Bounds on Abelian Currents in 4d CFTs

TL;DR

This work applies the four-dimensional conformal bootstrap to CFTs with a $U(1)$ global symmetry by analyzing the neutral sector of $\langle JJJJ\rangle$ and constructing the corresponding crossing equations. It develops a detailed tensor-structure classification for three-point functions and uses semidefinite programming to bound operator dimensions and central-charge-like quantities, notably $C_T$ and the abelian combination $P_J = (\lambda^-_{JJJ})^2 / C_J^3$, which encodes both the $t Hooft anomaly and current normalization. The results reveal non-perturbative constraints on $\Delta_{(0,0)}$, $\Delta_{(4,0)}$, $C_T$, and $P_J$, including characteristic kinks suggesting proximity to known theories such as the free complex scalar and the free Weyl fermion; Hofman–Maldacena bounds on the parameter $\gamma$ emerge when appropriate gaps are imposed. The framework lays groundwork for extending to non-abelian global symmetries and supersymmetric settings, offering a path to directly constrain anomalies and current data in 4D CFTs.

Abstract

We study four-dimensional conformal field theories (CFTs) with an abelian $U(1)$ global symmetry using the conformal bootstrap approach. We obtain numerical bounds on the scaling dimensions of low-lying operators, the stress-tensor central charge, and a particular combination of the 't Hooft anomaly and the current central charge. Our analysis provides the first non-perturbative constraints on four-dimensional CFTs with conserved abelian currents and establishes a framework that can be extended to theories with non-abelian global symmetries.

Figures (4)

• Figure 1: Bound on the scaling dimension $\Delta_{(4,0)}$ as a function of $\Delta_{(0,0)}$, at $n_\text{max}=16$. The allowed region lies below the blue curve. The red dots indicate the locations of free complex scalar and free Weyl fermion theories. The dashed line indicates the analytic lower bound $\Delta_{(4,0)}\geq 4$ found in Cordova:2017dhq using the ANEC.
• Figure 2: Upper bound on the coefficient $P_J \equiv C_J^{-3}\left(\lambda^-_{JJJ}\right)^2$, defined in \ref{['eq:P_J']}, as a function of $\Delta_{(0,0)}$. Different curves correspond to different values of $n_\text{max}$. The horizontal dashed line represents the analytic upper bound $P_J\leq 1$ for free theories. Free theories must lie below this line. Above this line the theory cannot be free and must be interacting.
• Figure 3: Lower bound on the stress-tensor central charge $C_T$ as a function of $\gamma$, at $n_\text{max}=16$. The different allowed regions correspond to different assumed values of the gap to the second lightest spin-2 operator after the stress-tensor $\Delta'_{(2,2)}$, ranging over $\Delta'_{(2,2)} = \{4,\, 4.5\,, 5,\, 6,\, 7,\, 8\}$. The two dashed vertical lines indicate the Hofman–Maldacena bounds on $\gamma$, given in \ref{['eq:HM_bound']}. The red dots show the free complex scalar and the free Weyl fermion theories.
• Figure 4: Lower bound on the stress-tensor central charge $C_T$ as a function of $\gamma$, at $n_\text{max}=16$. The different allowed regions correspond to different gaps $\Delta_{(0,0)}$ to the lightest scalar, ranging over the set $\{1.0,\, 1.5\,, 1.96,\, 2.42,\, 2.88,\, 3.33,\, 3.79,\, 4.25,\, 4.71,\, 5.17,\, 5.63,\, 6.08,\, 6.54\}.$ The smallest values of $\Delta_{(0,0)}$ give the weakest bounds. The two dashed vertical lines indicate the Hofman-Maldacena bounds on $\gamma$ given in \ref{['eq:HM_bound']}. The red dots denote the free complex scalar and the free Weyl fermion theories.