# Bulk viscosity at extreme limits: from kinetic theory to strings

## Abstract

In this paper we study bulk viscosity in a thermal QCD model with large number of colors at two extreme limits: the very weak and the very strong ’t Hooft couplings. The weak coupling scenario is based on kinetic theory, and one may go to the very strong coupling dynamics via an intermediate coupling regime. Although the former has a clear description in terms of kinetic theory, the intermediate coupling regime, which uses lattice results, suffers from usual technical challenges that render an explicit determination of bulk viscosity somewhat difficult. On the other hand, the very strong ’t Hooft coupling dynamics may be studied using string theories at both weak and strong string couplings using gravity duals in type IIB as well as M-theory respectively. In type IIB we provide the precise fluctuation modes of the metric in the gravity dual responsible for bulk viscosity, compute the speed of sound in the medium and analyze the ratio of the bulk to shear viscosities. In M-theory, where we uplift the type IIA mirror dual of the UV complete type IIB model, we study and compare both the bulk viscosity and the sound speed by analyzing the quasi-normal modes in the system at strong IIA string coupling. By deriving the spectral function, we show the consistency of our results both for the actual values of the parameters involved as well for the bound on the ratio of bulk to shear viscosities.

## Keywords

Holography and quark-gluon plasmas M-Theory Quark-Gluon Plasma## Notes

### **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.

## References

- [1]D. Teaney, J. Lauret and E.V. Shuryak,
*Flow at the SPS and RHIC as a quark gluon plasma signature*,*Phys. Rev. Lett.***86**(2001) 4783 [nucl-th/0011058] [INSPIRE]. - [2]P. Huovinen et al.,
*Radial and elliptic flow at RHIC: Further predictions*,*Phys. Lett.***B 503**(2001) 58 [hep-ph/0101136] [INSPIRE]. - [3]P.F. Kolb et al.,
*Centrality dependence of multiplicity, transverse energy and elliptic flow from hydrodynamics*,*Nucl. Phys.***A 696**(2001) 197 [hep-ph/0103234] [INSPIRE]. - [4]T. Hirano and K. Tsuda,
*Collective flow and two pion correlations from a relativistic hydrodynamic model with early chemical freezeout*,*Phys. Rev.***C 66**(2002) 054905 [nucl-th/0205043] [INSPIRE]. - [5]P.F. Kolb and R. Rapp,
*Transverse flow and hadrochemistry in Au+Au collisions at*\( \sqrt{S_{NN}} \) = 200*GeV*,*Phys. Rev.***C 67**(2003) 044903 [hep-ph/0210222] [INSPIRE]. - [6]P. Romatschke and U. Romatschke,
*Viscosity information from relativistic nuclear collisions: how perfect is the fluid observed at RHIC?*,*Phys. Rev. Lett.***99**(2007) 172301 [arXiv:0706.1522] [INSPIRE].ADSCrossRefGoogle Scholar - [7]M. Luzum and P. Romatschke,
*Conformal relativistic viscous hydrodynamics: applications to RHIC results at*\( \sqrt{s_{NN}} \) = 200*GeV*,*Phys. Rev.***C 78**(2008) 034915 [*Erratum ibid.***C 79**(2009) 039903] [arXiv:0804.4015] [INSPIRE]. - [8]K. Dusling and D. Teaney,
*Simulating elliptic flow with viscous hydrodynamics*,*Phys. Rev.***C 77**(2008) 034905 [arXiv:0710.5932] [INSPIRE].ADSGoogle Scholar - [9]H. Song and U.W. Heinz,
*Causal viscous hydrodynamics in 2+1 dimensions for relativistic heavy-ion collisions*,*Phys. Rev.***C 77**(2008) 064901 [arXiv:0712.3715] [INSPIRE].ADSGoogle Scholar - [10]PHENIX collaboration,
*Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: experimental evaluation by the PHENIX collaboration*,*Nucl. Phys.***A 757**(2005) 184 [nucl-ex/0410003] [INSPIRE]. - [11]B.B. Back et al.,
*The PHOBOS perspective on discoveries at RHIC*,*Nucl. Phys.***A 757**(2005) 28 [nucl-ex/0410022] [INSPIRE]. - [12]BRAHMS collaboration,
*Quark gluon plasma and color glass condensate at RHIC? The Perspective from the BRAHMS experiment*,*Nucl. Phys.***A 757**(2005) 1 [nucl-ex/0410020] [INSPIRE]. - [13]STAR collaboration,
*Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration*’*s critical assessment of the evidence from RHIC collisions*,*Nucl. Phys.***A 757**(2005) 102 [nucl-ex/0501009] [INSPIRE]. - [14]E. Shuryak,
*Physics of strongly coupled quark-gluon plasma*,*Prog. Part. Nucl. Phys.***62**(2009) 48 [arXiv:0807.3033] [INSPIRE].ADSCrossRefGoogle Scholar - [15]C. Gale, S. Jeon and B. Schenke,
*Hydrodynamic modeling of heavy-ion collisions*,*Int. J. Mod. Phys.***A 28**(2013) 1340011 [arXiv:1301.5893] [INSPIRE].ADSCrossRefGoogle Scholar - [16]U. Heinz and R. Snellings,
*Collective flow and viscosity in relativistic heavy-ion collisions*,*Ann. Rev. Nucl. Part. Sci.***63**(2013) 123 [arXiv:1301.2826] [INSPIRE].ADSCrossRefGoogle Scholar - [17]P. Romatschke,
*New developments in relativistic viscous hydrodynamics*,*Int. J. Mod. Phys.***E 19**(2010) 1 [arXiv:0902.3663] [INSPIRE].ADSCrossRefGoogle Scholar - [18]W. Florkowski, M.P. Heller and M. Spalinski,
*New theories of relativistic hydrodynamics in the LHC era*,*Rept. Prog. Phys.***81**(2018) 046001 [arXiv:1707.02282] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [19]S. Jeon and U. Heinz,
*Introduction to hydrodynamics*,*Int. J. Mod. Phys.***E 24**(2015) 1530010 [arXiv:1503.03931] [INSPIRE].ADSCrossRefGoogle Scholar - [20]J.M. Maldacena,
*The large N limit of superconformal field theories and supergravity*,*Int. J. Theor. Phys.***38**(1999) 1113 [*Adv. Theor. Math. Phys.***2**(1998) 231] [hep-th/9711200] [INSPIRE]. - [21]
- [22]P. Kovtun, D.T. Son and A.O. Starinets,
*Viscosity in strongly interacting quantum field theories from black hole physics*,*Phys. Rev. Lett.***94**(2005) 111601 [hep-th/0405231] [INSPIRE].ADSCrossRefGoogle Scholar - [23]Y. Kats and P. Petrov,
*Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory*,*JHEP***01**(2009) 044 [arXiv:0712.0743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [24]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Transport coefficients in high temperature gauge theories. 1. Leading log results*,*JHEP***11**(2000) 001 [hep-ph/0010177] [INSPIRE]. - [25]N. Christiansen, M. Haas, J.M. Pawlowski and N. Strodthoff,
*Transport coefficients in Yang-Mills theory and QCD*,*Phys. Rev. Lett.***115**(2015) 112002 [arXiv:1411.7986] [INSPIRE].ADSCrossRefGoogle Scholar - [26]A. Nakamura and S. Sakai,
*Transport coefficients of gluon plasma*,*Phys. Rev. Lett.***94**(2005) 072305 [hep-lat/0406009] [INSPIRE]. - [27]L.P. Csernai, J. Kapusta and L.D. McLerran,
*On the strongly-interacting low-viscosity matter created in relativistic nuclear collisions*,*Phys. Rev. Lett.***97**(2006) 152303 [nucl-th/0604032] [INSPIRE]. - [28]M. Prakash, M. Prakash, R. Venugopalan and G. Welke,
*Nonequilibrium properties of hadronic mixtures*,*Phys. Rept.***227**(1993) 321.ADSCrossRefGoogle Scholar - [29]R. Lang, N. Kaiser and W. Weise,
*Shear viscosity of a hot pion gas*,*Eur. Phys. J.***A 48**(2012) 109 [arXiv:1205.6648] [INSPIRE].ADSCrossRefGoogle Scholar - [30]P.B. Arnold, C. Dogan and G.D. Moore,
*The bulk viscosity of high-temperature QCD*,*Phys. Rev.***D 74**(2006) 085021 [hep-ph/0608012] [INSPIRE]. - [31]A. Buchel,
*Bulk viscosity of gauge theory plasma at strong coupling*,*Phys. Lett.***B 663**(2008) 286 [arXiv:0708.3459] [INSPIRE].ADSCrossRefGoogle Scholar - [32]S. Jeon,
*Hydrodynamic transport coefficients in relativistic scalar field theory*,*Phys. Rev.***D 52**(1995) 3591 [hep-ph/9409250] [INSPIRE]. - [33]P. Benincasa, A. Buchel and A.O. Starinets,
*Sound waves in strongly coupled non-conformal gauge theory plasma*,*Nucl. Phys.***B 733**(2006) 160 [hep-th/0507026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [34]S. Borsányi et al.,
*Recent results on the equation of state of QCD*, PoS(LATTICE 2014) 224 [arXiv:1410.7917] [INSPIRE]. - [35]A. Bazavov, P. Petreczky and J.H. Weber,
*Equation of state in*2 + 1*flavor QCD at high temperatures*,*Phys. Rev.***D 97**(2018) 014510 [arXiv:1710.05024] [INSPIRE].ADSGoogle Scholar - [36]D. Kharzeev and K. Tuchin,
*Bulk viscosity of QCD matter near the critical temperature*,*JHEP***09**(2008) 093 [arXiv:0705.4280] [INSPIRE].ADSCrossRefGoogle Scholar - [37]F. Karsch, D. Kharzeev and K. Tuchin,
*Universal properties of bulk viscosity near the QCD phase transition*,*Phys. Lett.***B 663**(2008) 217 [arXiv:0711.0914] [INSPIRE].ADSCrossRefGoogle Scholar - [38]G.D. Moore and O. Saremi,
*Bulk viscosity and spectral functions in QCD*,*JHEP***09**(2008) 015 [arXiv:0805.4201] [INSPIRE].ADSCrossRefGoogle Scholar - [39]S.S. Gubser, A. Nellore, S.S. Pufu and F.D. Rocha,
*Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics*,*Phys. Rev. Lett.***101**(2008) 131601 [arXiv:0804.1950] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [40]J. Noronha-Hostler, J. Noronha and C. Greiner,
*Transport coefficients of hadronic matter near T*_{c},*Phys. Rev. Lett.***103**(2009) 172302 [arXiv:0811.1571] [INSPIRE].ADSCrossRefGoogle Scholar - [41]G.S. Denicol, T. Kodama, T. Koide and P. Mota,
*Effect of bulk viscosity on elliptic flow near QCD phase transition*,*Phys. Rev.***C 80**(2009) 064901 [arXiv:0903.3595] [INSPIRE].ADSGoogle Scholar - [42]H. Song and U.W. Heinz,
*Extracting the QGP viscosity from RHIC data*—*A status report from viscous hydrodynamics*,*J. Phys.***G 36**(2009) 064033 [arXiv:0812.4274] [INSPIRE].ADSCrossRefGoogle Scholar - [43]G.S. Denicol, T. Kodama and T. Koide,
*The effect of shear and bulk viscosities on elliptic flow*,*J. Phys.***G 37**(2010) 094040 [arXiv:1002.2394] [INSPIRE].ADSCrossRefGoogle Scholar - [44]S. Ryu et al.,
*Importance of the bulk viscosity of QCD in ultrarelativistic heavy-ion collisions*,*Phys. Rev. Lett.***115**(2015) 132301 [arXiv:1502.01675] [INSPIRE].ADSCrossRefGoogle Scholar - [45]S. Ryu et al.,
*Effects of bulk viscosity and hadronic rescattering in heavy ion collisions at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider*,*Phys. Rev.***C 97**(2018) 034910 [arXiv:1704.04216] [INSPIRE].ADSGoogle Scholar - [46]J.-F. Paquet et al.,
*Production of photons in relativistic heavy-ion collisions*,*Phys. Rev.***C 93**(2016) 044906 [arXiv:1509.06738] [INSPIRE].ADSGoogle Scholar - [47]P. Bożek,
*Effect of bulk viscosity on interferometry correlations in ultrarelativistic heavy-ion collisions*,*Phys. Rev.***C 95**(2017) 054909 [arXiv:1702.01319] [INSPIRE].ADSGoogle Scholar - [48]A. Monnai, S. Mukherjee and Y. Yin,
*Phenomenological consequences of enhanced bulk viscosity near the QCD critical point*,*Phys. Rev.***C 95**(2017) 034902 [arXiv:1606.00771] [INSPIRE].ADSGoogle Scholar - [49]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Effective kinetic theory for high temperature gauge theories*,*JHEP***01**(2003) 030 [hep-ph/0209353] [INSPIRE]. - [50]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Transport coefficients in high temperature gauge theories. 2. Beyond leading log*,*JHEP***05**(2003) 051 [hep-ph/0302165] [INSPIRE]. - [51]J.-S. Gagnon and S. Jeon,
*Leading order calculation of electric conductivity in hot quantum electrodynamics from diagrammatic methods*,*Phys. Rev.***D 75**(2007) 025014 [*Erratum ibid.***D 76**(2007) 089902] [hep-ph/0610235] [INSPIRE]. - [52]J.-S. Gagnon and S. Jeon,
*Leading order calculation of shear viscosity in hot quantum electrodynamics from diagrammatic methods*,*Phys. Rev.***D 76**(2007) 105019 [arXiv:0708.1631] [INSPIRE].ADSGoogle Scholar - [53]P. Romatschke and D.T. Son,
*Spectral sum rules for the quark-gluon plasma*,*Phys. Rev.***D 80**(2009) 065021 [arXiv:0903.3946] [INSPIRE].ADSGoogle Scholar - [54]H.B. Meyer,
*A Calculation of the bulk viscosity in*SU(3)*gluodynamics*,*Phys. Rev. Lett.***100**(2008) 162001 [arXiv:0710.3717] [INSPIRE].ADSCrossRefGoogle Scholar - [55]H.B. Meyer,
*The bulk channel in thermal gauge theories*,*JHEP***04**(2010) 099 [arXiv:1002.3343] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [56]N.Yu. Astrakhantsev, V.V. Braguta and A.Yu. Kotov,
*Temperature dependence of the bulk viscosity within lattice simulation of*SU(3)*gluodynamics*,*Phys. Rev.***D 98**(2018) 054515 [arXiv:1804.02382] [INSPIRE].ADSGoogle Scholar - [57]I.R. Klebanov and M.J. Strassler,
*Supergravity and a confining gauge theory: duality cascades and χ*_{SB}*resolution of naked singularities*,*JHEP***08**(2000) 052 [hep-th/0007191] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [58]P. Ouyang,
*Holomorphic D*7*branes and flavored N*= 1*gauge theories*,*Nucl. Phys.***B 699**(2004) 207 [hep-th/0311084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [59]M. Mia, K. Dasgupta, C. Gale and S. Jeon,
*Five easy pieces: the dynamics of quarks in strongly coupled plasmas*,*Nucl. Phys.***B 839**(2010) 187 [arXiv:0902.1540] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [60]M. Mia, K. Dasgupta, C. Gale and S. Jeon,
*Toward large N thermal QCD from dual gravity: the heavy quarkonium potential*,*Phys. Rev.***D 82**(2010) 026004 [arXiv:1004.0387] [INSPIRE].ADSGoogle Scholar - [61]F. Chen, L. Chen, K. Dasgupta, M. Mia and O. Trottier,
*Ultraviolet complete model of large N thermal QCD*,*Phys. Rev.***D 87**(2013) 041901 [arXiv:1209.6061] [INSPIRE].ADSGoogle Scholar - [62]K. Dasgupta, J. Elituv, M. Emelin and A.-K. Trinh,
*Non-Kähler deformed conifold, ultra-violet completion and supersymmetric constraints in the baryonic branch*, arXiv:1805.03676 [INSPIRE]. - [63]K. Dasgupta, M. Emelin and E. McDonough,
*Non-Kähler resolved conifold, localized fluxes in M-theory and supersymmetry*,*JHEP***02**(2015) 179 [arXiv:1412.3123] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [64]M. Dhuria and A. Misra,
*Towards MQGP*,*JHEP***11**(2013) 001 [arXiv:1306.4339] [INSPIRE].ADSCrossRefGoogle Scholar - [65]A. Strominger, S.-T. Yau and E. Zaslow,
*Mirror symmetry is T duality*,*Nucl. Phys.***B 479**(1996) 243 [hep-th/9606040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [66]E. Witten,
*String theory dynamics in various dimensions*,*Nucl. Phys.***B 443**(1995) 85 [hep-th/9503124] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [67]A. Buchel,
*Violation of the holographic bulk viscosity bound*,*Phys. Rev.***D 85**(2012) 066004 [arXiv:1110.0063] [INSPIRE].ADSGoogle Scholar - [68]D.J. Gross and F. Wilczek,
*Asymptotically free gauge theories*—*I*,*Phys. Rev.***D 8**(1973) 3633 [INSPIRE].ADSGoogle Scholar - [69]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.ADSMathSciNetCrossRefGoogle Scholar - [70]S.C. Huot, S. Jeon and G.D. Moore,
*Shear viscosity in weakly coupled N*= 4*super Yang-Mills theory compared to QCD*,*Phys. Rev. Lett.***98**(2007) 172303 [hep-ph/0608062] [INSPIRE]. - [71]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Photon emission from ultrarelativistic plasmas*,*JHEP***11**(2001) 057 [hep-ph/0109064] [INSPIRE]. - [72]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Photon emission from quark gluon plasma: complete leading order results*,*JHEP***12**(2001) 009 [hep-ph/0111107] [INSPIRE]. - [73]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Photon and gluon emission in relativistic plasmas*,*JHEP***06**(2002) 030 [hep-ph/0204343] [INSPIRE]. - [74]E. Wang and U.W. Heinz,
*A generalized fluctuation dissipation theorem for nonlinear response functions*,*Phys. Rev.***D 66**(2002) 025008 [hep-th/9809016] [INSPIRE].ADSGoogle Scholar - [75]E. Wang and U.W. Heinz,
*Shear viscosity of hot scalar field theory in the real time formalism*,*Phys. Rev.***D 67**(2003) 025022 [hep-th/0201116] [INSPIRE].ADSGoogle Scholar - [76]A. Czajka et al.,
*Bulk viscosity of strongly interacting matter in the relaxation time approximation*,*Phys. Rev.***C 97**(2018) 044914 [arXiv:1712.05905] [INSPIRE].ADSMathSciNetGoogle Scholar - [77]G. ’t Hooft,
*Large N*, in the proceedings of*Phenomenology of large N*_{c}*QCD*, January 9–11, Tempe, U.S.A. (2002), hep-th/0204069 [INSPIRE]. - [78]O.K. Kalashnikov and V.V. Klimov,
*Infrared behavior of Green functions in Yang-Mills theory at finite temperatures*,*Sov. J. Nucl. Phys.***33**(1981) 443 [INSPIRE].Google Scholar - [79]V.V. Klimov,
*Spectrum of elementary Fermi excitations in quark gluon plasma*(in Russian),*Sov. J. Nucl. Phys.***33**(1981) 934 [*Yad. Fiz.***33**(1981) 1734] [INSPIRE]. - [80]E. Braaten and R.D. Pisarski,
*Soft amplitudes in hot gauge theories: a general analysis*,*Nucl. Phys.***B 337**(1990) 569 [INSPIRE].ADSCrossRefGoogle Scholar - [81]S. Caron-Huot,
*Asymptotics of thermal spectral functions*,*Phys. Rev.***D 79**(2009) 125009 [arXiv:0903.3958] [INSPIRE].ADSGoogle Scholar - [82]G. Baym and L.P. Kadanoff,
*Conservation laws and correlation functions*,*Phys. Rev.***124**(1961) 287 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [83]J.-P. Blaizot and E. Iancu,
*The quark gluon plasma: collective dynamics and hard thermal loops*,*Phys. Rept.***359**(2002) 355 [hep-ph/0101103] [INSPIRE]. - [84]P. Danielewicz,
*Quantum theory of nonequilibrium processes. 1.*,*Annals Phys.***152**(1984) 239 [INSPIRE]. - [85]J.M. Cornwall, R. Jackiw and E. Tomboulis,
*Effective action for composite operators*,*Phys. Rev.***D 10**(1974) 2428 [INSPIRE].ADSzbMATHGoogle Scholar - [86]E. Calzetta and B.L. Hu,
*Nonequilibrium quantum fields: closed time path effective action, Wigner function and Boltzmann equation*,*Phys. Rev.***D 37**(1988) 2878 [INSPIRE].ADSMathSciNetGoogle Scholar - [87]J. Berges,
*Introduction to nonequilibrium quantum field theory*,*AIP Conf. Proc.***739**(2004) 3 [hep-ph/0409233] [INSPIRE]. - [88]E.A. Calzetta, B.L. Hu and S.A. Ramsey,
*Hydrodynamic transport functions from quantum kinetic theory*,*Phys. Rev.***D 61**(2000) 125013 [hep-ph/9910334] [INSPIRE]. - [89]G. Aarts and J.M. Martinez Resco,
*Transport coefficients in large N*_{f}*gauge theories with massive fermions*,*JHEP***03**(2005) 074 [hep-ph/0503161] [INSPIRE]. - [90]M.E. Carrington and E. Kovalchuk,
*QED electrical conductivity using the 2PI effective action*,*Phys. Rev.***D 76**(2007) 045019 [arXiv:0705.0162] [INSPIRE].ADSGoogle Scholar - [91]M.E. Carrington and E. Kovalchuk,
*Leading order QED electrical conductivity from the 3PI effective action*,*Phys. Rev.***D 77**(2008) 025015 [arXiv:0709.0706] [INSPIRE].ADSGoogle Scholar - [92]M.E. Carrington and E. Kovalchuk,
*Leading order QCD shear viscosity from the three-particle irreducible effective action*,*Phys. Rev.***D 80**(2009) 085013 [arXiv:0906.1140] [INSPIRE].ADSGoogle Scholar - [93]A. Arrizabalaga and J. Smit,
*Gauge fixing dependence of*Φ*derivable approximations*,*Phys. Rev.***D 66**(2002) 065014 [hep-ph/0207044] [INSPIRE]. - [94]R. Kobes, G. Kunstatter and A. Rebhan,
*QCD plasma parameters and the gauge dependent gluon propagator*,*Phys. Rev. Lett.***64**(1990) 2992 [INSPIRE].ADSCrossRefGoogle Scholar - [95]H. Van Hees and J. Knoll,
*Renormalization of selfconsistent approximation schemes. 2. Applications to the sunset diagram*,*Phys. Rev.***D 65**(2002) 105005 [hep-ph/0111193] [INSPIRE]. - [96]H. van Hees and J. Knoll,
*Renormalization in selfconsistent approximation schemes at finite temperature. 3. Global symmetries*,*Phys. Rev.***D 66**(2002) 025028 [hep-ph/0203008] [INSPIRE]. - [97]J. Berges, S. Borsányi, U. Reinosa and J. Serreau,
*Nonperturbative renormalization for 2PI effective action techniques*,*Annals Phys.***320**(2005) 344 [hep-ph/0503240] [INSPIRE]. - [98]U. Reinosa and J. Serreau,
*2PI effective action for gauge theories: renormalization*,*JHEP***07**(2006) 028 [hep-th/0605023] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [99]U. Reinosa and J. Serreau,
*Ward Identities for the 2PI effective action in QED*,*JHEP***11**(2007) 097 [arXiv:0708.0971] [INSPIRE].ADSCrossRefGoogle Scholar - [100]D. Tong,
*Holographic conductivity*, lectures given at the*Cracow School on Theoretical Physics*, June 28–July 7, Cracow, Poland (2013).Google Scholar - [101]R. Argurio and M. Bertolini,
*Orientifolds and duality cascades: confinement before the wall*,*JHEP***02**(2018) 149 [arXiv:1711.08983] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [102]M. Attems et al.,
*Thermodynamics, transport and relaxation in non-conformal theories*,*JHEP***10**(2016) 155 [arXiv:1603.01254].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [103]M. Attems et al.,
*Thermodynamics, transport and relaxation in non-conformal theories*,*JHEP***10**(2016) 155 [arXiv:1603.01254] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [104]K. Dasgupta, M. Emelin, C. Gale and M. Richard,
*Renormalization group flow, stability and bulk viscosity in a large N thermal QCD model*,*Phys. Rev.***D 95**(2017) 086018 [arXiv:1611.07998] [INSPIRE].ADSMathSciNetGoogle Scholar - [105]M. Mia, F. Chen, K. Dasgupta, P. Franche and S. Vaidya,
*Non-extremality, chemical potential and the infrared limit of large N thermal QCD*,*Phys. Rev.***D 86**(2012) 086002 [arXiv:1202.5321] [INSPIRE].ADSGoogle Scholar - [106]C. Eling and Y. Oz,
*A Novel Formula for Bulk Viscosity from the Null Horizon Focusing Equation*,*JHEP***06**(2011) 007 [arXiv:1103.1657] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [107]K. Ohta and T. Yokono,
*Deformation of conifold and intersecting branes*,*JHEP***02**(2000) 023 [hep-th/9912266] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [108]K. Dasgupta, K. Oh and R. Tatar,
*Geometric transition, large N dualities and MQCD dynamics*,*Nucl. Phys.***B 610**(2001) 331 [hep-th/0105066] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [109]K. Dasgupta, K. Oh and R. Tatar,
*Open/closed string dualities and Seiberg duality from geometric transitions in M theory*,*JHEP***08**(2002) 026 [hep-th/0106040].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [110]K. Dasgupta, K.H. Oh, J. Park and R. Tatar,
*Geometric transition versus cascading solution*,*JHEP***01**(2002) 031 [hep-th/0110050].ADSMathSciNetCrossRefGoogle Scholar - [111]K. Dasgupta et al.,
*Infrared dynamics of a large N QCD model, the massless string sector and mesonic spectra*,*JHEP***07**(2015) 122 [arXiv:1409.0559] [INSPIRE].ADSCrossRefGoogle Scholar - [112]M. Dhuria and A. Misra,
*Transport coefficients of black MQGP M*3*-branes*,*Eur. Phys. J.***C 75**(2015) 16 [arXiv:1406.6076] [INSPIRE].ADSCrossRefGoogle Scholar - [113]M. Ionel and M. Min-Oo,
*Cohomogeneity one special lagrangian 3-folds in the deformed and the resolved conifolds*,*Illinois J. Math.***52**(2008) 839.MathSciNetCrossRefzbMATHGoogle Scholar - [114]M. Becker, K. Dasgupta, A. Knauf and R. Tatar,
*Geometric transitions, flops and non-Kähler manifolds. I.*,*Nucl. Phys.***B 702**(2004) 207 [hep-th/0403288] [INSPIRE]. - [115]S. Alexander et al.,
*In the realm of the geometric transitions*,*Nucl. Phys.***B 704**(2005) 231 [hep-th/0408192].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [116]M. Becker et al.,
*Geometric transitions, flops and non-Kähler manifolds. II*,*Nucl. Phys.***B 738**(2006) 124 [hep-th/0511099]. - [117]F. Chen et al.,
*Supersymmetric configurations, geometric transitions and new non-Kähler manifolds*,*Nucl. Phys.***B 852**(2011) 553 [arXiv:1007.5316] [INSPIRE].ADSzbMATHGoogle Scholar - [118]K. Becker, M. Becker, K. Dasgupta and R. Tatar,
*Geometric transitions, non-Kahler geometries and string vacua*,*Int. J. Mod. Phys.***A 20**(2005) 3442 [hep-th/0411039] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [119]K. Sil and A. Misra,
*On aspects of holographic thermal QCD at finite coupling*,*Nucl. Phys.***B 910**(2016) 754 [arXiv:1507.02692] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [120]A.M. Uranga,
*Brane configurations for branes at conifolds*,*JHEP***01**(1999) 022 [hep-th/9811004] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [121]K. Dasgupta and S. Mukhi,
*Brane constructions, conifolds and M-theory*,*Nucl. Phys.***B 551**(1999) 204 [hep-th/9811139] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [122]E. Witten,
*Solutions of four-dimensional field theories via M-theory*,*Nucl. Phys.***B 500**(1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [123]D. Tong,
*NS5-branes, T duality and world sheet instantons*,*JHEP***07**(2002) 013 [hep-th/0204186] [INSPIRE].ADSCrossRefGoogle Scholar - [124]A. Sen,
*Dynamics of multiple Kaluza-Klein monopoles in M and string theory*,*Adv. Theor. Math. Phys.***1**(1998) 115 [hep-th/9707042] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [125]P.K. Kovtun and A.O. Starinets,
*Quasinormal modes and holography*,*Phys. Rev.***D 72**(2005) 086009 [hep-th/0506184].ADSGoogle Scholar - [126]K. Sil, V. Yadav and A. Misra,
*Top-down holographic G-structure glueball spectroscopy at (N)LO in N and finite coupling*,*Eur. Phys. J.***C 77**(2017) 381 [arXiv:1703.01306].ADSCrossRefGoogle Scholar - [127]V. Yadav, A. Misra and K. Sil,
*Delocalized SYZ mirrors and confronting top-down*SU(3)*-structure holographic meson masses at finite g and N*_{c}*with P(article) D(ata) G(roup) Values*,*Eur. Phys. J.***C 77**(2017) 656 [arXiv:1707.02818]. - [128]K. Sil and A. Misra,
*New insights into properties of large-N holographic thermal QCD at finite gauge coupling at (the non-conformal/next-to) leading order in N*,*Eur. Phys. J.***C 76**(2016) 618 [arXiv:1606.04949] [INSPIRE].ADSCrossRefGoogle Scholar - [129]C.P. Herzog,
*The sound of M-theory*,*Phys. Rev.***D 68**(2003) 024013 [hep-th/0302086] [INSPIRE].ADSMathSciNetGoogle Scholar - [130]C.M. Bender and S.A. Orzag,
*Advanced mathematical methods for scientists and engineers I. Asymptotic methods and perturbation theory*, Springer, Germany (1999).Google Scholar - [131]P. Kovtun, D.T. Son and A.O. Starinets,
*Holography and hydrodynamics: Diffusion on stretched horizons*,*JHEP***10**(2003) 064 [hep-th/0309213] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [132]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].ADSMathSciNetCrossRefGoogle Scholar - [133]J. Casalderrey-Solana, S. Grozdanov and A.O. Starinets,
*Transport peak in the thermal spectral function of*\( \mathcal{N} \) = 4*supersymmetric Yang-Mills plasma at intermediate coupling*,*Phys. Rev. Lett.***121**(2018) 191603 [arXiv:1806.10997] [INSPIRE].ADSCrossRefGoogle Scholar - [134]T.W. Grimm, T.G. Pugh and M. Weissenbacher,
*On M-theory fourfold vacua with higher curvature terms*,*Phys. Lett.***B 743**(2015) 284 [arXiv:1408.5136] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [135]S. Caron-Huot et al.,
*Photon and dilepton production in supersymmetric Yang-Mills plasma*,*JHEP***12**(2006) 015 [hep-th/0607237] [INSPIRE].ADSCrossRefGoogle Scholar