Advertisement

Twisted Hilbert spaces of 3d supersymmetric gauge theories

  • Mathew Bullimore
  • Andrea Ferrari
Open Access
Regular Article - Theoretical Physics
  • 38 Downloads

Abstract

We study aspects of 3d \( \mathcal{N}=2 \) supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry.

Keywords

Supersymmetric Gauge Theory Supersymmetry and Duality Duality in Gauge 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]
    M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and Vermas, arXiv:1609.04406 [INSPIRE].
  2. [2]
    F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].MathSciNetGoogle Scholar
  4. [4]
    C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    S. Cecotti, C. Cordova and C. Vafa, Braids, Walls and Mirrors, arXiv:1110.2115 [INSPIRE].
  12. [12]
    S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
  13. [13]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. Iengo and D.-p. Li, Quantum mechanics and quantum Hall effect on Riemann surfaces, Nucl. Phys. B 413 (1994) 735 [hep-th/9307011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Klevtsov, X. Ma, G. Marinescu and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349 (2017) 819 [arXiv:1510.06720] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. Tong, A Quantum Hall fluid of vortices, JHEP 02 (2004) 046 [hep-th/0306266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    D. Tong and C. Turner, Quantum Hall effect in supersymmetric Chern-Simons theories, Phys. Rev. B 92 (2015) 235125 [arXiv:1508.00580] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    K. Wong, Spectral sequences and vacua in \( \mathcal{N}=2 \) gauged linear quantum mechanics with potentials, JHEP 03 (2016) 150 [arXiv:1511.05159] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Gaiotto, S-duality of boundary conditions and the Geometric Langlands program, Proc. Symp. Pure Math. 98 (2018) 139 [arXiv:1609.09030] [INSPIRE].Google Scholar
  21. [21]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Witten, Holomorphic Morse Inequalities, Teubner-Texte (1984).Google Scholar
  23. [23]
    B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Elitzur, Y. Frishman, E. Rabinovici and A. Schwimmer, Origins of Global Anomalies in Quantum Mechanics, Nucl. Phys. B 273 (1986) 93 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K. Intriligator and N. Seiberg, Aspects of 3d N = 2 Chern-Simons-Matter Theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4-D SYM to 2-D σ-models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
  30. [30]
    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
  31. [31]
    B. Collie and D. Tong, The Dynamics of Chern-Simons Vortices, Phys. Rev. D 78 (2008) 065013 [arXiv:0805.0602] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    D. Tong and K. Wong, Vortices and Impurities, JHEP 01 (2014) 090 [arXiv:1309.2644] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    E. Arbarello, M. Cornalba, P. Griffiths and J.D. Harris, Geometry of Algebraic Curves, vol. 267, Springer-Verlag New York (1985).Google Scholar
  34. [34]
    D. Eriksson and N.M. Romão, Kähler quantization of vortex moduli, arXiv:1612.08505 [INSPIRE].
  35. [35]
    I. MacDonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M.Bullimore, A. Ferrari and H. Kim, Twisted Hilbert Spaces: Topological Vacua and Instanton Corrections, work in progress.Google Scholar
  37. [37]
    A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE].
  38. [38]
    M. Bullimore and A. Ferrari, Twisted Hilbert Spaces and Line Operators, work in progress.Google Scholar
  39. [39]
    B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
  41. [41]
    T. Okazaki and S. Yamaguchi, Supersymmetric Boundary Conditions in 3D \( \mathcal{N}=2 \) Theories, Proc. Symp. Pure Math. 88 (2014) 343.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFTs, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  43. [43]
    M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry and Symplectic Duality in 3d \( \mathcal{N}=4 \) Gauge Theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  45. [45]
    A.M. Jaffe and C.H. Taubes, Vortices and monopoles. Structure of static gauge theories, Birkhaeuser (1980) [INSPIRE].
  46. [46]
    O. Garcia-Prada, Invariant connections and vortices, Commun. Math. Phys. 156 (1993) 527 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press (2004) [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematics Department, Durham University, Science LaboratoriesDurhamU.K.
  2. 2.Mathematical InstituteUniversity of OxfordOxfordU.K.

Personalised recommendations