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Seifert fibering operators in 3d \( \mathcal{N}=2 \) theories

  • Cyril ClossetEmail author
  • Heeyeon Kim
  • Brian Willett
Open Access
Regular Article - Theoretical Physics

Abstract

We study 3d \( \mathcal{N}=2 \) supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q)b with rational squashing parameter b2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

Keywords

Field Theories in Lower Dimensions Supersymmetric Gauge Theory Supersymmetry and Duality Topological Field Theories 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    Z. Komargodski and N. Seiberg, Comments on supercurrent multiplets, supersymmetric field theories and supergravity, JHEP 07 (2010) 017 [arXiv:1002.2228] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    T.T. Dumitrescu and N. Seiberg, Supercurrents and brane currents in diverse dimensions, JHEP 07 (2011) 095 [arXiv:1106.0031] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    S.M. Kuzenko et al., Three-dimensional \( \mathcal{N}=2 \) supergravity theories: From superspace to components, Phys. Rev. D 89 (2014) 085028 [arXiv:1312.4267] [INSPIRE].ADSGoogle Scholar
  7. [7]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  8. [8]
    C. Closset et al., Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    D. Gang, Chern-Simons theory on L(p, q) lens spaces and Localization, arXiv:0912.4664 [INSPIRE].
  12. [12]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    F. Benini, T. Nishioka and M. Yamazaki, 4D index to 3D index and 2D TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].ADSGoogle Scholar
  16. [16]
    J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    T. Dimofte, Complex Chern-Simons theory at level k via the 3d–3d correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    A. Kapustin and K. Vyas, A-models in three and four dimensions, arXiv:1002.4241 [INSPIRE].
  20. [20]
    S. Cecotti, D. Gaiotto and C. Vafa, tt * geometry in 3 and 4 dimensions, JHEP 05 (2014) 055 [arXiv:1312.1008] [INSPIRE].
  21. [21]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, From rigid supersymmetry to twisted holomorphic theories, Phys. Rev. D 90 (2014) 085006 [arXiv:1407.2598] [INSPIRE].ADSGoogle Scholar
  22. [22]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M. Aganagic, K. Costello, J. McNamara and C. Vafa, Topological Chern-Simons/matter theories, arXiv:1706.09977 [INSPIRE].
  24. [24]
    E. Witten, Topological Σ models, Commun. Math. Phys. 118 (1988) 411.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    A. Kapustin and B. Willett, Wilson loops in supersymmetric Chern-Simons-matter theories and duality, arXiv:1302.2164 [INSPIRE].
  26. [26]
    C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    P. Orlik, Seifert manifolds, Lecture notes in mathematics. Springer, Germany (1972).Google Scholar
  29. [29]
    P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    N.A. Nekrasov and S.L. Shatashvili, Bethe/gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  32. [32]
    S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett. A 6 (1991) 337 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    A.J. Niemi and G.W. Semenoff, Axial anomaly induced fermion fractionization and effective gauge theory actions in odd dimensional space-times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    L. Álvarez-Gaumé, S. Della Pietra and G.W. Moore, Anomalies and odd dimensions, Annals Phys. 163 (1985) 288 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    C. Closset et al., Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    E. Witten, Theparityanomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    N. Seiberg, T. Senthil, C. Wang and E. Witten, A duality web in 2 + 1 dimensions and condensed matter physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    K. Ohta and Y. Yoshida, Non-abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on Seifert manifold, Phys. Rev. D 86 (2012) 105018 [arXiv:1205.0046] [INSPIRE].ADSGoogle Scholar
  44. [44]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
  45. [45]
    M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nucl. Phys. B 408 (1993) 345 [hep-th/9305010] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    M. Blau and G. Thompson, On diagonalization in map(M, G), Commun. Math. Phys. 171 (1995) 639 [hep-th/9402097] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    M. Blau and G. Thompson, Chern-Simons theory on S1-bundles: abelianisation and q-deformed Yang-Mills theory, JHEP 05 (2006) 003 [hep-th/0601068] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    M. Blau and G. Thompson, Chern-Simons theory on Seifert 3-manifolds, JHEP 09 (2013) 033 [arXiv:1306.3381] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].ADSGoogle Scholar
  50. [50]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    B. Willett and I. Yaakov, N = 2 dualities and Z extremization in three dimensions, arXiv:1104.0487 [INSPIRE].
  52. [52]
    F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    O. Aharony, IR duality in d = 3 N = 2 supersymmetric USp(2N c) and U(N c) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3D dualities from 4D dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  57. [57]
    D. Martelli and J. Sparks, The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere, Nucl. Phys. B 866 (2013) 72 [arXiv:1111.6930] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on three-manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. [59]
    A. Tanaka, Localization on round sphere revisited, JHEP 11 (2013) 103 [arXiv:1309.4992] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    L.F. Alday, M. Fluder and J. Sparks, The large N limit of M2-branes on lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    Y. Imamura and D. Yokoyama, S 3 /Z n partition function and dualities, JHEP 11 (2012) 122 [arXiv:1208.1404] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  62. [62]
    Y. Imamura, H. Matsuno and D. Yokoyama, Factorization of the S 3 /n partition function, Phys. Rev. D 89 (2014) 085003 [arXiv:1311.2371] [INSPIRE].ADSGoogle Scholar
  63. [63]
    C. Toldo and B. Willett, Partition functions on 3d circle bundles and their gravity duals, JHEP 05 (2018) 116 [arXiv:1712.08861] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. [64]
    S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
  65. [65]
    Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  66. [66]
    D. Martelli, A. Passias and J. Sparks, The supersymmetric NUTs and bolts of holography, Nucl. Phys. B 876 (2013) 810 [arXiv:1212.4618] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992) 563.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    R. Lawrence and L. Rozansky, Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Comm. Math. Phys. 205 (1999) 287.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys. 253 (2004) 25 [hep-th/0207096] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  72. [72]
    C. Beasley and E. Witten, Non-abelian localization for Chern-Simons theory, J. Diff. Geom. 70 (2005) 183 [hep-th/0503126] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  73. [73]
    C. Closset, H. Kim, V. Mikhaylov and B. Willett, Comments on supersymmetric Chern-Simons theory on Seifert manifolds, work in progress.Google Scholar
  74. [74]
    Y. Fan, Localization and non-renormalization in Chern-Simons theory, arXiv:1805.11076 [INSPIRE].
  75. [75]
    S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
  76. [76]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  77. [77]
    T. Dimofte, S. Gukov, J. Lenells and D. Zagier, Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Num. Theor. Phys. 3 (2009) 363 [arXiv:0903.2472] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  78. [78]
    S. Garoufalidis and R. Kashaev, Evaluation of state integrals at rational points, Commun. Num. Theor. Phys. 09 (2015) 549 [arXiv:1411.6062] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  79. [79]
    T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  80. [80]
    C. Closset, H. Kim and B. Willett, \( \mathcal{N}=1 \) supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  81. [81]
    F. Aprile and V. Niarchos, \( \mathcal{N}=2 \) supersymmetric field theories on 3-manifolds with A-type boundaries, JHEP 07 (2016) 126 [arXiv:1604.01561] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  82. [82]
    T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFTs, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  83. [83]
    M. Furuta and B. Steer, Seifert fibred homology 3-spheres and the yang-mills equations on riemann surfaces with marked points, Adv. Math. 96 (1992) 38.MathSciNetzbMATHCrossRefGoogle Scholar
  84. [84]
    I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. 42 (1956) 359.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  85. [85]
    I. Satake, The Gauss-Bonnet theorem for v-manifolds, J. Math. Soc. Japan 9 (1957) 492.MathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    T. Kawasaki, The Riemann-Roch theorem for complex v-manifolds, Osaka J. Math. 16 (1979) 151.MathSciNetzbMATHGoogle Scholar
  87. [87]
    M. Brunella, On transversely holomorphic flows I, Inv. Math. 126 (1996) 265.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  88. [88]
    É. Ghys, On transversely holomorphic flows II, Inv. Math. 126 (1996) 281.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  89. [89]
    H. Geiges and J. Gonzalo Pérez, Generalised spin structures on 2-dimensional orbifolds, Osaka J. Math. 49 (2012) 449.MathSciNetzbMATHGoogle Scholar
  90. [90]
    H. Geiges and C. Lange, Seifert fibrations of lens spaces, arXiv:1608.06844 .
  91. [91]
    A. Hatcher, Notes on basic 3-manifold topology, http://pi.math.cornell.edu/hatcher/3M/3M.pdf.
  92. [92]
    H. Nishi, SU(N)-Chern-Simons invariants of seifert fibered 3-manifolds, Int. J. Math. 09 (1998) 295.MathSciNetzbMATHCrossRefGoogle Scholar
  93. [93]
    C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  94. [94]
    K. Hori and D. Tong, Aspects of non-abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  95. [95]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  96. [96]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric abelian vortex loops in 2 + 1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3D supersymmetric theories, JHEP 07 (2014) 137 [arXiv:1211.3409] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    N. Dorey and D. Tong, Mirror symmetry and toric geometry in three-dimensional gauge theories, JHEP 05 (2000) 018 [hep-th/9911094] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  100. [100]
    P.-S. Hsin and N. Seiberg, Level/rank duality and Chern-Simons-matter theories, JHEP 09 (2016) 095 [arXiv:1607.07457] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  101. [101]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  102. [102]
    K. Intriligator and N. Seiberg, Aspects of 3D N = 2 Chern-Simons-matter theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  103. [103]
    V. Niarchos, Seiberg duality in Chern-Simons theories with fundamental and adjoint matter, JHEP 11 (2008) 001 [arXiv:0808.2771] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  104. [104]
    H. Kim and J. Park, Aharony dualities for 3D theories with adjoint matter, JHEP 06 (2013) 106 [arXiv:1302.3645] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  105. [105]
    S. Benvenuti and S. Pasquetti, 3D \( \mathcal{N}=2 \) mirror symmetry, pq-webs and monopole superpotentials, JHEP 08 (2016) 136 [arXiv:1605.02675] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  106. [106]
    F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2 + 1 dimensions, JHEP 08 (2017) 086 [arXiv:1703.08460] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  107. [107]
    S. Giacomelli and N. Mekareeya, Mirror theories of 3D \( \mathcal{N}=2 \) SQCD, JHEP 03 (2018) 126 [arXiv:1711.11525] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  108. [108]
    D. Jafferis and X. Yin, A duality appetizer, arXiv:1103.5700 [INSPIRE].
  109. [109]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  110. [110]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N}=2 \) gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  111. [111]
    K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  112. [112]
    L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291.MathSciNetzbMATHCrossRefGoogle Scholar
  113. [113]
    M. Brion and M. Vergne, Arrangements of hyperplanes I: rational functions and Jeffrey-Kirwan residue, math/9903178.
  114. [114]
    L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995) 249 [hep-th/9504111] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  115. [115]
    L.D. Faddeev and R.M. Kashaev, Quantum dilogarithm, Mod. Phys. Lett. A 9 (1994) 427 [hep-th/9310070] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  116. [116]
    N. Kurokawa, Multiple sine functions and selberg zeta functions, Proc. Japan Acad. Ser. A 67 (1991) 61.MathSciNetzbMATHCrossRefGoogle Scholar
  117. [117]
    S.N.M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997) 1069.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  118. [118]
    F. van de Bult, Hyperbolic hypergeometric functions, http://math.caltech.edu/∼vdbult/Thesis.pdf.
  119. [119]
    K. Hikami, Generalized volume conjecture and the A-polynomials: the Neumann-Zagier potential function as a classical limit of quantum invariant, J. Geom. Phys. 57 (2007) 1895 [math/0604094] [INSPIRE].
  120. [120]
    B. Willett, Localization on three-dimensional manifolds, J. Phys. A 50 (2017) 443006 [arXiv:1608.02958] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  121. [121]
    S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys. A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  122. [122]
    S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  123. [123]
    S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  124. [124]
    E. Witten, Supersymmetric index of three-dimensional gauge theory, hep-th/9903005 [INSPIRE].
  125. [125]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  126. [126]
    L.D. Faddeev, R.M. Kashaev and A.Yu. Volkov, Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality, Commun. Math. Phys. 219 (2001) 199 [hep-th/0006156] [INSPIRE].
  127. [127]
    S.K. Hansen and T. Takata, Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions, math/0209403.
  128. [128]
    S. Banerjee and B. Wilkerson, Lambert series and q-functions near q = 1, arXiv:1602.01085.

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Theory DepartmentCERNGeneva 23Switzerland
  2. 2.Mathematical InstituteUniversity of OxfordOxfordU.K.
  3. 3.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.

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