Abstract
A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling τ in the upper-half plane and by the choice of the CFT in the decoupling limit τ → ∞. Upon performing an SL(2, ℤ) transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten’s SL(2, ℤ) action [1]. In particular the cusps on the real τ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk S transformation, and it also admits a decoupling limit which gives the O(2) model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an S-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the O(2) model. We also consider examples with other theories on the boundary, such as large-Nf Dirac fermions — for which the extrapolation to strong coupling can be done exactly order-by-order in 1/Nf — and a free complex scalar.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [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].
S. Teber, Electromagnetic current correlations in reduced quantum electrodynamics, Phys. Rev. D 86 (2012) 025005 [arXiv:1204.5664] [INSPIRE].
A.V. Kotikov and S. Teber, Note on an application of the method of uniqueness to reduced quantum electrodynamics, Phys. Rev. D 87 (2013) 087701 [arXiv:1302.3939] [INSPIRE].
S. Teber and A.V. Kotikov, Interaction corrections to the minimal conductivity of graphene via dimensional regularization, EPL 107 (2014) 57001 [arXiv:1407.7501] [INSPIRE].
S. Teber and A.V. Kotikov, The method of uniqueness and the optical conductivity of graphene: New application of a powerful technique for multiloop calculations, Theor. Math. Phys. 190 (2017) 446 [arXiv:1602.01962] [INSPIRE].
A.V. Kotikov and S. Teber, Critical behaviour of reduced QED 4,3 and dynamical fermion gap generation in graphene, Phys. Rev. D 94 (2016) 114010 [arXiv:1610.00934] [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary conformal field theory and a boundary central charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
D. Dudal, A.J. Mizher and P. Pais, Exact quantum scale invariance of three-dimensional reduced QED theories, Phys. Rev. D 99 (2019) 045017 [arXiv:1808.04709] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
L. Di Pietro, Z. Komargodski, I. Shamir and E. Stamou, Quantum electrodynamics in d = 3 from the ϵ expansion, Phys. Rev. Lett. 116 (2016) 131601 [arXiv:1508.06278] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QED d , F-theorem and the ϵ expansion, J. Phys. A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
S.M. Chester, M. Mezei, S.S. Pufu and I. Yaakov, Monopole operators from the 4 − ϵ expansion, JHEP 12 (2016) 015 [arXiv:1511.07108] [INSPIRE].
L. Janssen and Y.-C. He, Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality, Phys. Rev. B 96 (2017) 205113 [arXiv:1708.02256] [INSPIRE].
L. Di Pietro and E. Stamou, Scaling dimensions in QED 3 from the ϵ-expansion, JHEP 12 (2017) 054 [arXiv:1708.03740] [INSPIRE].
L. Di Pietro and E. Stamou, Operator mixing in the ϵ-expansion: Scheme and evanescent-operator independence, Phys. Rev. D 97 (2018) 065007 [arXiv:1708.03739] [INSPIRE].
Y. Ji and A.N. Manashov, Operator mixing in fermionic CFTs in noninteger dimensions, Phys. Rev. D 98 (2018) 105001 [arXiv:1809.00021] [INSPIRE].
N. Zerf, P. Marquard, R. Boyack and J. Maciejko, Critical behavior of the QED 3 -Gross-Neveu-Yukawa model at four loops, Phys. Rev. B 98 (2018) 165125 [arXiv:1808.00549] [INSPIRE].
S. Giombi, G. Tarnopolsky and I.R. Klebanov, On C J and C T in Conformal QED, JHEP 08 (2016) 156 [arXiv:1602.01076] [INSPIRE].
S.M. Chester, L.V. Iliesiu, M. Mezei and S.S. Pufu, Monopole Operators in U (1) Chern-Simons-Matter Theories, JHEP 05 (2018) 157 [arXiv:1710.00654] [INSPIRE].
J.A. Gracey, Fermion bilinear operator critical exponents at O(1/N 2) in the QED-Gross-Neveu universality class, Phys. Rev. D 98 (2018) 085012 [arXiv:1808.07697] [INSPIRE].
S. Benvenuti and H. Khachatryan, QED’s in 2+1 dimensions: complex fixed points and dualities, arXiv:1812.01544 [INSPIRE].
R. Boyack, A. Rayyan and J. Maciejko, Deconfined criticality in the QED 3 -Gross-Neveu-Yukawa model: the 1/N expansion revisited, arXiv:1812.02720.
R. Boyack et al., Transition between algebraic and ℤ2 quantum spin liquids at large n, Phys. Rev. B 98 (2018) 035137.
S. Benvenuti and H. Khachatryan, Easy-plane QED 3’s in the large N f limit, arXiv:1902.05767.
J.A. Gracey, Large N f quantum field theory, Int. J. Mod. Phys. A 33 (2019) 1830032 [arXiv:1812.05368] [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].
M.A. Metlitski and A. Vishwanath, Particle-vortex duality of two-dimensional Dirac fermion from electric-magnetic duality of three-dimensional topological insulators, Phys. Rev. B 93 (2016) 245151 [arXiv:1505.05142] [INSPIRE].
C. Wang and T. Senthil, Dual Dirac liquid on the surface of the electron topological insulator, Phys. Rev. X 5 (2015) 041031 [arXiv:1505.05141] [INSPIRE].
W.-H. Hsiao and D.T. Son, Duality and universal transport in mixed-dimension electrodynamics, Phys. Rev. B 96 (2017) 075127 [arXiv:1705.01102] [INSPIRE].
W.-H. Hsiao and D.T. Son, Self-dual ν = 1 bosonic quantum Hall state in mixed dimensional QED, arXiv:1809.06886 [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].
B. Rosenstein, B. Warr and S.H. Park, Dynamical symmetry breaking in four Fermi interaction models, Phys. Rept. 205 (1991) 59 [INSPIRE].
J. Zinn-Justin, Four fermion interaction near four-dimensions, Nucl. Phys. B 367 (1991) 105 [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].
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 11 (2009) 006 [arXiv:0904.0840] [INSPIRE].
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
C. Closset et al., Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal invariance in the long-range Ising model, Nucl. Phys. B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the long-range to short-range crossover and an infrared duality, J. Phys. A 50 (2017) 354002 [arXiv:1703.05325] [INSPIRE].
C. Behan, L. Rastelli, S. Rychkov and B. Zan, Long-range critical exponents near the short-range crossover, Phys. Rev. Lett. 118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
A. Karch and Y. Sato, Conformal manifolds with boundaries or defects, JHEP 07 (2018) 156 [arXiv:1805.10427] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, arXiv:1705.04278 [INSPIRE].
A. Bissi, T. Hansen and A. Söderberg, Analytic bootstrap for boundary CFT, JHEP 01 (2019) 010 [arXiv:1808.08155].
D. Mazáč, L. Rastelli and X. Zhou, An analytic approach to BCFT d, arXiv:1812.09314.
A. Kaviraj and M.F. Paulos, The functional bootstrap for boundary CFT, arXiv:1812.04034.
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [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].
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].
D.T. Son, Is the composite fermion a Dirac particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].
M. Baggio et al., Decoding a three-dimensional conformal manifold, JHEP 02 (2018) 062 [arXiv:1712.02698] [INSPIRE].
C. Herzog, K.-W. Huang and K. Jensen, Displacement operators and constraints on boundary central charges, Phys. Rev. Lett. 120 (2018) 021601 [arXiv:1709.07431] [INSPIRE].
C.P. Herzog, K.-W. Huang, I. Shamir and J. Virrueta, Superconformal Models for Graphene and Boundary Central Charges, JHEP 09 (2018) 161 [arXiv:1807.01700] [INSPIRE].
V.S. Alves, M. Gomes, S.V.L. Pinheiro and A.J. da Silva, The perturbative gross neveu model coupled to a Chern-Simons field: a renormalization group study, Phys. Rev. D 59 (1999) 045002 [hep-th/9810106] [INSPIRE].
W. Chen, G.W. Semenoff and Y.-S. Wu, Two loop analysis of nonAbelian Chern-Simons theory, Phys. Rev. D 46 (1992) 5521 [hep-th/9209005] [INSPIRE].
W. Chen, M.P.A. Fisher and Y.-S. Wu, Mott transition in an anyon gas, Phys. Rev. B 48 (1993) 13749 [cond-mat/9301037] [INSPIRE].
V.P. Spiridonov and F.V. Tkachov, Two loop contribution of massive and massless fields to the Abelian Chern-Simons term, Phys. Lett. B 260 (1991) 109 [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
A. Sen, S-duality improved superstring perturbation theory, JHEP 11 (2013) 029 [arXiv:1304.0458] [INSPIRE].
C. Beem, L. Rastelli, A. Sen and B.C. van Rees, Resummation and S-duality in N = 4 SYM, JHEP 04 (2014) 122 [arXiv:1306.3228] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
H. Kleinert et al., Five loop renormalization group functions of O(n) symmetric ϕ 4 theory and ϵ-expansions of critical exponents up to ϵ 5, Phys. Lett. B 272 (1991) 39 [Erratum ibid. B 319 (1993) 545] [hep-th/9503230] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Generalized F -theorem and the ϵ expansion, JHEP 12 (2015) 155 [arXiv:1507.01960] [INSPIRE].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement entropy of 3D conformal gauge theories with many flavors, JHEP 05 (2012) 036 [arXiv:1112.5342] [INSPIRE].
J. Braun, H. Gies, L. Janssen and D. Roscher, Phase structure of many-flavor QED 3, Phys. Rev. D 90 (2014) 036002 [arXiv:1404.1362] [INSPIRE].
S. Gukov, RG flows and bifurcations, Nucl. Phys. B 919 (2017) 583 [arXiv:1608.06638] [INSPIRE].
D.B. Kaplan, J.-W. Lee, D.T. Son and M.A. Stephanov, Conformality lost, Phys. Rev. D 80 (2009) 125005 [arXiv:0905.4752] [INSPIRE].
V. Gorbenko, S. Rychkov and B. Zan, Walking, weak first-order transitions and complex CFTs, JHEP 10 (2018) 108 [arXiv:1807.11512] [INSPIRE].
Z. Li, Solving QED 3 with conformal bootstrap, arXiv:1812.09281 [INSPIRE].
S. Giombi et al., 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].
G. Gur-Ari and R. Yacoby, Correlators of large N fermionic Chern-Simons vector models, JHEP 02 (2013) 150 [arXiv:1211.1866] [INSPIRE].
O. Aharony, S. Jain and S. Minwalla, Flows, fixed points and duality in Chern-Simons-Matter theories, JHEP 12 (2018) 058 [arXiv:1808.03317] [INSPIRE].
A. Dey et al., Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories, JHEP 11 (2018) 020 [arXiv:1808.04415] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Yukawa CFTs and emergent supersymmetry, PTEP 2016 (2016) 12C105 [arXiv:1607.05316] [INSPIRE].
T. Senthil et al., Quantum criticality beyond the landau-ginzburg-wilson paradigm, Phys. Rev. B 70 (2004) 144407.
P.-S. Hsin and N. Seiberg, Level/rank duality and Chern-Simons-Matter theories, JHEP 09 (2016) 095 [arXiv:1607.07457] [INSPIRE].
C. Córdova, P.-S. Hsin and N. Seiberg, Time-reversal symmetry, anomalies and dualities in (2 + 1)d, SciPost Phys. 5 (2018) 006 [arXiv:1712.08639] [INSPIRE].
Y.Q. Qin et al., Duality between the deconfined quantum-critical point and the bosonic topological transition, Phys. Rev. X 7 (2017) 031052 [arXiv:1705.10670] [INSPIRE].
L. Iliesiu, The Nèel-VBA quantum phase transition and the conformal bootstrap, talk given at the workshop Developments in Quantum Field Theory and Condensed Matter Physics , November 5-7, Simons Center for Geometry and Physics, Stony Brook, U.S.A. (2018).
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].
C. Wang et al., Deconfined quantum critical points: symmetries and dualities, Phys. Rev. X 7 (2017) 031051 [arXiv:1703.02426] [INSPIRE].
L.K. Hua and I. Reiner, On the generators of the symplectic modular group, Trans. Amer. Math. Soc. 65 (1949) 415.
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
Z. Komargodski and N. Seiberg, A symmetry breaking scenario for QCD 3, JHEP 01 (2018) 109 [arXiv:1706.08755] [INSPIRE].
J.Y. Lee et al., Emergent multi-flavor QED3 at the plateau transition between fractional Chern insulators: applications to graphene heterostructures, Phys. Rev. X 8 (2018) 031015 [arXiv:1802.09538] [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 et al., Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [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].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, arXiv:1807.02522 [INSPIRE].
M. Beccaria and A.A. Tseytlin, Vectorial AdS 5 /CFT 4 duality for spin-one boundary theory, J. Phys. A 47 (2014) 492001 [arXiv:1410.4457] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and interface CFTs from the conformal bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
D.T. Barfoot and D.J. Broadhurst, Z(2) × S 6 symmetry of the two loop diagram, Z. Phys. C 41 (1988) 81 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65.
D.J. Broadhurst, J.A. Gracey and D. Kreimer, Beyond the triangle and uniqueness relations: nonzeta counterterms at large N from positive knots, Z. Phys. C 75 (1997) 559 [hep-th/9607174] [INSPIRE].
Z.-W. Huang and J. Liu, NumExp: numerical ϵ-expansion of hypergeometric functions, Comput. Phys. Commun. 184 (2013) 1973 [arXiv:1209.3971] [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: 1902.09567
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Di Pietro, L., Gaiotto, D., Lauria, E. et al. 3d Abelian gauge theories at the boundary. J. High Energ. Phys. 2019, 91 (2019). https://doi.org/10.1007/JHEP05(2019)091
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)091