Advertisement

(0,2) dualities and the 4-simplex

  • Tudor Dimofte
  • Natalie M. PaquetteEmail author
Open Access
Regular Article - Theoretical Physics
  • 38 Downloads

Abstract

We propose that a simple, Lagrangian 2d \( \mathcal{N} \) = (0, 2) duality interface between the 3d \( \mathcal{N} \) = 2 XYZ model and 3d \( \mathcal{N} \) = 2 SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundary and 2d \( \mathcal{N} \) = (0, 2) interfaces. In particular, we identify IR dualities of interfaces associated to local changes of 4d triangulation, governed by the (3), (2, 4), and (2, 4) Pachner moves. We check these dualities using supersymmetric half-indices. We also describe how to produce stand-alone 2d theories (as opposed to interfaces) capturing the geometry of 4-simplices and Pachner moves by making additional field-theoretic choices, and find in this context that the Pachner moves recover abelian \( \mathcal{N} \) = (0, 2) trialities of Gadde-Gukov-Putrov. Our work provides new, explicit tools to explore the interplay between 2d dualities and 4-manifold geometry that has been developed in recent years.

Keywords

Duality in Gauge Field Theories Supersymmetric Gauge Theory Supersymmetry and Duality 

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]
    D. Gaiotto, N = 2 dualities, JHEP08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys.299 (2010) 163 [arXiv:0807.4723] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
  4. [4]
    T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys.98 (2011) 225 [arXiv:1006.0977] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    Y. Terashima and M. Yamazaki, SL(2, ℝ) Chern-Simons, Liouville and gauge theory on duality walls, JHEP08 (2011) 135 [arXiv:1103.5748] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Cecotti, C. Cordova and C. Vafa, Braids, walls and mirrors, arXiv:1110.2115 [INSPIRE].
  8. [8]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP07 (2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
  11. [11]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys.74 (2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
  13. [13]
    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
  14. [14]
    B. Assel, S. Schäfer-Nameki and J.-M. Wong, M 5-branes on S 2 × M 4: Nahms equations and 4d topological σ-models, JHEP09 (2016) 120 [arXiv:1604.03606] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
  16. [16]
    M. Dedushenko, S. Gukov and P. Putrov, Vertex algebras and 4-manifold invariants, arXiv:1705.01645 [INSPIRE].
  17. [17]
    O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys.B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
  18. [18]
    T. Dimofte, D. Gaiotto and R. van der Veen, RG domain walls and hybrid triangulations, Adv. Theor. Math. Phys.19 (2015) 137 [arXiv:1304.6721] [INSPIRE].
  19. [19]
    T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFTs, JHEP05 (2018) 060 [arXiv:1712.07654] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Gadde, S. Gukov and P. Putrov, Walls, lines and spectral dualities in 3d gauge theories, JHEP05 (2014) 047 [arXiv:1302.0015] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    H.-J. Chung, T. Dimofte, S. Gukov and P. Sulkowski, 3d-3d correspondence revisited, JHEP04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
  23. [23]
    K. Costello, T. Dimofte and D. Gaiotto, Boundary chiral algebras, to appear.Google Scholar
  24. [24]
    M. Bullimore and A. Ferrari, Twisted Hilbert spaces of 3d supersymmetric gauge theories, JHEP08 (2018) 018 [arXiv:1802.10120] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Gadde, S. Gukov and P. Putrov, Duality defects, arXiv:1404.2929 [INSPIRE].
  26. [26]
    B. Assel and S. Schäfer-Nameki, Six-dimensional origin of \( \mathcal{N} \) = 4 SYM with duality defects, JHEP12 (2016) 058 [arXiv:1610.03663] [INSPIRE].
  27. [27]
    C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP04 (2017) 111 [arXiv:1612.05640] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    C. Lawrie, D. Martelli and S. Schäfer-Nameki, Theories of class F and anomalies, JHEP10 (2018) 090 [arXiv:1806.06066] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    D. Gaiotto and T. Okazaki, Dualities of corner configurations and supersymmetric indices, arXiv:1902.05175 [INSPIRE].
  30. [30]
    E.E. Moise, Affine structures in 3-manifolds: V. the triangulation theorem and hauptvermutung, Ann. Masth.56 (1952) 96.MathSciNetCrossRefGoogle Scholar
  31. [31]
    T. Radó, Uber den begriff der riemannschen fläche, Acta Litt. Sci. Szeged2 (1925) 101.zbMATHGoogle Scholar
  32. [32]
    M.H. Freedman et al., The topology of four-dimensional manifolds, J. Diff. Geom.17 (1982) 357.MathSciNetCrossRefGoogle Scholar
  33. [33]
    R.C. Kirby, L. Siebenmann and L. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations. Princeton University Press, Princeton U.S.A. (1977).CrossRefGoogle Scholar
  34. [34]
    J. Milnor et al., Differential topology forty-six years later, Not. A.M.S.58 (2011) 804.MathSciNetzbMATHGoogle Scholar
  35. [35]
    C. Manolescu, Pin (2)-equivariant Seiberg-Witten floer homology and the triangulation conjecture, J. Amer. Math. Soc.29 (2016) 147 [arXiv:1303.2354].MathSciNetCrossRefGoogle Scholar
  36. [36]
    U. Pachner, Plhomeomorphic manifolds are equivalent by elementary shellings, Eur. J. Comb.12 (1991) 129.CrossRefGoogle Scholar
  37. [37]
    J.S. Carter, L.H. Kauffman and M. Saito, Structures and diagrammatics of four dimensional topological lattice field theories, Adv. Math.146 (1999) 39 [math/9806023].
  38. [38]
    R. Kashaev, On realizations of Pachner moves in 4D, Journal of Knot Theory and Its Ramifications24 (2015) 1541002 [arXiv:1504.01979] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  39. [39]
    R. Kashaev, A simple model of 4d-TQFT, arXiv:1405.5763 [INSPIRE].
  40. [40]
    W.P. Thurston, The geometry and topology of three-manifolds, Princeton University Princeton, Princeton U.S.A. (1979).Google Scholar
  41. [41]
    J.S. Carter and M. Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs volume 55, American Mathematical Society, Providence U.S.A. (1998).Google Scholar
  42. [42]
    J.A. Hillman, Four-manifolds, geometries and knots, Geometry & Topology Monographs volume 5. Geometry & Topology Publications, Coventry U.K. (2002).Google Scholar
  43. [43]
    J.G. Ratcliffe and S.T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Exp. Math.9 (2000) 101.MathSciNetCrossRefGoogle Scholar
  44. [44]
    D.D. Long and A.W. Reid, On the geometric boundaries of hyperbolic 4-manifolds, math/0007197.
  45. [45]
    A. Kolpakov and B. Martelli, Hyperbolic four-manifolds with one cusp, Geom. Funct. Anal.23 (2013) 1903.MathSciNetCrossRefGoogle Scholar
  46. [46]
    B. Martelli, Hyperbolic four-manifolds, arXiv:1512.03661.
  47. [47]
    R. Budney, B. A. Burton and J. Hillman, Triangulating a Cappell-Shaneson knot complement, arXiv:1109.3899.
  48. [48]
    A. Issa, Triangulating cappell-shaneson homotopy 4-spheres, Master Thesis, University of Melbourne, Melbourne, Australia (2017).Google Scholar
  49. [49]
    S. Matveev, Algorithmic topology and classification of 3-manifolds, 2nd edition, Algorithms and Computation in Mathematics volume 3, Springer, Berlin Germny (2007).Google Scholar
  50. [50]
    R. Piergallini, Standard moves for standard polyhedra and spines, Rend. Circ. Mat. Palermo Suppl. (1988) 391.Google Scholar
  51. [51]
    R. Benedetti and C. Petronio, A finite graphic calculus for 3-manifolds, Manuscr. Math.8 (1995) 291.MathSciNetCrossRefGoogle Scholar
  52. [52]
    G. Amendola, A calculus for ideal triangulations of three-manifolds with embedded arcs, Math. Nachr.278 (2005) 975.MathSciNetCrossRefGoogle Scholar
  53. [53]
    J.H. Rubinstein, H. Segerman and S. Tillmann, Traversing three-manifold triangulations and spines, [arXiv:1812.02806].
  54. [54]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  55. [55]
    S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
  56. [56]
    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys.129 (1990) 393 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    I. Brunner, J. Schulz and A. Tabler, Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models, JHEP06 (2019) 046 [arXiv:1904.07258] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    M. Roček, K. Roumpedakis and S. Seifnashri, 3D dualities and supersymmetry enhancement from domain walls, arXiv:1904.02722 [INSPIRE].
  59. [59]
    H. Jockers and P. Mayr, A 3D gauge theory/quantum k-theory correspondence, arXiv:1808.02040 [INSPIRE].
  60. [60]
    N.P. Warner, Supersymmetric, integrable boundary field theories, Nucl. Phys. Proc. Suppl.45A (1996) 154 [hep-th/9512183] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  62. [62]
    Y. Yoshida and K. Sugiyama, Localization of 3d \( \mathcal{N} \) = 2 supersymmetric theories on S 1 × D 2, arXiv:1409.6713 [INSPIRE].
  63. [63]
    Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
  64. [64]
    A. Kapustin and B. Willett, Generalized superconformal index for three dimensional field theories, arXiv:1106.2484 [INSPIRE].
  65. [65]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys.104 (2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    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].
  67. [67]
    W. Jaco and J.H. Rubinstein, Layered-triangulations of 3-manifolds, math/0603601.
  68. [68]
    C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys.B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
  69. [69]
    B. Feigin and S. Gukov, VOA[M4], arXiv:1806.02470 [INSPIRE].
  70. [70]
    H. Nakajima, Instantons and affine Lie algebras, Nucl. Phys. Proc. Suppl.46 (1996) 154 [alg-geom/9510003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  71. [71]
    I. Grojnowski, Instantons and affine algebras I: the Hilbert scheme and vertex operators, alg-geom/9506020 [INSPIRE].
  72. [72]
    T. Dimofte, M. Gabella and A.B. Goncharov, K-decompositions and 3d gauge theories, JHEP11 (2016) 151 [arXiv:1301.0192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    D. Gang and K. Yonekura, Symmetry enhancement and closing of knots in 3d/3d correspondence, JHEP07 (2018) 145 [arXiv:1803.04009] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    A. Gadde and S. Gukov, 2d index and surface operators, JHEP03 (2014) 080 [arXiv:1305.0266] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP)University of CaliforniaDavisU.S.A.
  2. 2.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.

Personalised recommendations