A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions

TL;DR

This paper introduces a method using decoupling operators to identify gauge groups in conformal field theories via bootstrap, demonstrated on scalar QED in various dimensions, advancing non-perturbative analysis of gauge theories.

Contribution

It proposes a novel approach employing decoupling operators to detect gauge group properties within the conformal bootstrap framework, applicable to theories in dimensions greater than two.

Findings

Successfully detected the gauge group rank using gap conditions.

Solved scalar QED in 3D and 2+ε dimensions via bootstrap.

Identified kinks and islands indicating critical points.

Abstract

We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions $d>2$. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of \emph{decoupling operator}, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of $2d$ Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a $U(1)$ gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both $d=3$ and $d=2+\epsilon$ dimensions.