AdS/CFT—Adding Probes

Abstract

In real experiments, one often adds “probes” to a system to examine its response. Or one adds impurities to a system to see how they change the properties of the system. In this chapter, we discuss how to add probes in AdS/CFT. As a typical example, we add “quarks” to gauge theories as probes and see the behavior of quark potentials.

Appendix: A Simple Example of the Confining Phase

In the text, we discussed the cutoff AdS spacetime as a toy model of the confining phase. Here, as an explicit example, we discuss the $S^1$-compactified ${\fancyscript{N}}=4$ SYM and its dual geometry.

AdS soliton   The SAdS$_5$ black hole is given by

$$\begin{aligned} ds_5^2&= \left( \frac{r}{L}\right) ^2 \big ({-}hdt^2+dx^2+dy^2+dz^2\big )+L^2\frac{dr^2}{h r^2}, \end{aligned}$$

(8.57)

$$\begin{aligned} h&= 1- \left( \frac{r_0}{r}\right) ^4. \end{aligned}$$

(8.58)

We now compactify the $z$-direction as $0 \le z <l$.

However, the compactified SAdS$_5$ black hole is not the only solution whose asymptotic geometry is $\mathbb {R}^{1,2} \times S^1$. The “double Wick rotation”

$$\begin{aligned} z' = it, \quad z = it' \end{aligned}$$

(8.59)

of the black hole gives the metric

$$\begin{aligned} ds_5^2 = \left( \frac{r}{L}\right) ^2 \big ({-}dt'^2+dx^2+dy^2+hdz'^2\big )+L^2\frac{dr^2}{h r^2}, \end{aligned}$$

(8.60)

which has the same asymptotic structure $\mathbb {R}^{1,2} \times S^1$. The geometry (8.60) is known as the AdS soliton [19].

As Euclidean geometries, they are the same, but they have different Lorentzian interpretations. The AdS soliton is not a black hole. Rather, because of the factor $h$ in front of $dz'^2$, the spacetime ends at $r=r_0$ just like the Euclidean black hole. From the discussion in the text, this geometry describes a confining phase.

For the SAdS black hole, the imaginary time direction has the periodicity $\beta = \pi L^2/r_0$ to avoid a conical singularity. Similarly, for the AdS soliton, $z'$ has the periodicity $l$ given by

$$\begin{aligned} l = \frac{\pi L^2}{r_0}. \end{aligned}$$

(8.61)

Wilson loop   Let us consider the quark potential in this geometry. Take the quark separation as ${\fancyscript{R}}$ in the $x$-direction. This corresponds to a Wilson loop on the $t'-x$ plane. Since the geometry ends at $r=r_0$, the formula (8.22) gives

$$\begin{aligned} E_t \propto \sqrt{-g_{t't'}g_{xx}}|_{r_0} \, {\fancyscript{R}}= \left( \frac{r_0}{L}\right) ^2 {\fancyscript{R}}, \end{aligned}$$

(8.62)

which is a confining potential.

In Sect. 8.2, we considered the Wilson loop in the SAdS black hole and discussed the Debye screening. Here, we consider a Wilson loop in the same Euclidean geometry, but the Wilson loop here is different from the one in Sect. 8.2:

• For the AdS soliton, we consider the Wilson loop on the $t'$-$x$ plane (temporal Wilson loop), but as the black hole, this is a Wilson loop on the $z$-$x$ plane or a spatial Wilson loop.

• For the black hole, we considered the temporal Wilson loop on the $t$-$x$ plane, but as the AdS soliton, this is a spatial Wilson loop on the $z'$-$x$ plane.

At high temperature $Tl>1$, the AdS soliton undergoes a first-order phase transition to the SAdS black hole (Sect. 14.2.1).