Correlators of local operators and 1/8 BPS Wilson loops on S^2 from 2d YM and matrix models

TL;DR

The paper solidifies a precise 4d-2d connection for correlators of 1/8 BPS Wilson loops on S^2 with local operators in N=4 SYM by mapping these observables to the zero-instanton sector of 2d YM on S^2. It derives a Gaussian two-matrix model (equivalently a complex matrix model) that resums ladder diagrams in 2d YM and computes exact planar correlators, which at strong coupling reproduce AdS5 × S5 string results for latitudes and two-longitudes. Perturbative checks demonstrate agreement between 4d and 2d calculations, while localization arguments justify the 2d description and operator mapping O_J(x) → Tr(i r * tilde F)^J. The combined weak and strong coupling analyses provide robust evidence that the zero-instanton sector of 2d YM captures the relevant dynamics of these protected observables, offering exact computational control and deepening insights into the gauge/string duality.

Abstract

We propose that, in N=4 Super Yang-Mills theory, correlation functions of certain 1/8 BPS Wilson loops and local operators inserted on a S^2 in space-time may be computed in terms of analogous observables in the "zero-instanton" sector of 2d Yang-Mills theory. The Wilson loops are mapped to the standard Wilson loops of the 2d theory, as recently conjectured, while the local operators are mapped to powers of the 2d field strength. We give several perturbative checks of the correspondence, and derive from 2d Yang-Mills a two-matrix model for the correlator of a local operator and a Wilson loop of arbitrary shape. We show that the strong coupling planar limit of the two-matrix model precisely agrees with a string theory calculation in AdS_5 x S^5.

Figures (4)

• Figure 1: The correlator of a Wilson loop $W(C)$ and a local operator $O_J(x)$ on $S^2$. The curve $C$ divides the $S^2$ into two regions which we denote as $S^+$ and $S^-$. Perturbatively, we find that the correlator only depends on whether the operator is inserted in the $S^+$ or $S^-$ region.
• Figure 2: The correlator $\left \langle O_J(x_0)W(C)\right \rangle$ at strong coupling. The plane represents the $S^2\in \mathbb{R}^4$ at the $AdS_5$ boundary, on which the Wilson loop $C$ resides. The operator $O_J(x_0)$ is inserted at a point $x_0\in S^2$, in either of the regions $S^+$ or $S^-$. The wavy line represents the supergravity mode dual to the local operator, which propagates from the insertion point $x_0$ to a point on the string worldsheet dual to the Wilson loop.
• Figure 3: The 1/4 BPS latitude (a) and two-longitudes (b) Wilson loops. The corresponding dual string solutions are given respectively in eq. \ref{['latitude']} and eq. \ref{['longitudes']}.
• Figure 4: A typical ladder diagram for the connected correlator of two circular Wilson loops on $S^2$. In this example, the points $z_1,z_4,z_7$ belong to $T(2,1)$ and $w_5,w_2,w_4$ are the corresponding points in $I(T(2,1))$. The points $z_2,z_3,z_5,z_6$ belong to $T(2,2)\cup I(T(2,2))$, and finally $w_1,w_3$ are in $T(1,1)\cup J(T(1,1))$.