Towards Bootstrapping QED$_3$

TL;DR

The paper initiates a nonperturbative conformal bootstrap study of 3d QED with N flavors by focusing on the four-point function of monopole operators with unit topological charge. It derives crossing equations for M_{1/2} operators transforming under SU(N) and implements a numerical SDP program to bound Δ_{M_1} and central charges c_T, c_J^f, c_J^t, finding a gap-dependent kink that tracks large-N expectations. The results provide a first nonperturbative handle on the IR fixed point structure of QED3 at small N and hint at potentially island-like regions upon exploring mixed correlators. The methodology combines a detailed treatment of SU(N) representations, reflection positivity, space-time parity constraints, and high-derivative semidefinite programming to probe strongly coupled gauge theories beyond perturbation theory.

Abstract

We initiate the conformal bootstrap study of Quantum Electrodynamics in $2+1$ space-time dimensions (QED$_{3}$) with $N$ flavors of charged fermions by focusing on the 4-point function of four monopole operators with the lowest unit of topological charge. We obtain upper bounds on the scaling dimension of the doubly-charged monopole operator, with and without assuming other gaps in the operator spectrum. Intriguingly, we find a (gap-dependent) kink in these bounds that comes reasonably close to the large $N$ extrapolation of the scaling dimensions of the singly-charged and doubly-charged monopole operators down to $N=4$ and $N=6$.

Figures (5)

• Figure 1: Bounds on basic $q=1$ monopole operator scaling dimension $\Delta_{M_1}$ in terms of basic $q=1/2$ monopole operator scaling dimension $\Delta_{M_{1/2}}$ in $d=3$ for $N=2,6$ (left,right) with gaps $\Delta'_1\geq.5,2,2.5,3,3.5,4$ for $N=2$ and $\Delta'_3\geq.5,2,2.5,3,3.5,4$ for $N=6$ in the uncharged sector in the same $SU(N)$ representation $\left(2^{N/2}\right)$ as $M_1$. These bounds were computed with $\ell_\text{max}=25$ and $\Lambda=19$. The black cross denotes the large $N$ expansion values of $(\Delta_{M_{1/2}},\Delta_{M_1})$.
• Figure 2: Bounds on basic $q=1$ monopole operator scaling dimension $\Delta_{M_1}$ in terms of basic $q=1/2$ monopole operator scaling dimension $\Delta_{M_{1/2}}$ in $d=3$ for $N=4$ with gaps $\Delta_2\geq.5,2,2.5,3,3.5,4$ in the uncharged sector in the same $SU(N)$ representation $\left(2^{N/2}\right)$ as $M_1$. The righthand plot focuses on the $\Delta_2\geq3$ case, and shows that placing an additional gap $\Delta_{M'_1}$ above $\Delta_{M_1}$ creates a peninsula around the kinks seen in the lefthand plots. These bounds were computed with $\ell_\text{max}=25$ and $\Lambda=19$. The black cross denotes the large $N$ expansion values of $(\Delta_{M_{1/2}},\Delta_{M_1})$.
• Figure 3: Bounds on stress tensor central charge $c_T$ in terms of basic $q=1/2$ monopole operator scaling dimension $\Delta_{M_{1/2}}$ in $d=3$ for $N=2,4,6$ with gaps $\Delta'_2\geq.5,2,2.5,3,3.5,4$ for $N=2$, $\Delta_4\geq.5,2,2.5,3,3.5,4$ for $N=4$, and $\Delta'_6\geq2,2.5,3,3.5,4$ for $N=6$ in the uncharged sector in the same $SU(N)$ representation $\left(2^{N/2}\right)$ as $M_1$. These bounds were computed with $\ell_\text{max}=25$ and $\Lambda=19$. The black crosses denote the large $N$ expansion values of $c_T$.
• Figure 4: Bounds on topological $U(1)$ current charge $c_J^t$ in terms of basic $q=1/2$ monopole operator scaling dimension $\Delta_{M_{1/2}}$ in $d=3$ for $N=2,4,6$ with gaps $\Delta'_2\geq.5,2,2.5,3,3.5,4$ for $N=2$, $\Delta_4\geq.5,2,2.5,3,3.5,4$ for $N=4$, and $\Delta'_6\geq2,2.5,3,3.5,4$ for $N=6$ in the uncharged sector in the same $SU(N)$ representation $\left(2^{N/2}\right)$ as $M_1$. These bounds were computed with $\ell_\text{max}=25$ and $\Lambda=19$. The black crosses denote the large $N$ expansion values of $c_J^t$.
• Figure 5: Bounds on $SU(N)$ flavor current charge $c_J^f$ in terms of basic $q=1/2$ monopole operator scaling dimension $\Delta_{M_{1/2}}$ in $d=3$ for $N=2,4,6$ with gaps $\Delta'_2\geq.5,2,2.5,3,3.5,4$ for $N=2$, $\Delta_4\geq.5,2,2.5,3,3.5,4$ for $N=4$, and $\Delta'_6\geq2,2.5,3,3.5,4$ for $N=6$ in the uncharged sector in the same $SU(N)$ representation $\left(2^{N/2}\right)$ as $M_1$. These bounds were computed with $\ell_\text{max}=25$ and $\Lambda=19$. The black crosses denote the large $N$ expansion values of $c_J^f$.