Advertisement

A topologically twisted index for three-dimensional supersymmetric theories

  • Francesco BeniniEmail author
  • Alberto Zaffaroni
Open Access
Regular Article - Theoretical Physics

Abstract

We provide a general formula for the partition function of three-dimensional \( \mathcal{N}=2 \) gauge theories placed on S2 ×S1 with a topological twist along S2, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields. The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes. The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries. We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities. Finally, our formula generalizes to Ω-backgrounds, as well as two-dimensional theories on S2 and four-dimensional theories on S2 × T 2. In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants.

Keywords

Supersymmetry and Duality Chern-Simons Theories Nonperturbative Effects Differential and Algebraic Geometry 

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]
    E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [INSPIRE].
  2. [2]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].zbMATHGoogle Scholar
  4. [4]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional \( \mathcal{N}=2 \) gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Cordova and S.-H. Shao, An Index Formula for Supersymmetric Quantum Mechanics, arXiv:1406.7853 [INSPIRE].
  9. [9]
    L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Brion and M. Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. École Norm. Sup. (4) 32 (1999) 715.Google Scholar
  11. [11]
    A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004) 453.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    O. Aharony, IR duality in d = 3 \( \mathcal{N}=2 \) supersymmetric USp(2N (c)) and U(N (c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Pasquetti, Factorisation of \( \mathcal{N}=2 \) Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    S. Cecotti, D. Gaiotto and C. Vafa, tt geometry in 3 and 4 dimensions, JHEP 05 (2014) 055 [arXiv:1312.1008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N}=\left(2,2\right) \) Gauge Theories on S2 and Vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Y. Yoshida, Factorization of 4d \( \mathcal{N}=1 \) superconformal index, arXiv:1403.0891 [INSPIRE].
  21. [21]
    W. Peelaers, Higgs branch localization of \( \mathcal{N}=1 \) theories on S3 × S1, JHEP 08 (2014) 060 [arXiv:1403.2711] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    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
  23. [23]
    A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    C. Closset and I. Shamir, The \( \mathcal{N}=1 \) Chiral Multiplet on T 2 × S2 and Supersymmetric Localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Nishioka and I. Yaakov, Generalized indices for \( \mathcal{N}=1 \) theories in four-dimensions, JHEP 12 (2014) 150 [arXiv:1407.8520] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, arXiv:1501.01310 [INSPIRE].
  27. [27]
    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].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    C. Closset and S. Cremonesi, Comments on \( \mathcal{N}=\left(2,\;2\right) \) supersymmetry on two-manifolds, JHEP 07 (2014) 075 [arXiv:1404.2636] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    A. Almuhairi and J. Polchinski, Magnetic AdS × R2: supersymmetry and stability, arXiv:1108.1213 [INSPIRE].
  33. [33]
    D. Kutasov and J. Lin, (0,2) Dynamics From Four Dimensions, Phys. Rev. D 89 (2014) 085025 [arXiv:1310.6032] [INSPIRE].ADSGoogle Scholar
  34. [34]
    A.N. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions, Phys. Rev. Lett. 52 (1984) 18 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    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
  36. [36]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of \( \mathcal{N}=2 \) supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    H.-C. Kao, K.-M. Lee and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories, Phys. Lett. B 373 (1996) 94 [hep-th/9506170] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  41. [41]
    E. Witten, Supersymmetric index of three-dimensional gauge theory, hep-th/9903005 [INSPIRE].
  42. [42]
    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    A. Kapustin and B. Willett, Wilson loops in supersymmetric Chern-Simons-matter theories and duality, arXiv:1302.2164 [INSPIRE].
  44. [44]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York U.S.A. (1997).CrossRefzbMATHGoogle Scholar
  45. [45]
    E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    D. Jafferis and X. Yin, A Duality Appetizer, arXiv:1103.5700 [INSPIRE].
  48. [48]
    C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    S. Kim, The Complete superconformal index for \( \mathcal{N}=6 \) Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
  52. [52]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    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
  54. [54]
    M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d \( \mathcal{N}=2 \) theories, PTEP 2014 (2014) 123B02 [arXiv:1312.3627] [INSPIRE].zbMATHGoogle Scholar
  55. [55]
    C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702 [INSPIRE].
  56. [56]
    E. Witten, Phases of \( \mathcal{N}=2 \) theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional \( \mathcal{N}=\left(2,2\right) \) Theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    F. Benini, D.S. Park and P. Zhao, Cluster algebras from dualities of 2d \( \mathcal{N}=\left(2,2\right) \) quiver gauge theories, arXiv:1406.2699 [INSPIRE].
  59. [59]
    J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, arXiv:1407.1852 [INSPIRE].
  60. [60]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    D.R. Morrison and M.R. Plesser, Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys. B 440 (1995) 279 [hep-th/9412236] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP (Nc) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    S. Hellerman, A. Henriques, T. Pantev, E. Sharpe and M. Ando, Cluster decomposition, T-duality and gerby CFTs, Adv. Theor. Math. Phys. 11 (2007) 751 [hep-th/0606034] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties, JHEP 11 (2012) 166 [arXiv:1205.3192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Two-Sphere Partition Functions and Gromov-Witten Invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  68. [68]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    D. Gaiotto, \( \mathcal{N}=2 \) dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  72. [72]
    F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and \( \mathcal{N}=2 \) superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and \( \mathcal{N}=1 \) dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  75. [75]
    S.L. Cacciatori and D. Klemm, Supersymmetric AdS4 black holes and attractors, JHEP 01 (2010) 085 [arXiv:0911.4926] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  76. [76]
    G. Dall’Agata and A. Gnecchi, Flow equations and attractors for black holes in \( \mathcal{N}=2 \) U(1) gauged supergravity, JHEP 03 (2011) 037 [arXiv:1012.3756] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    K. Hristov and S. Vandoren, Static supersymmetric black holes in AdS4 with spherical symmetry, JHEP 04 (2011) 047 [arXiv:1012.4314] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  78. [78]
    K. Hristov, A. Tomasiello and A. Zaffaroni, Supersymmetry on Three-dimensional Lorentzian Curved Spaces and Black Hole Holography, JHEP 05 (2013) 057 [arXiv:1302.5228] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    T.T. Wu and C.N. Yang, Dirac Monopole Without Strings: Monopole Harmonics, Nucl. Phys. B 107 (1976) 365 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  80. [80]
    K. Hori and M. Romo, Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary, arXiv:1308.2438 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Delta Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Blackett LaboratoryImperial College LondonLondonUnited Kingdom
  3. 3.Dipartimento di FisicaUniversità di Milano-BicoccaMilanoItaly
  4. 4.INFN, sezione di Milano-BicoccaMilanoItaly

Personalised recommendations