D1-brane correction to a line operator index

TL;DR

This work analyzes finite-$k$ corrections to line operator indices in ${\cal N}=4$ $U(N)$ SYM via the AdS/CFT correspondence. While the large-$k$ limit is captured by D3-brane fluctuations, the symmetric representation exhibits a mismatch with gauge-theory indices, motivating the study of D1-brane contributions. The authors show that D1-branes stretched along a diameter of the D3-brane’s $S^2$ introduce BPS boundary modes whose spectrum yields a leading correction with $i_{D1}=-i_{\rm F1}=-x-y+q$ and $I_{D1,1}=\frac{(1-x)(1-y)}{1-q}$. They perform both a representation-theory argument and an explicit direct-mode analysis, including unitarity considerations for non-unitary boundary modes, and discuss broader implications for the AdS/CFT dictionary of line operators and finite-$N$ corrections.

Abstract

Wilson line operators in the rank $k$ totally symmetric tensor representation of ${\cal N}=4$ $U(N)$ sypersymmetric Yang-Mills theories are expected to be realized as D3-branes expanded in $AdS_5$. Although there is a mismatch between the corresponding line operator indices even in the large $N$ and large $k$ limit, it is possible to calculate the finite $k$ correction on the AdS side as the contribution from D1-branes. We analyze D1-brane fluctuation modes and calculate the leading finite $k$ correction to the line operator index on the AdS side.

Figures (3)

• Figure 1: The tubular D3-brane and a D1-brane along the diameter.
• Figure 2: Irreducible representations of $osp(2|4)$
• Figure 3: The mode spectrum is shown in two cases: (a) $\sinh\sigma_*<1$ and (b) $\sinh\sigma_*>1$. For the modes shown as circles, we treat the corresponding expansion coefficients in a standard way. Namely, they are treated as annihilation operators. For the modes shown as squares, the corresponding coefficients are treated as creation operators.