Abstract
We discuss in detail level/rank duality in three-dimensional Chern-Simons theories and various related dualities in three-dimensional Chern-Simons-matter theories. We couple the dual Lagrangians to appropriate background fields (including gauge fields, spin c connections and the metric). The non-trivial maps between the currents and the line operators in the dual theories is accounted for by mixing of these fields. In order for the duality to be valid we must add finite counterterms depending on these background fields. This analysis allows us to resolve a number of puzzles with these dualities, to provide derivations of some of them, and to find new consistency conditions and relations between them. In addition, we find new level/rank dualities of topological Chern-Simons theories and new dualities of Chern-Simons-matter theories, including new boson/boson and fermion/fermion dualities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Nakanishi and A. Tsuchiya, Level rank duality of WZW models in conformal field theory, Commun. Math. Phys. 144 (1992) 351 [INSPIRE].
F. Xu, Algebraic coset conformal field theories, Commun. Math. Phys. 211 (2000) 1 [math/9810035] [INSPIRE].
K.-H. Rehren, Algebraic Conformal QFT, in proceedings of The 3rd Meeting of the French-Italian Research Team on Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Salerno, Italy (2009).
C. Pauly, Strange duality revisited, Math. Res. Lett. 21 (2014) 1353 [INSPIRE].
V. Ostrik and M. Sun, Level-Rank Duality Via Tensor Categories, Commun. Math. Phys. 326 (2014) 49 [arXiv:1208.5131].
P. Goddard, A. Kent and D.I. Olive, Virasoro Algebras and Coset Space Models, Phys. Lett. B 152 (1985) 88 [INSPIRE].
G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group Level Duality in WZW Models and Chern-Simons Theory, Phys. Lett. B 246 (1990) 417 [INSPIRE].
E.J. Mlawer, S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352 (1991) 863 [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory and 2-D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].
M.R. Douglas, Chern-Simons-Witten theory as a topological Fermi liquid, hep-th/9403119 [INSPIRE].
E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, hep-th/9312104 [INSPIRE].
N. Seiberg and E. Witten, Gapped Boundary Phases of Topological Insulators via Weak Coupling, arXiv:1602.04251 [INSPIRE].
N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, arXiv:1606.01989 [INSPIRE].
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].
A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
J. Park and K.-J. Park, Seiberg-like Dualities for 3d N = 2 Theories with SU(N) gauge group, JHEP 10 (2013) 198 [arXiv:1305.6280] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, D = 3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, Correlation Functions of Large-N Chern-Simons-Matter Theories and Bosonization in Three Dimensions, JHEP 12 (2012) 028 [arXiv:1207.4593] [INSPIRE].
M.A. Vasiliev, Holography, Unfolding and Higher-Spin Theory, J. Phys. A 46 (2013) 214013 [arXiv:1203.5554] [INSPIRE].
S. Jain, S. Minwalla and S. Yokoyama, Chern Simons duality with a fundamental boson and fermion, JHEP 11 (2013) 037 [arXiv:1305.7235] [INSPIRE].
G. Gur-Ari and R. Yacoby, Three Dimensional Bosonization From Supersymmetry, JHEP 11 (2015) 013 [arXiv:1507.04378] [INSPIRE].
D. Radicevic, Disorder Operators in Chern-Simons-Fermion Theories, JHEP 03 (2016) 131 [arXiv:1511.01902] [INSPIRE].
O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 02 (2016) 093 [arXiv:1512.00161] [INSPIRE].
A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, arXiv:1606.01893 [INSPIRE].
J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and superconductors, arXiv:1606.01912 [INSPIRE].
E. Witten, SL(2, \( \mathrm{\mathbb{Z}} \)) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
C. Xu and Y.-Z. You, Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator, Phys. Rev. B 92 (2015) 220416 [arXiv:1510.06032] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
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].
M.A. Metlitski, S-duality of u(1) gauge theory with θ = π on non-orientable manifolds: Applications to topological insulators and superconductors, arXiv:1510.05663 [INSPIRE].
J.M. Maldacena, G.W. Moore and N. Seiberg, D-brane charges in five-brane backgrounds, JHEP 10 (2001) 005 [hep-th/0108152] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
L. Fidkowski, X. Chen and A. Vishwanath, Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model, Phys. Rev. X 3 (2013) 041016 [arXiv:1305.5851] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
L. Álvarez-Gaumé, S. Della Pietra and G.W. Moore, Anomalies and Odd Dimensions, Annals Phys. 163 (1985) 288 [INSPIRE].
E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
S. Kachru, M. Mulligan, G. Torroba and H. Wang, Mirror symmetry and the half-filled Landau level, Phys. Rev. B 92 (2015) 235105 [arXiv:1506.01376] [INSPIRE].
T. Senthil and M.P.A. Fisher, Competing orders, non-linear σ-models and topological terms in quantum magnets, Phys. Rev. B 74 (2006) 064405 [cond-mat/0510459] [INSPIRE].
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: 1607.07457
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hsin, PS., Seiberg, N. Level/rank duality and Chern-Simons-matter theories. J. High Energ. Phys. 2016, 95 (2016). https://doi.org/10.1007/JHEP09(2016)095
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2016)095