Abstract
Many examples of low-energy dualities have been found in supersymmetric gauge theories with four supercharges, both in four and in three space-time dimensions. In these dualities, two theories that are different at high energies have the same low-energy limit. In this paper we clarify the relation between the dualities in four and in three dimensions. We show that every four dimensional duality gives rise to a three dimensional duality between theories that are similar, but not identical, to the dimensional reductions of the four dimensional dual gauge theories to three dimensions. From these specific three dimensional dualities one can flow to many other low-energy dualities, including known three dimensional dualities and many new ones. We discuss in detail the case of three dimensional SU(N c) supersymmetric QCD theories, showing how to derive new duals for these theories from the four dimensional duality.
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References
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].
V. Niarchos, Seiberg dualities and the 3d/4d connection, JHEP 07 (2012) 075 [arXiv:1205.2086] [INSPIRE].
K.A. Intriligator and N. Seiberg, Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(N (c)) gauge theories, Nucl. Phys. B 444 (1995) 125 [hep-th/9503179] [INSPIRE].
I. Affleck, J.A. Harvey and E. Witten, Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].
O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2N (c)) and U(N (c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP (N (c)) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].
A. Karch, Seiberg duality in three-dimensions, Phys. Lett. B 405 (1997) 79 [hep-th/9703172] [INSPIRE].
S. Elitzur, A. Giveon and D. Kutasov, Branes and N = 1 duality in string theory, Phys. Lett. B 400 (1997) 269 [hep-th/9702014] [INSPIRE].
S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, Brane dynamics and N =1 supersymmetric gauge theory, Nucl. Phys. B 505(1997) 202[hep-th/9704104] [INSPIRE].
A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
F. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
V. Spiridonov and G. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots and vortices, arXiv:1107.5788 [INSPIRE].
V. Spiridonov and G. Vartanov, Elliptic Hypergeometry of Supersymmetric Dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].
V. Spiridonov and G. Vartanov, Superconformal indices for N = 1 theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [INSPIRE].
R. Eager, J. Schmude and Y. Tachikawa, Superconformal Indices, Sasaki-Einstein Manifolds and Cyclic Homologies, arXiv:1207.0573 [INSPIRE].
F. Dolan, V. Spiridonov and G. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
A. Gadde and W. Yan, Reducing the 4d Index to the S 3 Partition Function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
D. Kutasov and A. Schwimmer, On duality in supersymmetric Yang-Mills theory, Phys. Lett. B 354 (1995) 315 [hep-th/9505004] [INSPIRE].
D. Kutasov, A. Schwimmer and N. Seiberg, Chiral rings, singularity theory and electric-magnetic duality, Nucl. Phys. B 459 (1996) 455 [hep-th/9510222] [INSPIRE].
K. Intriligator and N. Seiberg, Aspects of 3d N = 2 Chern-Simons-Matter Theories, arXiv:1305.1633 [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact Terms, Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
B.R. Safdi, I.R. Klebanov and J. Lee, A Crack in the Conformal Window, arXiv:1212.4502 [INSPIRE].
O. Aharony and I. Shamir, On O(N c) D = 3 N = 2 supersymmetric QCD Theories, JHEP 12 (2011) 043 [arXiv:1109.5081] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, arXiv:1305.0318 [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, arXiv:1307.0511 [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
C. Hwang, K.-J. Park and J. Park, Evidence for Aharony duality for orthogonal gauge groups, JHEP 11 (2011) 011 [arXiv:1109.2828] [INSPIRE].
D. Kutasov, A Comment on duality in N = 1 supersymmetric nonAbelian gauge theories, Phys. Lett. B 351 (1995) 230 [hep-th/9503086] [INSPIRE].
V. Niarchos, Seiberg Duality in Chern-Simons Theories with Fundamental and Adjoint Matter, JHEP 11 (2008) 001 [arXiv:0808.2771] [INSPIRE].
V. Niarchos, R-charges, Chiral Rings and RG Flows in Supersymmetric Chern-Simons-Matter Theories, JHEP 05 (2009) 054 [arXiv:0903.0435] [INSPIRE].
T. Morita and V. Niarchos, F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories, Nucl. Phys. B 858 (2012) 84 [arXiv:1108.4963] [INSPIRE].
A. Kapustin, H. Kim and J. Park, Dualities for 3d Theories with Tensor Matter, JHEP 12 (2011) 087 [arXiv:1110.2547] [INSPIRE].
H. Kim and J. Park, Aharony Dualities for 3d Theories with Adjoint Matter, JHEP 06 (2013) 106 [arXiv:1302.3645] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, work in progress.
K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N =(2,2) Theories, JHEP 05(2007) 079 [hep-th/0609032] [INSPIRE].
K. Hori, Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories, arXiv:1104.2853 [INSPIRE].
A. Gadde and S. Gukov, 2d Index and Surface operators, arXiv:1305.0266 [INSPIRE].
J. Park and K.-J. Park, Seiberg-like Dualities for 3d N = 2 Theories with SU(N) gauge group, arXiv:1305.6280 [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
A. Niemi and G. Semenoff, Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].
A. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
B. Zupnik and D. Pak, Topologically massive gauge theories in superspace, Sov. Phys. J. 31 (1988) 962 [INSPIRE].
E. Ivanov, Chern-Simons matter systems with manifest N = 2 supersymmetry, Phys. Lett. B 268 (1991) 203 [INSPIRE].
A.M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857 [hep-th/9402044] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
J. de Boer, K. Hori, Y. Oz and Z. Yin, Branes and mirror symmetry in N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 502 (1997) 107 [hep-th/9702154] [INSPIRE].
I. Shamir, Aspects of three dimensional Seiberg duality, MSc Thesis submitted to the Weizmann Institute of Science, April 2010.
A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
E. Witten, Supersymmetric index of three-dimensional gauge theory, hep-th/9903005 [INSPIRE].
S.G. Naculich, H. Riggs and H. Schnitzer, Group level duality in WZW models and Chern-Simons theory, Phys. Lett. B 246 (1990) 417 [INSPIRE].
M. Camperi, F. Levstein and G. Zemba, The Large-N limit of Chern-Simons gauge theory, Phys. Lett. B 247 (1990) 549 [INSPIRE].
E. Mlawer, S.G. Naculich, H. Riggs and H. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352 (1991) 863 [INSPIRE].
T. Nakanishi and A. Tsuchiya, Level rank duality of WZW models in conformal field theory, Commun. Math. Phys. 144 (1992) 351 [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].
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. Kapustin, Three-dimensional Avatars of Seiberg Duality, talk given at Simons Summer Workshop in Mathematics and Physics 2011, http://media.scgp.stonybrook.edu/video/video.php?f=20110810 1 qtp.mp4.
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
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].
Y. Imamura and D. Yokoyama, N=2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
N. Seiberg, Modifying the Sum Over Topological Sectors and Constraints on Supergravity, JHEP 07 (2010) 070 [arXiv:1005.0002] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
F. van de Bult, Hyperbolic Hypergeometric Functions, Ph.D. Thesis, University of Amsterdam (2007) [http://www.its.caltech.edu/˜vdbult/Thesis.pdf].
G. Felder and A. Varchenko, The elliptic gamma function and SL(3, \( \mathbb{Z} \)) × Z 3, Adv. Math. 156 (2000) 44 [math/9907061].
V. Spiridonov and G. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP 06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
M. Cvetič, T.W. Grimm and D. Klevers, Anomaly Cancellation And Abelian Gauge Symmetries In F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].
B. Willett and I. Yaakov, N=2 Dualities and Z Extremization in Three Dimensions, arXiv:1104.0487 [INSPIRE].
A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for Superconformal Field Theories in 3,5 and 6 Dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
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].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, arXiv:1112.5179 [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
C. Hwang, H.-C. Kim and J. Park, Factorization of the 3d superconformal index, arXiv:1211.6023 [INSPIRE].
C. Krattenthaler, V. Spiridonov and G. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [INSPIRE].
Y. Imamura and D. Yokoyama, S 3 /Z n partition function and dualities, JHEP 11 (2012) 122 [arXiv:1208.1404] [INSPIRE].
C. Hwang, H. Kim, K.-J. Park and J. Park, Index computation for 3d Chern-Simons matter theory: test of Seiberg-like duality, JHEP 09 (2011) 037 [arXiv:1107.4942] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
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Aharony, O., Razamat, S.S., Seiberg, N. et al. 3d dualities from 4d dualities. J. High Energ. Phys. 2013, 149 (2013). https://doi.org/10.1007/JHEP07(2013)149
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DOI: https://doi.org/10.1007/JHEP07(2013)149