Abstract
We argue that an effective field theory of local fluid elements captures the constraints on hydrodynamic transport stemming from the presence of quantum anomalies in the underlying microscopic theory. Focussing on global current anomalies for an arbitrary flavour group, we derive the anomalous constitutive relations in arbitrary even dimensions. We demonstrate that our results agree with the constraints on anomaly governed transport derived hitherto using a local version of the second law of thermodynamics. The construction crucially uses the anomaly inflow mechanism and involves a novel thermofield double construction. In particular, we show that the anomalous Ward identities necessitate non-trivial interaction between the two parts of the Schwinger-Keldysh contour.
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References
V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, arXiv:1107.5780 [INSPIRE].
T. Schäfer and D. Teaney, Nearly perfect fluidity: from cold atomic gases to hot quark gluon plasmas, Rept. Prog. Phys. 72 (2009) 126001 [arXiv:0904.3107] [INSPIRE].
J. Erdmenger, M. Haack, M. Kaminski and A. Yarom, Fluid dynamics of R-charged black holes, JHEP 01 (2009) 055 [arXiv:0809.2488] [INSPIRE].
N. Banerjee et al., Hydrodynamics from charged black branes, JHEP 01 (2011) 094 [arXiv:0809.2596] [INSPIRE].
A. Vilenkin, Parity violating currents in thermal radiation, Phys. Lett. B 80 (1978) 150 [INSPIRE].
A. Vilenkin, Macroscopic parity violating effects: neutrino fluxes from rotating black holes and in rotating thermal radiation, Phys. Rev. D 20 (1979) 1807 [INSPIRE].
K. Landsteiner, E. Megias and F. Pena-Benitez, Anomalous transport from Kubo formulae, Lect. Notes Phys. 871 (2013) 433 [arXiv:1207.5808] [INSPIRE].
S. Bhattacharyya, S. Lahiri, R. Loganayagam and S. Minwalla, Large rotating AdS black holes from fluid mechanics, JHEP 09 (2008) 054 [arXiv:0708.1770] [INSPIRE].
D.T. Son and P. Surowka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].
D.E. Kharzeev and H.J. Warringa, Chiral magnetic conductivity, Phys. Rev. D 80 (2009) 034028 [arXiv:0907.5007] [INSPIRE].
I. Amado, K. Landsteiner and F. Pena-Benitez, Anomalous transport coefficients from Kubo formulas in holography, JHEP 05 (2011) 081 [arXiv:1102.4577] [INSPIRE].
K. Landsteiner, E. Megias and F. Pena-Benitez, Gravitational anomaly and transport, Phys. Rev. Lett. 107 (2011) 021601 [arXiv:1103.5006] [INSPIRE].
R. Loganayagam, Anomaly induced transport in arbitrary dimensions, arXiv:1106.0277 [INSPIRE].
D.E. Kharzeev and H.-U. Yee, Anomalies and time reversal invariance in relativistic hydrodynamics: the second order and higher dimensional formulations, Phys. Rev. D 84 (2011) 045025 [arXiv:1105.6360] [INSPIRE].
M. Torabian and H.-U. Yee, Holographic nonlinear hydrodynamics from AdS/CFT with multiple/non-Abelian symmetries, JHEP 08 (2009) 020 [arXiv:0903.4894] [INSPIRE].
K. Jensen, Chiral anomalies and AdS/CMT in two dimensions, JHEP 01 (2011) 109 [arXiv:1012.4831] [INSPIRE].
R. Loganayagam and P. Surowka, Anomaly/transport in an ideal Weyl gas, JHEP 04 (2012) 097 [arXiv:1201.2812] [INSPIRE].
Y. Neiman and Y. Oz, Relativistic hydrodynamics with general anomalous charges, JHEP 03 (2011) 023 [arXiv:1011.5107] [INSPIRE].
K. Jensen, Triangle anomalies, thermodynamics and hydrodynamics, Phys. Rev. D 85 (2012) 125017 [arXiv:1203.3599] [INSPIRE].
V. Nair, R. Ray and S. Roy, Fluids, anomalies and the chiral magnetic effect: a group-theoretic formulation, Phys. Rev. D 86 (2012) 025012 [arXiv:1112.4022] [INSPIRE].
D. Capasso, V. Nair and J. Tekel, The isospin asymmetry in anomalous fluid dynamics, Phys. Rev. D 88 (2013) 085025 [arXiv:1307.7610] [INSPIRE].
N. Banerjee et al., Constraints on fluid dynamics from equilibrium partition functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
N. Banerjee, S. Dutta, S. Jain, R. Loganayagam and T. Sharma, Constraints on anomalous fluid in arbitrary dimensions, JHEP 03 (2013) 048 [arXiv:1206.6499] [INSPIRE].
S. Jain and T. Sharma, Anomalous charged fluids in 1 + 1d from equilibrium partition function, JHEP 01 (2013) 039 [arXiv:1203.5308] [INSPIRE].
M. Valle, Hydrodynamics in 1 + 1 dimensions with gravitational anomalies, JHEP 08 (2012) 113 [arXiv:1206.1538] [INSPIRE].
R. Banerjee, Exact results in two dimensional chiral hydrodynamics with diffeomorphism and conformal anomalies, arXiv:1303.5593 [INSPIRE].
R. Banerjee, P. Chakraborty, S. Dey, B.R. Majhi and A.K. Mitra, Two dimensional hydrodynamics with gauge and gravitational anomalies, arXiv:1307.1313 [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, arXiv:1310.7024 [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, arXiv:1311.2935 [INSPIRE].
T. Azeyanagi, R. Loganayagam, G.S. Ng and M.J. Rodriguez, Holographic thermal helicity, arXiv:1311.2940 [INSPIRE].
A. Taub, General relativistic variational principle for perfect fluids, Phys. Rev. 94 (1954) 1468 [INSPIRE].
B. Carter, Elastic perturbation theory in general relativity and a variation principle for a rotating solid star, Comm. Math. Phys. 30 (1973) 261.
B. Carter, Covariant theory of conductivity in ideal fluid or solid media, Lect. Notes Math. 1385 (1989) 1.
J.D. Brown, Action functionals for relativistic perfect fluids, Class. Quant. Grav. 10 (1993) 1579 [gr-qc/9304026] [INSPIRE].
H. Leutwyler, Phonons as goldstone bosons, Helv. Phys. Acta 70 (1997) 275 [hep-ph/9609466] [INSPIRE].
R. Jackiw, V. Nair, S. Pi and A. Polychronakos, Perfect fluid theory and its extensions, J. Phys. A 37 (2004) R327 [hep-ph/0407101] [INSPIRE].
S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, Null energy condition and superluminal propagation, JHEP 03 (2006) 025 [hep-th/0512260] [INSPIRE].
S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya and M. Rangamani, Non-dissipative hydrodynamics: effective actions versus entropy current, JHEP 02 (2013) 153 [arXiv:1211.1020] [INSPIRE].
S. Dubovsky, L. Hui and A. Nicolis, Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions, arXiv:1107.0732 [INSPIRE].
A. Nicolis and D.T. Son, Hall viscosity from effective field theory, arXiv:1103.2137 [INSPIRE].
F.M. Haehl and M. Rangamani, Comments on Hall transport from effective actions, JHEP 10 (2013) 074 [arXiv:1305.6968] [INSPIRE].
D. Nickel and D.T. Son, Deconstructing holographic liquids, New J. Phys. 13 (2011) 075010 [arXiv:1009.3094] [INSPIRE].
S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: first order effects, Phys. Rev. D 88 (2013) 105001 [arXiv:1211.6461] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, arXiv:1305.3670 [INSPIRE].
K. Jensen et al., Parity-violating hydrodynamics in 2 + 1 dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
R. Loganayagam, Entropy current in conformal hydrodynamics, JHEP 05 (2008) 087 [arXiv:0801.3701] [INSPIRE].
P. Romatschke, Relativistic viscous fluid dynamics and non-equilibrium entropy, Class. Quant. Grav. 27 (2010) 025006 [arXiv:0906.4787] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A theory of first order dissipative superfluid dynamics, arXiv:1105.3733 [INSPIRE].
S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [INSPIRE].
C.G. Callan Jr. and J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE].
T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].
C. Eling, Y. Oz, S. Theisen and S. Yankielowicz, Conformal anomalies in hydrodynamics, JHEP 05 (2013) 037 [arXiv:1301.3170] [INSPIRE].
S. Weinberg, The quantum theory of fields: modern applications, volume 2, Cambridge University Press, Cambridge U.K. (1996).
J.A. Harvey, TASI 2003 lectures on anomalies, hep-th/0509097 [INSPIRE].
W.A. Bardeen and B. Zumino, Consistent and covariant anomalies in gauge and gravitational theories, Nucl. Phys. B 244 (1984) 421 [INSPIRE].
S. Bhattacharyya et al., Local fluid dynamical entropy from gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].
M. Nakahara, Geometry, topology, and physics, CRC Press, U.S.A. (2003).
P. Mora, R. Olea, R. Troncoso and J. Zanelli, Transgression forms and extensions of Chern-Simons gauge theories, JHEP 02 (2006) 067 [hep-th/0601081] [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
C. Herzog and D. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].
K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, Equilibrium and nonequilibrium formalisms made unified, Phys. Rept. 118 (1985) 1 [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
L. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].
J. Maciejko, An introduction to nonequilibrium many-body theory, lecture notes (2007).
A. Kamenev and A. Levchenko, Keldysh technique and nonlinear σ-model: basic principles and applications, arXiv:0901.3586 [INSPIRE].
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Haehl, F.M., Loganayagam, R. & Rangamani, M. Effective actions for anomalous hydrodynamics. J. High Energ. Phys. 2014, 34 (2014). https://doi.org/10.1007/JHEP03(2014)034
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DOI: https://doi.org/10.1007/JHEP03(2014)034