Gauge-Gravity duality and hydrodynamics

Abstract

Gauge/gravity correspondence is a conjectured equivalence \cite{1} between a $(d+1)$-dimensional theory of gravity, such as string theory, and a $d$-dimensional theory with no gravity at all. The most precise statement of the correspondence is that of a 10-dimensional Type IIB string theory on $\text{AdS}_5 \times S^5$ with $N$ units of five-form flux through $S^5$. The parameters of the theory are the equal radii of $\text{AdS}5$ and $S^5$, $L$, and the ten-dimensional Newton's constant $G{10}$ given by

?? = ( 4 ?? ?? ?? ?? ) 1 / 4 ?? ?? , ?? 10 = 8 ?? 6 ?? ?? 2 ?? ?? 8 , L=(4?g s ?

N) 1/4 l s ?

,G 10 ?

=8? 6 g s 2 ?

l s 8 ?

,

where $g_s$ is the string coupling and $l_s$ is the string length. On the other side of the correspondence, we have a 4-dimensional super-Yang-Mills theory with maximal $N=4$ supersymmetry and $SU(N)$ gauge group. The only parameter of the theory is the Yang-Mills coupling constant $g_\text{YM}$. The parameters of the gravity theory are related to the parameters of SYM theory as

?? = 4 ?? ?? ?? ?? = ?? YM 2 ?? = ?? , ?? YM 2 = 4 ?? ?? ?? , ?=4?g s ?

N=g YM 2 ?

N=A,g YM 2 ?

=4?g s ?

,

where $A = g_\text{YM}^2 N$ is the ’t Hooft coupling. The gauge/gravity duality implies that the spectrum as well as all the physical observables of these two theories are equivalent to each other.

In the strongest form of the conjecture, the correspondence should hold for all values of $N$ and all regimes of the string coupling $g_s$. However, certain limits of the conjecture are more tractable. Since $G_{10} \sim 1/N^2$, in the ’t Hooft limit on the SYM side where $A$ is fixed as $N \to \infty$, quantum effects are suppressed. This corresponds to classical string theory on $\text{AdS}_5 \times S^5$. Higher stringy corrections are also small if the curvature of the background is much smaller than the string scale, $L \gg l_s$, which happens if $A \gg 1$. Thus we conclude that the strong ’t Hooft coupling limit of the large-$N$ gauge theory is dual to the two-derivative action of classical Type IIB supergravity on $\text{AdS}_5 \times S^5$.

So far we have described the $N=4$ super Yang-Mills theory at zero temperature. This setup can be generalized to non-zero temperature and non-zero chemical potential. Finite temperature in the gauge theory corresponds to studying a black brane (or black hole) solution in the bulk, and non-zero chemical potential corresponds to studying a Reissner-Nordström black hole in the bulk. We can even study near-equilibrium physics by turning on small time-dependent perturbations about the equilibrium configurations. In the domain of linear response theory, this allows us to study physical phenomena such as transport properties of the dual CFT.

The basic object to study transport phenomena in a field theory is the retarded two-point Green’s function, $G_R$,

?? ?? ( ?? , ?? ) = ?? ?? ( ?? ) ? [ ?? ( ?? , ?? ) , ?? ( 0 , 0 ) ] ? , G R ?

(t,x)=i?(t)?[O(t,x),O(0,0)]?,

where $\mathcal{O}(t, x)$ are operators of a $d$-dimensional quantum field theory, such as the stress tensor $T_{\mu\nu}$ or current $J_\mu$. Gauge/gravity correspondence provides a prescription for evaluating $G_R$ by studying the classical equations of motion in the bulk, which capture the response of the gauge theory to linear perturbations.

For a field theory at equilibrium temperature $T$ and chemical potential $\mu$, if we perturb the system slightly, the near-equilibrium behavior can be described by an effective hydrodynamic model. The evolution of the field theory is governed by the conservation of the stress tensor and current $J_\mu$. The perturbed fluid equilibrates through dissipation, and the response of these perturbations is encoded in transport coefficients such as shear viscosity $\eta$, electrical conductivity $\sigma$, etc. The gauge/gravity duality allows one to compute these transport coefficients via the retarded Green’s functions using Kubo formulas.

Recently, there has been interest in constructing holographic duals to model macroscopic low-energy phenomena. These duals provide new insights because the phenomena are strongly coupled in the field theory but semiclassical in gravity. A universal result is that the ratio of shear viscosity $\eta$ to entropy density $s$ for theories with isotropic gravity duals in the two-derivative approximation is

?? ?? = 1 4 ?? . s ? ?

= 4? 1 ?

.

This ratio has been evaluated for well-known theories like $N=4$ SYM, phenomenological gravity models, and even anisotropic backgrounds \cite{2,3}.

The gauge/gravity correspondence has been extended to non-AdS near-horizon geometries, e.g., Dp-branes ($p>2$), and non-conformal field theories \cite{4,5}. For 1+1 dimensional field theories, where there is no shear, studying non-conformal field theories is necessary to obtain non-trivial hydrodynamic coefficients. D1-branes provide the simplest symmetric non-conformal 1+1 dimensional field theory with a gravity dual. The dual description involves a charged black hole solution of an Einstein-Maxwell-dilaton system in 3 dimensions, obtained from a consistent truncation of a spinning D1-brane in 10 dimensions. Using the fluid/gravity correspondence, transport properties of the D1-brane theory were computed. The ratio of bulk viscosity to entropy density is independent of chemical potential and equals $1/4\pi$ (dimensionless units). Critical behavior appears in the charge diffusion mode at the boundary of thermodynamic stability ($k=3/2$), with a square-root branch cut near the critical point, yielding a critical exponent $1/2$, indicative of mean-field behavior. Similar behavior is observed for shear viscosity and conductivity of single-charged D3-branes. Transport coefficients of the D1-brane theory are proportional to those of the M2-brane up to a scaling factor.

Brane

?? ??

?? ??

??

??

Single-charged D1

( 2 ?? + 3 ) 2 16 ?? ?? 3

1

1 16 ?? ?? 3

?

Equal-charged D1

?

?

?

?

Equal-charged M2

?

?

?

? Brane Single-charged D1 Equal-charged D1 Equal-charged M2 ?

D c ?

16?G 3 ?

(2k+3) 2 ?

? ? ?

D s ?

1 ? ? ?

? 16?G 3 ?

1 ?

? ? ?

? ? ? ? ?

?

Gauge/gravity duality can also derive sum rules constraining spectral densities of strongly coupled systems \cite{7}:

? ?? ?? ? ?? ( ?? ) = constant , ?? ( ?? ) = Im ? ?? ?? ?? ( ?? ) , ?d??(?)=constant,?(?)=ImG ii ?

(?),

where the RHS depends on thermodynamic variables like energy, pressure, and chemical potential. These sum rules encode unitarity and causality and relate short-distance and long-distance physics. For example, shear and bulk sum rules in QCD are connected to hydrodynamic behavior and asymptotic freedom, while the Ferrell-Glover sum rule in BCS theory determines superconducting skin depth.

Parity-odd transport coefficients at first order in the derivative expansion are fixed by the microscopic theory \cite{10}. At second order, parity-odd coefficients constrain spectral densities and affect the dispersion of chiral modes. Using equilibrium partition function methods, several second-order coefficients can be expressed in terms of anomaly coefficients, shear viscosity, charge diffusivity, and thermodynamic functions. The remaining coefficients are constrained by relations involving the anomaly and bulk viscosity.

?? 2 ? ?? 1 = ?? ( anomaly , ?? , ?? ?? , ?? , ?? , ?? , ?? ) ,

? ?

A 2 ?

?A 1 ?

=f(anomaly,?,D c ?

,E,P,T,v), ? ?

where $C$ is the gauge anomaly coefficient and $C_2$ relates to the mixed gauge-gravitational anomaly. Other coefficients are constrained by three relations involving the anomaly and bulk viscosity.