Abstract
By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M3. This generalization is applicable to both the 3d \( \mathcal{N} \) = 2 and \( \mathcal{N} \) = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M3. This is carried out in detail for M3 a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M3, which matches the Witten index computation that takes the higher-form symmetries into account.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
Y. Terashima and M. Yamazaki, SL(2, ℝ) Chern-Simons, Liouville and Gauge Theory on Duality Walls, JHEP08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
S. Cecotti, C. Cordova and C. Vafa, Braids, Walls and Mirrors, arXiv:1110.2115 [INSPIRE].
J. Eckhard, S. Schäfer-Nameki and J.-M. Wong, An \( \mathcal{N} \) = 1 3d-3d Correspondence, JHEP07 (2018) 052 [arXiv:1804.02368] [INSPIRE].
L.F. Alday, P. Benetti Genolini, M. Bullimore and M. van Loon, Refined 3d-3d Correspondence, JHEP04 (2017) 170 [arXiv:1702.05045] [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys.355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
E. Witten, Supersymmetric index of three-dimensional gauge theory, hep-th/9903005 [INSPIRE].
C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP03 (2017) 074 [arXiv:1701.03171] [INSPIRE].
C. Closset, H. Kim and B. Willett, Seifert fibering operators in 3d \( \mathcal{N} \) = 2 theories, JHEP11 (2018) 004 [arXiv:1807.02328] [INSPIRE].
C. Closset and H. Kim, Three-dimensional \( \mathcal{N} \) = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review, Int. J. Mod. Phys.A 34 (2019) 1930011 [arXiv:1908.08875] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP12 (1998) 012 [hep-th/9812012] [INSPIRE].
D.S. Freed and C. Teleman, Relative quantum field theory, Commun. Math. Phys.326 (2014) 459 [arXiv:1212.1692] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, Invent. Math.3 (1967) 308.
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II, Invent. Math.4 (1967) 87.
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, Prog. Math.319 (2016) 155 [arXiv:1306.4320] [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
D. Pei, 3d-3d Correspondence for Seifert Manifolds, Ph.D. Thesis, Caltech (2016) [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
M. Dedushenko, S. Gukov, H. Nakajima, D. Pei and K. Ye, 3d TQFTs from Argyres-Douglas theories, arXiv:1809.04638 [INSPIRE].
M.C.N. Cheng, S. Chun, F. Ferrari, S. Gukov and S.M. Harrison, 3d Modularity, JHEP10 (2019) 010 [arXiv:1809.10148] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys.13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
P.-S. Hsin, H.T. Lam and N. Seiberg, Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d, SciPost Phys.6 (2019) 039 [arXiv:1812.04716] [INSPIRE].
D. Jafferis and X. Yin, A Duality Appetizer, arXiv:1103.5700 [INSPIRE].
S.S. Razamat and B. Willett, Star shaped quivers with flux, to appear.
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl.192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/Gauge correspondence on curved spaces, JHEP01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math.96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
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].
B. Assel, S. Schäfer-Nameki and J.-M. Wong, M5-branes on S2 × M4: Nahm’s equations and 4d topological σ-models, JHEP09 (2016) 120 [arXiv:1604.03606] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, 4-manifolds and topological modular forms, arXiv:1811.07884 [INSPIRE].
H.-J. Chung, T. Dimofte, S. Gukov and P. Sułkowski, 3d-3d Correspondence Revisited, JHEP04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys.98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
P. Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol. 291, Springer-Verlag, Berlin-New York (1972).
H.M. Pedersen, Splice diagram determining singularity links and universal abelian covers, Geom. Dedicata150 (2011) 75.
M. Scharlemann, Heegaard splittings of compact 3-manifolds, in Handbook of geometric topology, pp. 921–953, North-Holland, Amsterdam (2002).
N. Saveliev, Lectures on the topology of 3-manifolds, De Gruyter Textbook, Walter de Gruyter & Co., Berlin (1999) [DOI].
D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The theta-Angle in N = 4 Super Yang-Mills Theory, JHEP06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys.135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
F. Aprile, S. Pasquetti and Y. Zenkevich, Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles, JHEP04 (2019) 138 [arXiv:1812.08142] [INSPIRE].
A. Kapustin, H. Kim and J. Park, Dualities for 3d Theories with Tensor Matter, JHEP12 (2011) 087 [arXiv:1110.2547] [INSPIRE].
A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory, Nucl. Phys.B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
B. Willett and I. Yaakov, N = 2 Dualities and Z Extremization in Three Dimensions, arXiv:1104.0487 [INSPIRE].
D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP05 (2013) 126 [arXiv:1304.0779] [INSPIRE].
J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry and supersymmetric gauge theories, Lett. Math. Phys.104 (2014) 527 [arXiv:1202.4698] [INSPIRE].
F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality in (2 + 1)d, JHEP04 (2017) 135 [arXiv:1702.07035] [INSPIRE].
D. Gang and K. Yonekura, Symmetry enhancement and closing of knots in 3d/3d correspondence, JHEP07 (2018) 145 [arXiv:1803.04009] [INSPIRE].
B. Willett, Higher form symmetries in 3d N = 2 gauge theories, to appear.
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
S. Monnier, The anomaly field theories of six-dimensional (2, 0) superconformal theories, Adv. Theor. Math. Phys.22 (2018) 2035 [arXiv:1706.01903] [INSPIRE].
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys.22 (1997) 103 [hep-th/9610234] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, The Uncertainty of Fluxes, Commun. Math. Phys.271 (2007) 247 [hep-th/0605198] [INSPIRE].
S.S. Razamat and B. Willett, Down the rabbit hole with theories of class \( \mathcal{S} \) , JHEP10 (2014) 099 [arXiv:1403.6107] [INSPIRE].
D. Pei and K. Ye, A 3d-3d appetizer, JHEP11 (2016) 008 [arXiv:1503.04809] [INSPIRE].
F. Benini, D. Gang and L.A. Pando Zayas, Rotating Black Hole Entropy from M5 Branes, arXiv:1909.11612 [INSPIRE].
S. Gukov, Gauge theory and knot homologies, Fortsch. Phys.55 (2007) 473 [arXiv:0706.2369] [INSPIRE].
B.S. Acharya and C. Vafa, On domain walls of N = 1 supersymmetric Yang-Mills in four-dimensions, hep-th/0103011 [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
D. Xie, M5 brane and four dimensional N = 1 theories I, JHEP04 (2014) 154 [arXiv:1307.5877] [INSPIRE].
A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris257 (1963) 7.
V. Bashmakov, J. Gomis, Z. Komargodski and A. Sharon, Phases of \( \mathcal{N} \) = 1 theories in 2 + 1 dimensions, JHEP07 (2018) 123 [arXiv:1802.10130] [INSPIRE].
J. Milnor, On the 3-dimensional Brieskorn manifolds M (p, q, r), in Knots, groups, and 3-manifolds (Papers dedicated to the memory of R.H. Fox), L.P. Neuwirth ed., Princeton University Press (1975).
M.J. Hopkins, Quadratic and bilinear forms, Harvard University, Lecture Notes (2015).
G.W. Mackey, Graphs, singularities, and finite groups, Usp. Mat. Nauk38 (1983) 159.
W. Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I. (1980).
L. Ben Abdelghani and S. Boyer, A calculation of the Culler-Shalen seminorms associated to small Seifert Dehn fillings, Proc. Lond. Math. Soc.83 (2001) 235.
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1910.14086
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Eckhard, J., Kim, H., Schäfer-Nameki, S. et al. Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence. J. High Energ. Phys. 2020, 101 (2020). https://doi.org/10.1007/JHEP01(2020)101
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2020)101