What does the string/gauge correspondence teach us about Wilson loops?

TL;DR

The work surveys how Wilson loop expectation values can be extracted from string/gauge dualities across AdS$_5\times$S$^5$, non-conformal D$_p$-brane backgrounds, and finite-temperature geometries. It develops a unified classical framework for static Wilson loops in curved backgrounds, proves a general theorem that yields leading and subleading confinement behavior, and derives sufficient conditions for area-law confinement. The analysis extends to non-supersymmetric type 0 models, includes quantum fluctuations and Luscher terms, and discusses the impact of a nontrivial $B_{\mu\nu}$ background on Wilson loops. It also addresses finite-temperature screening, area laws in 3d and 4d YM, monopole ('t Hooft) lines, exact aspects in flat space, and baryon configurations via wrapped brane vertices, providing a broad set of predictions for confining gauge theories from holography. Overall, the work strengthens the gauge/gravity correspondence as a tool to understand confinement, quantify corrections from fluctuations, and connect string theoretic constructs to observable features like quark potentials and baryon stability in strongly coupled gauge theories.

Abstract

In these lectures we describe the attempt to extract the expectation values of Wilson loops from the string/gauge correspondence. We start with the original calculation in $AdS_5$. It is then extended to the non-conformal background of $D_p$ in the near horizon limit. We discuss the computation at finite temperature. Supergravity models that admit confinement in 3d and 4d are described. A theorem that determines the classical values of loops associated with a generalized background is derived.In particular we determine sufficient conditions for confining behavior. We then apply the theorem to various string models including type 0 ones. We describe an attempt to go beyond the classical string picture by incorporating quadratic quantum fluctuations. In particular we address the BPS configuration of a single quark, the supersymmetric determinant of $AdS_5\times S^$ and a setup that corresponds to a confining gauge theory. We determine the form of the Wilson loop for actions that include non trivial $B_{μν}$ field. The issue of an exact determination of the value of the stringy Wilson loop is discussed. We end with a brief review of the baryons from the string/gauge correspondence Lectures presented in the ``Advanced School on Supersymmetry in the theories of fields, strings and branes'' Santiago de Compostela-99.

Figures (10)

• Figure 1: The basic setup of the Wilson loop
• Figure 2: The Wilson loop setup for the $AdS_5\times S^5$
• Figure 3: "Duality" between SYM with 16 supersymmetries and the supergravity of $D_p$ in the near horizon limit..
• Figure 4:
• Figure 6: The energy $E$ of the quark-anti-quark pair as a function of $L$ for a given $T$. The solid line corresponds to the numerical calculation up to $L_C$, the dashed line indicates the expected behavior for large $L$.

Theorems & Definitions (1)

• Theorem 1