Line Operator Index on S1 $\times$ S3

TL;DR

This work constructs and tests a comprehensive line-operator superconformal index for 4d ${\cal N}=2$ theories on $S^1\times S^3$, incorporating both 1-loop and monopole bubbling effects. By detailing the 1-loop structure for ${\cal N}=4$ SYM and extending to general ${\cal N}=2$ theories, it derives explicit bubbling-corrected indices and demonstrates their consistency with S-duality across multiple gauge groups and representations. The authors further connect line-operator indices to Verlinde loop operators, validating the framework through concrete SU(2) and SU(3) examples, including ${\cal N}=2$ theories with four flavors, and show holographic matching with AdS5×S5 via fundamental strings and D5-branes. Together, these results provide a robust, cross-checked toolkit for exact line defect observables in 4d SCFTs with implications for dualities and holography.

Abstract

We derive a general formula of an index for N = 2 superconformal field theories on S1 \times S3 with insertions of BPS Wilson line or 't Hooft line operator at the north pole and their anti-counterpart at the south pole of S3. One-loop and monopole bubbling effects are taken into account in the computation. As examples, we calculate the indices for N = 4 theories and N = 2 SU(2) theory with Nf = 4, and find good agreements between indices of line operators related by S-duality. The relation between Verlinde loop operators and the indices is explored. The holographic correspondence between the fundamental (anti-symmetric) Wilson line operator and the fundamental string (D5 brane) in AdS5\timesS5 is confirmed by the index comparison.

Figures (5)

• Figure 1: A monopole and an antimonopole on $S^3$
• Figure 2: A simple brane picture of monopole bubbling. Black lines represent D3-branes (0123) and horizontal lines represent D1-branes (04). Infinitely stretched D1s (red line) ending on the D3s correspond to 't Hooft line operators in the field theory on D3s. A line operator with magnetic charge $B=(2,0)$ (left) can be screened by a massless monopole (brown line) and have reduced charge $v=(1,1)$ (right).
• Figure 3: $(i_s, j_s)$ denote the location of $s$ in $Y_{\alpha}$. In this example, $i_s=1, j_s=2$, and the arm-and leg-length are denoted by the white and black disk, $A_{Y_{\alpha}}(s)=1, L_{Y_{ \alpha}}(s)=1$ (left). The arm- and leg-length can be defined for $Y_{ \beta}$ such that $s \notin Y_{ \beta}$. In this example, the arm-length $A_{Y_{ \beta}} (s)= 5-2=3$, and the leg-length $L_{Y_{ \beta}}(s)= 3-1=2$ (right).
• Figure 4: $p$ infinitely stretched D1-branes (04, red lines) ending on two D3-branes (0123, black lines) correspond to a 't Hooft operator with magnetic charge $\textrm{diag}. (p, 0)$, of which the traceless part is $B= \textrm{daig}. ( \frac{p}{2}, - \frac{p}{2})$ in the $SU(2)$ theory (left). Once $\frac{p-s}{2}$ massless D1-branes (04, brown lines) end on the tops of infinitely stretched D1-branes, the magnetic charge is reduced to $\textrm{diag}.(p- \frac{p-s}{2}, \frac{p-s}{2})$, of which the traceless part is $v=\textrm{diag}.(\frac{s}{2}, - \frac{s}{2})$ (right).
• Figure 5: Holographic description of the quark-antiquark ($q\bar{q}$) system in $U(N)$${\cal N}=4$ SYM. The figure describes a constant (global) time slice of $AdS_5$. A fundamental string (F1) is stretched along $\rho$ direction (radial direction in the figure) and meets the boundary (located at $\rho = \infty$) of $AdS_5$ at two points, the south and north poles on $S^3$.