Confinement in $(1+1)$ dimensions: a holographic perspective from I-branes

Abstract

In this paper we holographically study the strongly coupled dynamics of the field theory on I-branes (D5 branes intersecting on a line). In this regime, the field theory becomes $(2 + 1)$ dimensional with 16 supercharges. The dual background has an IR singularity. We resolve this singularity by compactifying the theory on a circle, preserving 4 supercharges. We study various aspects: confinement, symmetry breaking, Entanglement Entropy, etc. We also discuss a black membrane solution and make some comments on the string $σ$-model on our backgrounds.

Figures (7)

• Figure 1: Dual Theory in ($1+1$) dimensions with chiral fermions running in the links. This encodes the field theory at weak coupling. The inflow from the bulk of the branes is understood.
• Figure 2: Upper left: Plot comparing the exact expression for the quark separation $L_{QQ}$ in eq.(\ref{['lqq2']}) with the approximate one $\hat{L}_{QQ}$ in eq.(\ref{['lqqapp']}). The minimal separation as $r_0$ grows large hints at a LST behaviour. Upper right: Plot of $E_{QQ}(r_0)$ with $r_0$ in units of $r_+$. The plot is made in the BPS limit with $m=0$. Bottom left: Parametric plot of $E_{QQ}(L_{QQ})$ in the BPS bound $m=0\,,\, r_+=e_B=e_A=1$. Bottom right: the profiles of different strings as they explore the bulk. The longer the separation $L_{QQ}$, the more the string approaches $\frac{r_0}{r_+}\sim 1$. This is usual of the backgrounds dual to a confining QFT behaviour.
• Figure 3: Left: Plot comparing the exact expression for the quark separation $L_{MM}$ in eq.(\ref{['Lmmexact']}) with the approximate one $\hat{L}_{MM}$ in eq.(\ref{['aproxLmm']}), both in the BPS limit. Right: the profiles of different strings as they enter the bulk. Strings with small separation of the monopole pair penetrate deeper into the bulk. There is a maximum separation for the pair of monopoles, associated with the Little String Theory scale.
• Figure 4: Left: after introducing a cutoff at $r_{MAX}$, the figure shows the double-valued character of $L_{MM}$. Removing the cutoff recovers the LST behaviour (on the left panel of Figure \ref{['figura2']}). Right: with the UV cutoff at $r_{MAX}=20$, we see that $E_{MM}$ as a function of $L_{MM}$ has the 'upwards' concavity (indicating stability). The presence of the phase transition to a disconnected configuration is observed.
• Figure 5: Left: Plot comparing the exact expression for the separation of the entangled regions $L_{EE}$ in eq.(\ref{['LEEexact']}) with the approximate one $\hat{L}_{EE}$ in eq.(\ref{['LEEE']}). Right: The plot of $S_{EE}(r_0)$.