Musings

Let’s start with a stack of $N_c\gg 1$ D4-branes, wrapped on a circle ($x_4\sim x_4+2\pi R$). The theory on the worldvolume of the D4 branes is a 5-dimensional supersymmetric $U(N_c)$ gauge theory, whose 't Hooft coupling is $${\lambda }_5=g_5^2{N}_c=(2\pi {)}^2{g}_s{\ell }_s{N}_c$$ The UV completion of this theory is not 5-dimensional, but 6-dimensional, the $A_{N_c-1}$ (2,0) superconformal theory in $D=6$. At length scales much longer than $R$, the theory is effectively 4-dimensional. With supersymmetry-preserving boundary condition on the $S^1$, it would reduce to $N=4$ SYM. Instead, we will choose the bounding spin structure for the fermions on the $S^1$, which breaks supersymmetry. The 4D fermions pick up masses of order $1/R$, and the scalars in the adjoint of $\mathrm{SU}(N_c)$ pick up 1-loop mass-squareds of order $$\frac{{\lambda }_4}{16{\pi }^2}\frac{1}{R^2}$$ where ${\lambda }_4=\frac{{\lambda }_5}{2\pi R}$ is the 4D 't Hooft coupling. The six neutral scalars (corresponding to the center-of-mass postion of the D4-branes in $x_5,\dots x_9$ and the $U(1)$ Wilson line on the circle) remain massless, but decouple.

This theory reduces to pure Yang-Mills in 4D. In the limit where ${\lambda }_4\ll 1$ the coupling is weak at the cutoff scale, $1/R$, and the theory confines at a scale exponentially smaller than the cutoff. The supergravity approximation is valid in the opposite limit, where ${\lambda }_5/R\gg 1$.

The near-horizon geometry is

where $$\begin{array}{rlr}{u}_0& =(\pi g_s{N}_c{)}^{1/3}{\ell }_s,& u_{\Lambda }={\left(\frac{2}{3R}\right)}^2{u}_0^3,\\ f(u)& =1-{\left(\frac{u_{\Lambda }}{u}\right)}^3,\\ G^{(4)}& =2\pi N_c\frac{{\mathrm{d}}^4{\Omega }_4}{V_4},& e^{\varphi }=g_s{\left(\frac{u}{u_0}\right)}^{3/4}\end{array}$$ which is weakly-curved¹ at $u=u_{\Lambda }$ for ${\lambda }_5/R\gg 1$.

For large ${\lambda }_5/R$, the gravity model undergoes a Hawking-Page transition, which is the gravitational dual of the confinement/deconfinement transition, at a temperature $T=\frac{1}{2\pi R}$. Wick rotate $t\to i\tau$, and periodically identify $\tau \sim \tau +1/T$. In the low-temperature phase, the geometry (1), in which the $x_4$-circle shrinks to zero size at $u=u_{\Lambda }$, dominates. In the high temperature phase, it’s the geometry where the $\tau$-circle shrinks to zero size at $$u_T={\left(\frac{4\pi T}{3}\right)}^2{u}_0^3$$ which dominates. When $T=\frac{1}{2\pi R}$, the two circles have the same radius, and there’s an obvious symmetry under $x_4\leftrightarrow \tau$. But the two geometries are not continuously connected, and the transition is 1^st order.

Now, following Sakai and Sugimoto, we add $N_f\ll N_c$ D8-branes at $x_4=0$ and $N_f$ $\overline{\text{D8}}$-branes at $x_4=L$, for $0<L\le \pi R$. This configuration is, of course, unstable (because of the mutual attraction of the D8- and $\overline{\text{D8}}$-branes). But that instability can be systematically suppressed by taking $g_s\to 0$ and $g_s{N}_f\to 0$ while holding $g_s{N}_c$ fixed. In this “probe” limit, the introduction of the D8 branes does not affect the geometry (1). Nor does it affect the “bulk” thermodynamics (the number of degrees of freedom of the gauge theory scales like $N_c^2\gg N_c{N}_f$.

Quantizing the 4-8 strings, the ground state of the NS sector is massive, while the ground state of the R sector is comprised of massless fermions in the $(N_f,N_c)$ representation of $U(N_f)\times U(N_c)$, “quarks”. In the probe limit, the gauge coupling on the D8-branes vanishes, and the theory has a $U(N_f)\times U(N_f)$ global symmetry.

Now let’s look at the near-horizon geometry, dual to the low-energy gauge theory. In the low-temperature phase (1), the $x_4$ circle shrinks to zero size at $u=u_{\Lambda }$. At that point, the D8- and $\overline{\text{D8}}$-branes merge smoothly. Rather than separate stacks of D8- and $\overline{\text{D8}}$-branes, there is only a single stack which starts out, asymptotically on the cigar, as a stack of D8-branes located at $x_4=0$. It curves around at the tip of the cigar, and emerges asymptotically as a stack of $\overline{\text{D8}}$-branes located at $x_4=L$. The would-be $U(N_f)\times U(N_f)$ symmetry is broken to $U(N_f{)}_{\text{diag}}$.

Correspondingly, among the mesons of the theory (strings stretched between the D8- and $\overline{\text{D8}}$-branes) are a set of massless Goldstone bosons, localized near the tip of the cigar. The pion decay constant, as found by Sakai and Sugimoto, is $$f_{\pi }^2=\frac{1}{27{\pi }^4}\frac{{\lambda }_4{N}_c}{R^2}$$

In the high temperature (deconfined) phase, the $x_4$ circle is not capped off. Instead it’s the $\tau$ circle that is capped off at $u=u_T$. So there exists a solution in which the one has separate stacks of D8- and $\overline{\text{D8}}$-branes at $x_4=0,L$. But there’s also a solution where the two stacks bend around and rejoin at some $u=u_{\text{min}}(T)$. In the former solution, chiral symmetry is unbroken; in the latter, it is broken to $U(N_f)$, just as in the confining phase.

Which solution is favoured is an energetic calculation, comparing the DBI action for the two solutions. The result is that the transition temperature is $T_{\chi }=0.154/L$. This, of course, hold only so long as $T_{\chi }>T_d$. $$\begin{array}{rl}{T}_d& =\frac{1}{2\pi R}\\ T_{\chi }& =\{\begin{array}{ll}0.154/L& T_{\chi }>T_d(\text{or, equivalently,}L/R<0.97)\\ T_d& \text{otherwise}\end{array}\end{array}$$

Phase diagram of the Sakai-Sugimoto model.

So the phase diagram found by Aharony et al looks like the figure at left. All of the phase transitions are first order. In the pale blue region below $\mathrm{TR}=1/2\pi$, the theory confines and chiral symmetry is broken. In the pink region, the theory is deconfined but, nonetheless, chiral symmetry is broken. Above all the lines, chiral symmetry is unbroken and the theory is deconfined.

Physically, the existence of a phase with chiral symmetry-breaking but without confinement seems rather exotic. It’s a feature of the large ${\lambda }_4$ limit that deserves to be better-understood. Also, the $x_4\leftrightarrow \tau$ duality at $T=\frac{1}{2\pi R}$ persists as we lower ${\lambda }_4$ (though the supergravity approximation breaks down). However, for small ${\lambda }_4$, we expect the deconfinement transition at a temperature $T_d\ll \frac{1}{2\pi R}$. What happens at $T=\frac{1}{2\pi R}$? Is there a phase transition? Finally: before adding the probe D8-branes, the theory had a UV completion as a 6 dimensional quantum field theory. What about after adding the D8-branes? Is there a UV completion, besides the full 10 string theory?