Abstract
We consider the dynamics of 2+1 dimensional SU(N) gauge theory with Chern-Simons level k and N f fundamental fermions. By requiring consistency with previously suggested dualities for N f ≤ 2k as well as the dynamics at k = 0 we propose that the theory with N f > 2k breaks the U(N f ) global symmetry spontaneously to U(N f /2 + k) × U(N f /2 − k). In contrast to the 3+1 dimensional case, the symmetry breaking takes place in a range of quark masses and not just at one point. The target space never becomes parametrically large and the Nambu-Goldstone bosons are therefore not visible semi-classically. Such symmetry breaking is argued to take place in some intermediate range of the number of flavors, 2k < N f < N∗(N, k), with the upper limit N∗ obeying various constraints. The Lagrangian for the Nambu-Goldstone bosons has to be supplemented by nontrivial Wess-Zumino terms that are necessary for the consistency of the picture, even at k = 0. Furthermore, we suggest two scalar dual theories in this range of N f . A similar picture is developed for SO(N) and Sp(N) gauge theories. It sheds new light on monopole condensation and confinement in the SO(N) & Spin(N) theories.
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References
N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
N. Seiberg, T. Senthil, C. Wang and E. Witten, A duality web in 2 + 1 dimensions and condensed matter physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].
P.-S. Hsin and N. Seiberg, Level/rank duality and Chern-Simons-Matter theories, JHEP 09 (2016) 095 [arXiv:1607.07457] [INSPIRE].
O. Aharony, F. Benini, P.-S. Hsin and N. Seiberg, Chern-Simons-Matter dualities with SO and U Sp gauge groups, JHEP 02 (2017) 072 [arXiv:1611.07874] [INSPIRE].
A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
C. Closset et al., Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset et al., Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
K. Intriligator and N. Seiberg, Aspects of 3D N = 2 Chern-Simons-Matter theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [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, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and superconductors, JHEP 05 (2017) 159 [arXiv:1606.01912] [INSPIRE].
D. Radicevic, D. Tong and C. Turner, Non-abelian 3D bosonization and quantum Hall states, JHEP 12 (2016) 067 [arXiv:1608.04732] [INSPIRE].
S. Kachru, M. Mulligan, G. Torroba and H. Wang, Bosonization and mirror symmetry, Phys. Rev. D 94 (2016) 085009 [arXiv:1608.05077] [INSPIRE].
S. Kachru, M. Mulligan, G. Torroba and H. Wang, Nonsupersymmetric dualities from mirror symmetry, Phys. Rev. Lett. 118 (2017) 011602 [arXiv:1609.02149] [INSPIRE].
A. Karch, B. Robinson and D. Tong, More abelian dualities in 2 + 1 dimensions, JHEP 01 (2017) 017 [arXiv:1609.04012] [INSPIRE].
M.A. Metlitski, A. Vishwanath and C. Xu, Duality and bosonization of (2 + 1)-dimensional Majorana fermions, Phys. Rev. B 95 (2017) 205137 [arXiv:1611.05049] [INSPIRE].
F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].
M.E. Peskin, Mandelstam ’t Hooft duality in abelian lattice models, Annals Phys. 113 (1978) 122 [INSPIRE].
C. Dasgupta and B.I. Halperin, Phase transition in a lattice model of superconductivity, Phys. Rev. Lett. 47 (1981) 1556 [INSPIRE].
M. Barkeshli and J. McGreevy, Continuous transition between fractional quantum Hall and superfluid states, Phys. Rev. B 89 (2014) 235116 [arXiv:1201.4393] [INSPIRE].
D.T. Son, Is the composite fermion a Dirac particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].
A.C. Potter, M. Serbyn and A. Vishwanath, Thermoelectric transport signatures of Dirac composite fermions in the half-filled Landau level, Phys. Rev. X 6 (2016) 031026 [arXiv:1512.06852] [INSPIRE].
C. Wang and T. Senthil, Composite Fermi liquids in the lowest Landau level, Phys. Rev. B 94 (2016) 245107 [arXiv:1604.06807] [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].
O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [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].
A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
B. Willett and I. Yaakov, N = 2 dualities and Z extremization in three dimensions, arXiv:1104.0487 [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3D Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [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, 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, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
S. Giombi and X. Yin, On higher spin gauge theory and the critical O(N) model, Phys. Rev. D 85 (2012) 086005 [arXiv:1105.4011] [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 et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [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].
S. Giombi and X. Yin, The higher spin/vector model duality, J. Phys. A 46 (2013) 214003 [arXiv:1208.4036] [INSPIRE].
O. Aharony, S. Giombi, G. Gur-Ari, J. Maldacena and R. Yacoby, The thermal free energy in large-N Chern-Simons-Matter theories, JHEP 03 (2013) 121 [arXiv:1211.4843] [INSPIRE].
S. Jain et al., Phases of large-N vector Chern-Simons theories on S 2 × S 1, JHEP 09 (2013) 009 [arXiv:1301.6169] [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].
S. Jain et al., Unitarity, crossing symmetry and duality of the S-matrix in large-N Chern-Simons theories with fundamental matter, JHEP 04 (2015) 129 [arXiv:1404.6373] [INSPIRE].
K. Inbasekar et al., Unitarity, crossing symmetry and duality in the scattering of \( \mathcal{N}=1 \) SUSY matter Chern-Simons theories, JHEP 10 (2015) 176 [arXiv:1505.06571] [INSPIRE].
S. Minwalla and S. Yokoyama, Chern-Simons bosonization along RG flows, JHEP 02 (2016) 103 [arXiv:1507.04546] [INSPIRE].
G. Gur-Ari, S.A. Hartnoll and R. Mahajan, Transport in Chern-Simons-Matter theories, JHEP 07 (2016) 090 [arXiv:1605.01122] [INSPIRE].
T. Appelquist and D. Nash, Critical behavior in (2 + 1)-dimensional QCD, Phys. Rev. Lett. 64 (1990) 721 [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4 , walls and dualities in 2 + 1 dimensions, arXiv:1708.06806 [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on abelian Higgs models and persistent order, arXiv:1705.04786 [INSPIRE].
C. Vafa and E. Witten, Restrictions on symmetry breaking in vector-like gauge theories, Nucl. Phys. B 234 (1984) 173 [INSPIRE].
C. Vafa and E. Witten, Eigenvalue inequalities for fermions in gauge theories, Commun. Math. Phys. 95 (1984) 257 [INSPIRE].
C. Vafa and E. Witten, Parity conservation in QCD, Phys. Rev. Lett. 53 (1984) 535 [INSPIRE].
D.K. Hong and H.-U. Yee, Holographic aspects of three dimensional QCD from string theory, JHEP 05 (2010) 036 [Erratum ibid. 08 (2010) 120] [arXiv:1003.1306] [INSPIRE].
D. Radicevic, Disorder operators in Chern-Simons-fermion theories, JHEP 03 (2016) 131 [arXiv:1511.01902] [INSPIRE].
E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
E. Witten, Current algebra, baryons and quark confinement, Nucl. Phys. B 223 (1983) 433 [INSPIRE].
D.S. Freed, Z. Komargodski and N. Seiberg, The sum over topological sectors and θ in the 2+1-dimensional ℂℙ1 σ-model, arXiv:1707.05448 [INSPIRE].
F. Wilczek and A. Zee, Linking numbers, spin and statistics of solitons, Phys. Rev. Lett. 51 (1983) 2250 [INSPIRE].
D.S. Freed, Pions and generalized cohomology, J. Diff. Geom. 80 (2008) 45 [hep-th/0607134] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
C. Cordova, P.-S. Hsin and N. Seiberg, Global symmetries, counterterms and duality in Chern-Simons matter theories with orthogonal gauge groups, arXiv:1711.10008 [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
Z. Komargodski, T. Sulejmanpasic and M. Unsal, Walls, anomalies and (de)confinement in quantum anti-ferromagnets, arXiv:1706.05731 [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].
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Komargodski, Z., Seiberg, N. A symmetry breaking scenario for QCD3. J. High Energ. Phys. 2018, 109 (2018). https://doi.org/10.1007/JHEP01(2018)109
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DOI: https://doi.org/10.1007/JHEP01(2018)109