# Gauge-Gravity Duality: A brief overview

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Gauge-Gravity Duality: A brief overview
Andrei Starinets Rudolf Peierls Centre for Theoretical Physics Oxford University ICMS workshop “Numerical relativity beyond astrophysics” Edinburgh 12 July 2011

Some references: O.Aharony, S.Gubser, J.Maldacena, H.Ooguri, Y.Oz, hep-th/ J.Casalderrey-Solana, H.Liu, D.Mateos, K.Rajagopal, U.Wiedemann, [hep-th] D.T.Son and A.O.S., “Viscosity, Black Holes, and Quantum Field Theory”, [hep-th] P.K.Kovtun and A.O.S., “Quasinormal modes and holography”, hep-th/

S. Hartnoll “Lectures on holographic methods for condensed matter physics”, [hep-th] C. Herzog “Lectures on holographic superfluidity and superconductivity”, [hep-th] M. Rangamani “Gravity and hydrodynamics: Lectures on the fluid-gravity correspondence”, [hep-th] S.Sachdev “Condensed matter and AdS/CFT”, [hep-th]

What is string theory?

Equations such as describe the low energy limit of string theory As long as the dilaton is small, and thus the string interactions are suppressed, this limit corresponds to classical 10-dim Einstein gravity coupled to certain matter fields such as Maxwell field, p-forms, dilaton, fermions Validity conditions for the classical (super)gravity approximation - curvature invariants should be small: - quantum loop effects (string interactions = dilaton) should be small: In AdS/CFT duality, these two conditions translate into and

From brane dynamics to AdS/CFT correspondence
Open strings picture: dynamics of coincident D3 branes at low energy is described by Closed strings picture: dynamics of coincident D3 branes at low energy is described by conjectured exact equivalence Maldacena (1997); Gubser, Klebanov, Polyakov (1998); Witten (1998)

(super)conformal field theory = coupling doesn’t run
supersymmetric YM theory Gliozzi,Scherk,Olive’77 Brink,Schwarz,Scherk’77 Field content: Action: (super)conformal field theory = coupling doesn’t run

conjectured exact equivalence Generating functional for correlation functions of gauge-invariant operators String partition function In particular Classical gravity action serves as a generating functional for the gauge theory correlators

AdS/CFT correspondence: the role of J
For a given operator , identify the source field , e.g. satisfies linearized supergravity e.o.m. with b.c. The recipe: To compute correlators of , one needs to solve the bulk supergravity e.o.m. for and compute the on-shell action as a functional of the b.c. Warning: e.o.m. for different bulk fields may be coupled: need self-consistent solution Then, taking functional derivatives of gives

Holography at finite temperature and density
Nonzero expectation values of energy and charge density translate into nontrivial background values of the metric (above extremality)=horizon and electric potential = CHARGED BLACK HOLE (with flat horizon) temperature of the dual gauge theory chemical potential of the dual theory

Example: charge diffusion
Hydrodynamics: fundamental d.o.f. = densities of conserved charges Need to add constitutive relations! Example: charge diffusion Conservation law Constitutive relation [Fick’s law (1855)] Diffusion equation Dispersion relation Expansion parameters:

4-dim gauge theory – large N, strong coupling 10-dim gravity
M,J,Q Holographically dual system in thermal equilibrium M, J, Q T S Gravitational background fluctuations Deviations from equilibrium ???? and B.C. Quasinormal spectrum

First-order transport (kinetic) coefficients
Shear viscosity Bulk viscosity Charge diffusion constant Supercharge diffusion constant Thermal conductivity Electrical conductivity * Expect Einstein relations such as to hold

Second-order hydrodynamics
Hydrodynamics is an effective theory, valid for sufficiently small momenta First-order hydro eqs are parabolic. They imply instant propagation of signals. This is not a conceptual problem since hydrodynamics becomes “acausal” only outside of its validity range but it is very inconvenient for numerical work on Navier-Stokes equations where it leads to instabilities [Hiscock & Lindblom, 1985] These problems are resolved by considering next order in derivative expansion, i.e. by adding to the hydro constitutive relations all possible second-order terms compatible with symmetries (e.g. conformal symmetry for conformal plasmas)

Second-order conformal hydrodynamics (in d dimensions)

Second-order transport (kinetic) coefficients
(for theories conformal at T=0) Relaxation time Second order trasport coefficient Second order trasport coefficient Second order trasport coefficient Second order trasport coefficient In non-conformal theories such as QCD, the total number of second-order transport coefficients is quite large

Predictions of the second-order conformal hydrodynamics
Sound dispersion: Kubo:

In quantum field theory, the dispersion relations such as
appear as poles of the retarded correlation functions, e.g. - in the hydro approximation -

The role of quasinormal modes
G.T.Horowitz and V.E.Hubeny, hep-th/ D.Birmingham, I.Sachs, S.N.Solodukhin, hep-th/ D.T.Son and A.O.S., hep-th/ ; P.K.Kovtun and A.O.S., hep-th/ I. Computing the retarded correlator: inc.wave b.c. at the horizon, normalized to 1 at the boundary II. Computing quasinormal spectrum: inc.wave b.c. at the horizon, Dirichlet at the boundary

Sound and supersymmetric sound in
In 4d CFT Sound mode: Supersound mode: Quasinormal modes in dual gravity Graviton: Gravitino:

Quasinormal spectra of black holes/branes
Schwarzschild black hole (asymptotically flat) AdS-Schwarzschild black brane

Sound dispersion in analytic approximation analytic approximation

First-order transport coefficients in N = 4 SYM
in the limit Shear viscosity Bulk viscosity for non-conformal theories see Buchel et al; G.D.Moore et al Gubser et al. Charge diffusion constant Supercharge diffusion constant (G.Policastro, 2008) Thermal conductivity Electrical conductivity

Shear viscosity in SYM perturbative thermal gauge theory
S.Huot,S.Jeon,G.Moore, hep-ph/ Correction to : Buchel, Liu, A.S., hep-th/ Buchel, [hep-th]; Myers, Paulos, Sinha, [hep-th]

Universality of Theorem:
For a thermal gauge theory, the ratio of shear viscosity to entropy density is equal to in the regime described by a dual gravity theory Remarks: Extended to non-zero chemical potential: Benincasa, Buchel, Naryshkin, hep-th/ Extended to models with fundamental fermions in the limit Mateos, Myers, Thomson, hep-th/ String/Gravity dual to QCD is currently unknown

The absorption argument D. Son, P. Kovtun, A.S., hep-th/ Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/ “Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem P. Kovtun, D.Son, A.S., hep-th/ , A.S., [hep-th], P.Kovtun, A.S., hep-th/ , A.Buchel, J.Liu, hep-th/

Computing transport coefficients from dual gravity
Assuming validity of the gauge/gravity duality, all transport coefficients are completely determined by the lowest frequencies in quasinormal spectra of the dual gravitational background (D.Son, A.S., hep-th/ , P.Kovtun, A.S., hep-th/ ) This determines kinetics in the regime of a thermal theory where the dual gravity description is applicable Transport coefficients and quasiparticle spectra can also be obtained from thermal spectral functions

Hydrodynamic properties of strongly interacting hot plasmas in 4 dimensions
can be related (for certain models!) to fluctuations and dynamics of 5-dimensional black holes

Beyond near-equilibrium regime

Computing real-time correlation functions from gravity
To extract transport coefficients and spectral functions from dual gravity, we need a recipe for computing Minkowski space correlators in AdS/CFT The recipe of [D.T.Son & A.S., 2001] and [C.Herzog & D.T.Son, 2002] relates real-time correlators in field theory to Penrose diagram of black hole in dual gravity Quasinormal spectrum of dual gravity = poles of the retarded correlators in 4d theory [D.T.Son & A.S., 2001]

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