Abstract
We study Abelian Maxwell-Chern-Simons theory in three-dimensional AdS black hole backgrounds for both integer and non-integer Chern-Simons coupling. Such theories can be derived from various string theory constructions, which we review in the present work. In particular we find exact solutions in the low frequency, low momentum limit, ω, k ≪ T (hydrodynamic limit). Using the holographic principle, we translate our results into correlation functions of vector and scalar operators in the dual strongly coupled 1 + 1-dimensional quantum field theory with a chiral anomaly at non-zero temperature T . Starting from the conformal case we show applicability of the hydrodynamic limit and discuss extensions to the non-conformal case. Correlation functions in the conformal case are compared to an exact field theoretic computation.
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References
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics. 2. Sound waves, JHEP 12 (2002) 054 [hep-th/0210220] [INSPIRE].
A. Adams, L.D. Carr, T. Schäfer, P. Steinberg and J.E. Thomas, Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality, New J. Phys. 14 (2012) 115009 [arXiv:1205.5180] [INSPIRE].
J. Voit, One-dimensional Fermi liquids, Rep. Prog. Phys. 58 (1995) 977 [cond-mat/9510014].
R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12 (1957) 570.
D.T. Son and P. Surowka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].
R. Loganayagam, Anomaly induced transport in arbitrary dimensions, arXiv:1106.0277 [INSPIRE].
K. Jensen et al., Parity-violating hydrodynamics in 2 + 1 dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
N. Banerjee et al., Constraints on fluid dynamics from equilibrium partition functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
D.E. Kharzeev and D.T. Son, Testing the chiral magnetic and chiral vortical effects in heavy ion collisions, Phys. Rev. Lett. 106 (2011) 062301 [arXiv:1010.0038] [INSPIRE].
K. Landsteiner, Anomalous transport of Weyl fermions in Weyl semimetals, Phys. Rev. B 89 (2014) 075124 [arXiv:1306.4932] [INSPIRE].
D. Anninos, S.A. Hartnoll and N. Iqbal, Holography and the Coleman-Mermin-Wagner theorem, Phys. Rev. D 82 (2010) 066008 [arXiv:1005.1973] [INSPIRE].
T. Andrade, J.I. Jottar and R.G. Leigh, Boundary conditions and unitarity: the Maxwell-Chern-Simons System in AdS 3 /CFT 2, JHEP 05 (2012) 071 [arXiv:1111.5054] [INSPIRE].
S. Jain and T. Sharma, Anomalous charged fluids in 1 + 1d from equilibrium partition function, JHEP 01 (2013) 039 [arXiv:1203.5308] [INSPIRE].
V. Balasubramanian, I. Garcia-Etxebarria, F. Larsen and J. Simon, Helical Luttinger liquids and three dimensional black holes, Phys. Rev. D 84 (2011) 126012 [arXiv:1012.4363] [INSPIRE].
K. Jensen, Chiral anomalies and AdS/CMT in two dimensions, JHEP 01 (2011) 109 [arXiv:1012.4831] [INSPIRE].
M. Fujita, W. Li, S. Ryu and T. Takayanagi, Fractional quantum Hall effect via holography: Chern-Simons, edge states and hierarchy, JHEP 06 (2009) 066 [arXiv:0901.0924] [INSPIRE].
X.G. Wen, Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B 41 (1990) 12838 [INSPIRE].
X. Wen, Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons, Oxford University Press, Oxford U.K. (2007).
L.-Y. Hung and A. Sinha, Holographic quantum liquids in 1 + 1 dimensions, JHEP 01 (2010) 114 [arXiv:0909.3526] [INSPIRE].
X. Gao, M. Kaminski, H.-B. Zeng and H.-Q. Zhang, Non-equilibrium field dynamics of an honest holographic superconductor, JHEP 11 (2012) 112 [arXiv:1204.3103] [INSPIRE].
P. Kraus, Lectures on black holes and the AdS 3 /CFT 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
G.V. Dunne, Aspects of Chern-Simons theory, hep-th/9902115 [INSPIRE].
E. D’Hoker, P. Kraus and A. Shah, RG flow of magnetic brane correlators, JHEP 04 (2011) 039 [arXiv:1012.5072] [INSPIRE].
T. Giamarchi, Quantum physics in one dimension, International Series of Monographs on Physics. Clarendon Press, U.K. (2004).
J.L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17 (1984) L385 [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
P. Kovtun and A. Ritz, Universal conductivity and central charges, Phys. Rev. D 78 (2008) 066009 [arXiv:0806.0110] [INSPIRE].
M. Valle, Hydrodynamics in 1 + 1 dimensions with gravitational anomalies, JHEP 08 (2012) 113 [arXiv:1206.1538] [INSPIRE].
J.M. Luttinger, An exactly soluble model of a many-fermion system, J. Math. Phys. 4 (1963) 1154.
J. Voit, A brief introduction to Luttinger liquids, AIP Conf. Proc. 544 (2000) 309 [cond-mat/0005114].
F.D.M. Haldane, Luttinger liquid theory of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas, J. Phys. C 14 (1981) 2585 [INSPIRE].
V. Meden and K. Schönhammer, Spectral functions for the Tomonaga-Luttinger model, Phys. Rev. B 46 (1992) 15753.
J. Voit, Charge-spin separation and the spectral properties of Luttinger liquids, Phys. Rev. B 47 (1993) 6740 [cond-mat/9310048].
X.G. Wen, Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B 41 (1990) 12838 [INSPIRE].
S. Gukov, E. Martinec, G.W. Moore and A. Strominger, The search for a holographic dual to AdS 3 × S 3 × S 3 × S 1, Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].
S. Gukov, E. Martinec, G.W. Moore and A. Strominger, Chern-Simons gauge theory and the AdS 3 /CFT 2 correspondence, hep-th/0403225 [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Karch and A. O’Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024 [arXiv:0705.3870] [INSPIRE].
A. O’Bannon, Hall conductivity of flavor fields from AdS/CFT, Phys. Rev. D 76 (2007) 086007 [arXiv:0708.1994] [INSPIRE].
J.A. Harvey and A.B. Royston, Gauge/gravity duality with a chiral N = (0, 8) string defect, JHEP 08 (2008) 006 [arXiv:0804.2854] [INSPIRE].
J.L. Davis, P. Kraus and A. Shah, Gravity dual of a quantum Hall plateau transition, JHEP 11 (2008) 020 [arXiv:0809.1876] [INSPIRE].
D. Mateos, R.C. Myers and R.M. Thomson, Thermodynamics of the brane, JHEP 05 (2007) 067 [hep-th/0701132] [INSPIRE].
P. Karndumri and E.O. Colgáin, 3D Supergravity from wrapped D3-branes, JHEP 10 (2013) 094 [arXiv:1307.2086] [INSPIRE].
S. Detournay and M. Guica, Stringy Schrödinger truncations, JHEP 08 (2013) 121 [arXiv:1212.6792] [INSPIRE].
N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D3 branes and holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner, Flavor superconductivity from gauge/gravity duality, JHEP 10 (2009) 067 [arXiv:0903.1864] [INSPIRE].
A.A. Tseytlin, On non-Abelian generalization of Born-Infeld action in string theory, Nucl. Phys. B 501 (1997) 41 [hep-th/9701125] [INSPIRE].
A. Hashimoto and W. Taylor, Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action, Nucl. Phys. B 503 (1997) 193 [hep-th/9703217] [INSPIRE].
O. Domenech, M. Montull, A. Pomarol, A. Salvio and P.J. Silva, Emergent gauge fields in holographic superconductors, JHEP 08 (2010) 033 [arXiv:1005.1776] [INSPIRE].
M. Montull, O. Pujolàs, A. Salvio and P.J. Silva, Flux periodicities and quantum hair on holographic superconductors, Phys. Rev. Lett. 107 (2011) 181601 [arXiv:1105.5392] [INSPIRE].
M. Montull, O. Pujolàs, A. Salvio and P.J. Silva, Magnetic response in the holographic insulator/superconductor transition, JHEP 04 (2012) 135 [arXiv:1202.0006] [INSPIRE].
A. Salvio, Holographic superfluids and superconductors in dilaton-gravity, JHEP 09 (2012) 134 [arXiv:1207.3800] [INSPIRE].
D. Marolf and S.F. Ross, Boundary conditions and new dualities: vector fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
S. Hyun, U duality between three-dimensional and higher dimensional black holes, J. Korean Phys. Soc. 33 (1998) S532 [hep-th/9704005] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
W. Mueck and K.S. Viswanathan, Conformal field theory correlators from classical field theory on Anti-de Sitter space. 2. Vector and spinor fields, Phys. Rev. D 58 (1998) 106006 [hep-th/9805145] [INSPIRE].
S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [INSPIRE].
M. Fujita, M. Kaminski and A. Karch, SL(2, ℤ) action on AdS/BCFT and Hall conductivities, JHEP 07 (2012) 150 [arXiv:1204.0012] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
D.T. Son and A.O. Starinets, Viscosity, black holes and quantum field theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
P. Minces and V.O. Rivelles, Chern-Simons theories in the AdS/CFT correspondence, Phys. Lett. B 455 (1999) 147 [hep-th/9902123] [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].
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].
R.C. Myers, A.O. Starinets and R.M. Thomson, Holographic spectral functions and diffusion constants for fundamental matter, JHEP 11 (2007) 091 [arXiv:0706.0162] [INSPIRE].
C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Holographic meson melting, JHEP 04 (2007) 031 [hep-th/0612169] [INSPIRE].
J. Erdmenger, M. Kaminski and F. Rust, Holographic vector mesons from spectral functions at finite baryon or isospin density, Phys. Rev. D 77 (2008) 046005 [arXiv:0710.0334] [INSPIRE].
J. Erdmenger, M. Kaminski and F. Rust, Isospin diffusion in thermal AdS/CFT with flavor, Phys. Rev. D 76 (2007) 046001 [arXiv:0704.1290] [INSPIRE].
J. Erdmenger, M. Kaminski, P. Kerner and F. Rust, Finite baryon and isospin chemical potential in AdS/CFT with flavor, JHEP 11 (2008) 031 [arXiv:0807.2663] [INSPIRE].
M. Kaminski, Flavor superconductivity & superfluidity, Lect. Notes Phys. 828 (2011) 349 [arXiv:1002.4886] [INSPIRE].
M. Kaminski, Holographic quark gluon plasma with flavor, Fortsch. Phys. 57 (2009) 3 [arXiv:0808.1114] [INSPIRE].
H.-U. Yee and I. Zahed, Holographic two dimensional QCD and Chern-Simons term, JHEP 07 (2011) 033 [arXiv:1103.6286] [INSPIRE].
M. Fujita, M5-brane Defect and QHE in AdS 4 × N (1, 1)/N =3 SCFT, Phys. Rev. D 83 (2011) 105016 [arXiv:1011.0154] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N =6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
D. Gaiotto and A. Tomasiello, The gauge dual of Romans mass, JHEP 01 (2010) 015 [arXiv:0901.0969] [INSPIRE].
O. Aharony, D. Jafferis, A. Tomasiello and A. Zaffaroni, Massive type IIA string theory cannot be strongly coupled, JHEP 11 (2010) 047 [arXiv:1007.2451] [INSPIRE].
C.L. Kane and M.P.A. Fisher, Transport in a one-channel Luttinger liquid, Phys. Rev. Lett. 68 (1992) 1220.
X. Wen, Edge transport properties of the fractional quantum Hall states and weak impurity scattering of one-dimensional ‘Charge density wave’, Phys. Rev. B 44 (1991) 5708.
S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT correspondence, Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [INSPIRE].
M. Fujita, Y. Hikida, S. Ryu and T. Takayanagi, Disordered systems and the replica method in AdS/CFT, JHEP 12 (2008) 065 [arXiv:0810.5394] [INSPIRE].
T. Faulkner and N. Iqbal, Friedel oscillations and horizon charge in 1D holographic liquids, JHEP 07 (2013) 060 [arXiv:1207.4208] [INSPIRE].
W. Harrison, Solid state theory, Dover Publications, U.S.A. (1970).
D. Grumiller and N. Johansson, Gravity duals for logarithmic conformal field theories, J. Phys. Conf. Ser. 222 (2010) 012047 [arXiv:1001.0002] [INSPIRE].
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Chang, HC., Fujita, M. & Kaminski, M. From Maxwell-Chern-Simons theory in AdS 3 towards hydrodynamics in 1 + 1 dimensions. J. High Energ. Phys. 2014, 118 (2014). https://doi.org/10.1007/JHEP10(2014)118
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DOI: https://doi.org/10.1007/JHEP10(2014)118