Exploring Confinement in Anti-de Sitter Space

TL;DR

This paper investigates the confinement-deconfinement transition in Yang-Mills theory on four-dimensional Anti-de Sitter space, using perturbation theory to analyze boundary conditions, operator anomalous dimensions, and implications for the transition mechanism.

Contribution

It provides the first perturbative analysis of the transition in AdS space, computing anomalous dimensions and boundary effects, and relates these to the bulk beta function.

Findings

Singlet scalar operator has a larger negative anomalous dimension than the adjoint.

The correction to the coefficient C_J suggests the singlet reaches marginality before C_J=0.

Neumann boundary condition yields a positive anomalous dimension, consistent with flat space extrapolation.

Abstract

We study Yang-Mills theory on four dimensional Anti-de Sitter space. The Dirichlet boundary condition cannot exist at arbitrarily large radius because it would give rise to colored asymptotic states in flat space. As observed in [1] this implies a deconfinement-confinement transition as the radius is increased. We gather hints on the nature of this transition using perturbation theory. We compute the anomalous dimensions of the lightest scalar operators in the boundary theory, finding that the singlet gets a larger negative anomalous dimension compared to the adjoint. We also compute the correction to the coefficient $C_J$ and we estimate that the singlet operator reaches marginality before the value of the coupling at which $C_J=0$. These results favor the scenario of merger and annihilation as the most promising candidate for the transition. For the Neumann boundary condition, the lightest scalar operator is found to have a positive anomalous dimension, in agreement with the idea that this boundary condition extrapolates smoothly to flat space. The perturbative calculations are made possible by a drastic simplification of the gauge field propagator in Fried-Yennie gauge. We also derive a general result for the leading-order anomalous dimension of the displacement operator for a generic perturbation in Anti-de Sitter, showing that it is related to the beta function of bulk couplings.