Abstract
We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions d>2d>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 decoupling operator, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of 2d2d 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)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=3d=3 and d=2+\epsilond=2+ϵ dimensions.
Highlights
Coupling gapless particles with gauge fields is one of the few known ways to obtain an interacting conformal field theory in dimensions d > 2
One cannot help to wonder if a similar physics exists for higher dimensional conformal field theories (CFTs), and can it be further utilized in the bootstrap study? We provide a positive answer to this by exploring gauge theories and in particular, their relations with the 2d WZW CFTs
The color number Nc of the gauge group plays the role of the WZW level (i.e. k) in WZW CFTs. We explore another related observation in higher dimensional CFTs, namely the equation of motion (EOM) can lead to the phenomenon of operator missing in the CFT spectrum [15, 16, 36, 37]
Summary
Coupling gapless particles with gauge fields is one of the few known ways to obtain an interacting conformal field theory in dimensions d > 2. The phenomenon that these CFTs appear at kinks of bootstrap bound is closely related to the existence of a family of CFTs sharing the same global symmetry and similar operator spectrum Compared to their cousins, the Ising CFT and SU(2) WZW are special because they have null operators at low levels. One can impose the condition of φ3 being missing by adding a large gap above φ in the O(N ) vector channel, this is how one obtains the famous bootstrap island of WF CFTs [14, 16] We will push this idea further by exploring the consequence of EOMs on high level missing operators. We will discuss the idea of decoupling operators and their bootstrap application in the context of a prototypical gauge theory, namely the scalar QED It is described by Nf flavor critical bosons coupled to a U(1) gauge field, L. Note added: Upon the completion of this work we became aware of an independent work [55] that overlaps with ours
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