262. (2010) arXiv:1004.5122
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
Carlsson, E., Nekrasov, N., Okounkov, A.: Five Dimensional Gauge Theories and Vertex Operators. arXiv:1308.2465 (2013)
Zenkevich Y.: Generalized macdonald polynomials, spectral duality for conformal blocks and agt correspondence in five dimensions. JHEP 05, 131. (2015) arXiv:1412.8592
Zenkevich, Y.: Quantum Spectral Curve for (q,t)-matrix Model. arXiv:1507.00519 (2015)
Morozov A., Zenkevich Y.: Decomposing Nekrasov decomposition. JHEP 02, 098. (2016) arXiv:1510.01896
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
Mironov A., Morozov A., Zenkevich Y.: Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings. JHEP 05, 121. (2016) arXiv:1603.00304
Aganagic, M., Haouzi, N., Shakirov, S.: \({A_n}\)-Triality. arXiv:1403.3657 (2014)
Lockhart, G., Vafa, C.: Superconformal Partition Functions and Non-perturbative Topological Strings. arXiv:1210.5909 (2012)
Morozov A.: Integrability and matrix models. Phys. Usp. 37, 1–55. (1994) arXiv:hep-th/9303139
Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D gravity and random matrices. Phys. Rept. 254, 1–133. (1995) arXiv:hep-th/9306153
Odake, S.: Beyond CFT: Deformed Virasoro and elliptic algebras. In: Theoretical Physics at the End of the Twentieth Century. Proceedings, Summer School, Banff, Canada, June 27–July 10, 1999, pp. 307–449 (1999). arXiv:hep-th/9910226
Itoyama H., Oota T., Yoshioka R.: q -vertex operator from 5d nekrasov function. J. Phys. A Math. Theor. 49, 345201 (2016)
Yoshida, Y., Sugiyama, K.: Localization of 3d \({\mathcal{N}=2}\) supersymmetric theories on \({s^1 \times d^2}\). arXiv:1409.6713 (2014)
Benini F., Eager R., Hori K., Tachikawa Y.: Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups. Lett. Math. Phys. 104, 465–493. (2014) arXiv:1305.0533
Benini F., Eager R., Hori K., Tachikawa Y.: Elliptic genera of 2d \({{\mathcal{N}}}\) = 2 gauge theories. Commun. Math. Phys. 333, 1241–1286. (2015) arXiv:1308.4896
Gadde A., Gukov S.: 2d index and surface operators. JHEP 03, 080. (2014) arXiv:1305.0266
Hosomichi, K.: A review on SUSY gauge theories on \({\mathbf S^3}\). In: J. Teschner, (ed.) New Dualities of Supersymmetric Gauge Theories, pp. 307–338. arXiv:1412.7128 (2016)
Jafferis D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 05, 159. (2012) arXiv:1012.3210
Tanaka A.: Comments on knotted 1/2 BPS Wilson loops. JHEP 07, 097. (2012) arXiv:1204.5975
Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Virasoro type symmetries in solvable models, In: Extended and Quantum Algebras and their Applications to Physics Tianjin, China, August 19–24, 1996. arXiv:hep-th/9612233 (1996)
Ohkubo, Y., Awata, H., Fujino, H.: Crystallization of Deformed Virasoro Algebra, Ding–Iohara–Miki Algebra and 5d agt Correspondence. arXiv:1512.08016 (2015)
Bouwknegt, P., Pilch, K.: The deformed virasoro algebra at roots of unity. Commun. Math. Phys. 196:249–288 (1998)
Itoyama H., Oota T., Yoshioka R.: 2d–4d connection between q-virasoro/w block at root of unity limit and instanton partition function on ale space. Nucl. Phys. B 877, 506–537. (2013) arXiv:1308.2068
Itoyama H., Oota T., Yoshioka R.: q-Virasoro/W algebra at root of unity and parafermions. Nucl. Phys. B 889, 25–35. (2014) arXiv:1408.4216
Frenkel, E., Reshetikhin, N.: Deformations of W-algebras Associated to Simple Lie Algebras. arXiv:q-alg/9708006 (1997)
Frenkel E., Reshetikhin N., Semenov-Tian-Shansky M.: Drinfeld–Sokolov reduction for difference operators and deformations of -algebras i. the case of virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998)
Semenov-Tian-Shansky A.M., Sevostyanov V.A.: Drinfeld–sokolov reduction for differenceoperators and deformations of w-algebras ii. the general semisimple case. Commun. Math. Phys. 192, 631–647 (1998)
Mironov A., Morozov A.: On the origin of Virasoro constraints in matrix models: Lagrangian approach. Phys. Lett. B 252, 47–52 (1990)
Nedelin, A., Zabzine, M.: q-Virasoro Constraints in Matrix Models. arXiv:1511.03471 (2015)
Dijkgraaf R., Verlinde H.L., Verlinde E.P.: Loop equations and Virasoro constraints in nonperturbative 2-D quantum gravity. Nucl. Phys. B 348, 435–456. (1991)
Marino M.: Chern–Simons theory, matrix integrals, and perturbative three manifold invariants. Commun. Math. Phys. 253, 25–49. (2004) arXiv:hep-th/0207096
Aganagic M., Klemm A., Marino M., Vafa C.: Matrix model as a mirror of Chern–Simons theory. JHEP 02, 010. (2004) arXiv:hep-th/0211098
Dubinkin O.: On the Virasoro constraints for torus knots. J. Phys. A 47, 485203. (2014) arXiv:1307.7909
Iqbal A., Kozcaz C.: Refined Hopf link revisited. JHEP 04, 046. (2012) arXiv:1111.0525
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
Hatsuda, Y.: ABJM on ellipsoid and topological strings. JHEP 07, 026. arXiv:1601.02728 (2016)
Moore G.W., Seiberg N.: Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177 (1989)
Sugishita S., Terashima S.: Exact results in supersymmetric field theories on manifolds with boundaries. JHEP 11, 021 (2013)
Faizal M., Luo Y., Smith D.J., Tan M.-C., Zhao Q.: Gauge and supersymmetry invariance of n = 2 boundary Chern–Simons theory. Nucl. Phys. B 914, 577–598. (2017) arXiv:1601.05429
Aprile F., Niarchos V.: \({\mathcal{N}=2}\) supersymmetric field theories on 3-manifolds with A-type boundaries. JHEP 07, 176. (2016) arXiv:1604.01561
Drukker N., Marino M., Putrov P.: From weak to strong coupling in ABJM theory. Commun. Math. Phys. 306, 511–563. (2011) arXiv:1007.3837
Drukker N., Marino M., Putrov P.: Nonperturbative aspects of ABJM theory. JHEP 11, 141. (2011) arXiv:1103.4844
Marino, M., Putrov, P.: ABJM theory as a Fermi gas. J. Stat. Mech. 1203, P03001. arXiv:1110.4066 (2012)
Codesido S., Grassi A., Marino M.: Exact results in \({\mathcal{N}=8}\) Chern–Simons-matter theories and quantum geometry. JHEP 07, 011. (2015) arXiv:1409.1799
Grassi A., Hatsuda Y., Marino M.: Quantization conditions and functional equations in ABJ(M) theories. J. Phys. A 49, 115401. (2016) arXiv:1410.7658
Assel B., Drukker N., Felix J.: Partition functions of 3d \({\hat D}\)-quivers and their mirror duals from 1d free fermions. JHEP 08, 071. (2015) arXiv:1504.07636
Benvenuti S., Pasquetti S.: 3d N = 2 mirror symmetry, pq-webs and monopole superpotentials. JHEP 08, 136. (2016) arXiv:1605.02675
Gomis J., Le Floch B.: M2-brane surface operators and gauge theory dualities in Toda. JHEP 04, 183. (2016) arXiv:1407.1852
Kim H.-C., Kim S.-S., Lee K.: 5-dim superconformal index with enhanced En global symmetry. JHEP 10, 142. (2012) arXiv:1206.6781
Terashima S.: Supersymmetric gauge theories on S 4 x S 1. Phys. Rev. D 89, 125001. (2014) arXiv:1207.2163
Iqbal A., Vafa C.: BPS degeneracies and superconformal index in diverse dimensions. Phys. Rev. D 90, 105031. (2014) arXiv:1210.3605
Källén J., Zabzine M.: Twisted supersymmetric 5D Yang–Mills theory and contact geometry. JHEP 05, 125. (2012) arXiv:1202.1956
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
Imamura Y.: Perturbative partition function for squashed S 5. Prog. Theor. Exp. Phys. 2013(7), 073B01. (2003) arXiv:1210.6308
Kim, H.-C., Kim, J., Kim, S.: Instantons on the 5-Sphere and M5-Branes. arXiv:1211.0144 (2012)
Kim H.-C., Kim S.: M5-branes from gauge theories on the 5-sphere. JHEP 05, 144. (2013) arXiv:1206.6339
Minahan J.A., Nedelin A., Zabzine M.: 5D super Yang–Mills theory and the correspondence to AdS7/CFT6. J. Phys. A 46, 355401. (2013) arXiv:1304.1016
Qiu J., Zabzine M.: 5D super Yang–Mills on \({Y^{p,q}}\) Sasaki–Einstein manifolds. Commun. Math. Phys. 333, 861–904. (2015) arXiv:1307.3149
Qiu J., Zabzine M.: Factorization of 5D super Yang–Mills theory on \({Y^{p,q}}\) spaces. Phys. Rev. D 89, 065040. (2014) arXiv:1312.3475
Qiu J., Tizzano L., Winding J., Zabzine M.: Gluing Nekrasov partition functions. Commun. Math. Phys. 337, 785–816. (2015) arXiv:1403.2945
Vafa C.: Superstrings and topological strings at large N. J. Math. Phys. 42, 2798–2817. (2001) arXiv:hep-th/0008142
Cachazo, F., Intriligator, K.A., Vafa, C.: A large N duality via a geometric transition. Nucl. Phys. B 603, 3–41. arXiv:hep-th/0103067 (2001)
Gaiotto D., Witten E.: Supersymmetric boundary conditions in N = 4 super Yang–Mills theory. J. Stat. Phys. 135, 789–855. (2009) arXiv:0804.2902
Gaiotto D., Witten E.: S-duality of boundary conditions in N = 4 super Yang–Mills theory. Adv. Theor. Math. Phys. 13, 721–896. (2009) arXiv:0807.3720
Cecotti, S., Cordova, C., Vafa, C.: Braids, Walls, and Mirrors. arXiv:1110.2115 (2011)
Gaiotto D.: N=2 dualities. JHEP 08, 034. (2012) arXiv:0904.2715
Terashima Y., Yamazaki M.: SL(2,R) Chern–Simons, Liouville, and gauge theory on duality walls. JHEP 08, 135. (2011) arXiv:1103.5748
Galakhov D.V., Mironov A., Morozov A., Smirnov A., Mironov A.D., Morozov A.Yu. et al.: Three-dimensional extensions of the Alday–Gaiotto–Tachikawa relation. Theor. Math. Phys. 172, 939–962. (2012) arXiv:1104.2589
Teschner J., Vartanov G.: 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories. Lett. Math. Phys. 104, 527–551. (2014) arXiv:1202.4698
Teschner J., Vartanov G.S.: Supersymmetric gauge theories, quantization of \({\mathcal{M}_{\mathrm{flat}}}\), and conformal field theory. Adv. Theor. Math. Phys. 19, 1–135. (2015) arXiv:1302.3778
Le Floch, B.: S-duality Wall of SQCD from Toda Braiding. arXiv:1512.09128 (2015)
Ponsot, B., Teschner, J.: Liouville Bootstrap Via Harmonic Analysis on a Noncompact Quantum Group. arXiv:hep-th/9911110 (1999)
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
Teschner J.: Liouville theory revisited. Class. Quant. Grav. 18, R153–R222. (2001) arXiv:hep-th/0104158
Nieri, F.: An Elliptic Virasoro Symmetry in 6d. arXiv:1511.00574 (2015)
Iqbal, A., Kozcaz, C., Yau, S.-T.: Elliptic Virasoro Conformal Blocks. arXiv:1511.00458
Romelsberger C.: Counting chiral primaries in N = 1, d = 4 superconformal field theories. Nucl. Phys. B 747, 329–353. (2006) arXiv:hep-th/0510060
Romelsberger, C.: Calculating the Superconformal Index and Seiberg Duality. arXiv:0707.3702 (2007)
Kinney, J., Maldacena, J.M., Minwalla, S., Raju, S.: An index for 4 dimensional super conformal theories. Commun. Math. Phys. 275, 209–254. arXiv:hep-th/0510251 (2007)
Dolan, F.A., Osborn, H.: Applications of the superconformal index for protected operators and q-hypergeometric identities to n = 1 dual theories. Nucl. Phys. B 818, 137–178. arXiv:0801.4947 (2009)
Rastelli, L., Razamat, S.S.: The Superconformal Index of Theories of Class \({\mathcal {S}}\). In: Teschner, J. (ed.) New Dualities of Supersymmetric Gauge Theories, pp. 261–305. arXiv:1412.7131 (2016)
Yoshida, Y.: Factorization of 4d N = 1 Superconformal Index. arXiv:1403.0891 (2014)
Peelaers W.: Higgs branch localization of \({\mathcal{N}}\) = 1 theories on S3 x S1. JHEP 08, 060. (2014) arXiv:1403.2711
Chen H.-Y., Chen H.-Y.: Heterotic surface defects and dualities from 2d/4d indices. JHEP. 10, 004 (2014) arXiv:1407.4587
Razamat S.S., Razamat S.S.: S-duality and the N = 2 lens space index. JHEP 10, 048. (2013) arXiv:1306.1543
Closset C., Shamir I.: The \({\mathcal{N}=1}\) chiral multiplet on \({T^2\times S^2}\) and supersymmetric localization. JHEP 03, 040. (2014) arXiv:1311.2430
Assel B., Cassani D., Martelli D.: Localization on Hopf surfaces. JHEP 08, 123. (2014) arXiv:1405.5144
Nishioka T., Yaakov I.: Generalized indices for \({\mathcal{N}\,{=}\, 1}\) theories in four-dimensions. JHEP 12, 150. (2014) arXiv:1407.8520
Honda, M., Yoshida, Y.: Supersymmetric index on \({T^2 \times S^2}\) and elliptic genus. arXiv:1504.04355 (2015)
Gadde A., Razamat S.S., Willett B.: On the reduction of 4d \({\mathcal{N}=1}\) theories on \({{\mathbb{S}}^2}\). JHEP 11, 163. (2015) arXiv:1506.08795
Gadde A., Rastelli L., Razamat S.S., Yan W.: The 4d superconformal index from q-deformed 2d Yang–Mills. Phys. Rev. Lett. 106, 241602 (2011) arXiv:1104.3850
Gaiotto D., Rastelli L., Razamat S.S.: Bootstrapping the superconformal index with surface defects. JHEP 01, 022. (2013) arXiv:1207.3577
Razamat S.S.: On the \({\mathcal{N} =}\) 2 superconformal index and eigenfunctions of the elliptic RS model. Lett. Math. Phys. 104, 673–690. (2014) arXiv:1309.0278
Faddeev L.D.: Modular double of quantum group. Math. Phys. Stud. 21, 149–156. (2000) arXiv:math/9912078
Kharchev S., Lebedev D., Semenov-Tian-Shansky M.: Unitary representations of u(q) (sl(2, r)), the modular double, and the multiparticle q deformed toda chains. Commun. Math. Phys. 225, 573–609. (2002) arXiv:hep-th/0102180
Narukawa A.: The modular properties and the integral representations of the multiple elliptic gamma functions. Adv. Math. 189(2), 247–267. (2004) arXiv:math/0306164
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. A. Nekrasov
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.
About this article
Cite this article
Nedelin, A., Nieri, F. & Zabzine, M. q-Virasoro Modular Double and 3d Partition Functions. Commun. Math. Phys. 353, 1059–1102 (2017). https://doi.org/10.1007/s00220-017-2882-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2882-1