Bootstrapping the O(N) Vector Models

TL;DR

This paper extends the conformal bootstrap to 3D CFTs with O(N) symmetry to bound the dimensions of the lowest singlet S and symmetric tensor T in the φ_i × φ_j OPE and to bound the central charge c. It introduces rational conformal-block representations via a pole-recursion method and formulates the problem as a semidefinite program, enabling precise numerical bounds across N. The results show that the critical O(N) vector models saturate the bootstrap bounds and align with large-N predictions, yielding sharp estimates for Δ_S, Δ_T, and c that improve understanding of these theories. The work provides both methodological advances in block computation and concrete physical predictions relevant to lattice tests and holographic interpretations of O(N) vector models.

Abstract

We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $φ_i \times φ_j$ OPE, where $φ_i$ is a fundamental of O(N). Comparing these bounds to previous determinations of critical exponents in the O(N) vector models, we find strong numerical evidence that the O(N) vector models saturate the bootstrap constraints at all values of N. We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N.

Figures (5)

• Figure 1: Configuration of points for radial quantization in the $\rho$-coordinate Hogervorst:2013sma.
• Figure 2: Upper bounds on the dimension of the lowest dimension singlet $S$ in the $\phi\times\phi$ OPE, where $\phi$ transforms as a vector under an $O(N)$ global symmetry group. Here, we show $N=1,2,3,4,5,6,10,20$. The blue error bars represent the best available analytical and Monte Carlo determinations of the operator dimensions $(\Delta_\phi,\Delta_{S})$ in the $O(N)$ vector models for $N=1,2,3,4,5,6$ (with $N=1$ being the 3D Ising Model). The black crosses show the predictions in Eq. (\ref{['eq:largeNdimensions']}) from the large-$N$ expansion for $N=10,20,..., 100$. In this expansion, $\Delta_\phi$ has been determined to three-loop order, while $\Delta_{S}$ is at two-loop order. The dashed line interpolates the large-$N$ prediction for $N\in(4,\infty)$.
• Figure 3: Upper bounds on the dimension of the lowest dimension symmetric tensor $T$ in the $\phi\times\phi$ OPE, where $\phi$ transforms as a vector under an $O(N)$ global symmetry group, for $N=2,3,4,5,6,10,20$. We additionally assume that the lowest dimension singlet $S$ has $\Delta_S\geq 1$. The blue error bars represent the best available analytical and Monte Carlo determinations of the operator dimensions $(\Delta_\phi,\Delta_{T})$ in the $O(N)$ vector models for $N=2,3,4,5$. Note in particular that previous predictions for the $O(5)$ model are essentially ruled out by our bounds. The black crosses show the predictions in Eq. (\ref{['eq:largeNdimensions']}) from the large-$N$ expansion for $N=10,20,..., 100$. The dashed line interpolates the large-$N$ prediction for $N\in(4,\infty)$.
• Figure 4: Lower bounds on the central charge for theories containing a scalar $\phi$ transforming as a vector under $O(N)$. We additionally assume that $\Delta_S,\Delta_T\geq 1$. The black crosses show the predictions in Eq. (\ref{['eq:largeNdimensions']}) from the large-$N$ expansion for $N=10,20,..., 100$. The dashed line shows the asymptotic behavior of the central charge as a function of $\Delta_\phi$ as $N\to\infty$.
• Figure 5: Lower bounds on the central charge for theories containing a scalar $\phi$ transforming as a vector under $O(N)$. In this figure we assume that the lowest dimension singlet scalar operators saturate the bounds we found in subsection \ref{['sec:boundsonsinglets']}, while the symmetric tensor scalar operators are assumed to have dimensions $\Delta_T \ge 1$. The black crosses show the predictions in Eq. (\ref{['eq:largeNdimensions']}) from the large-$N$ expansion for $N=10,20,..., 100$. The dashed line shows the asymptotic behavior of the central charge as a function of $\Delta_\phi$ as $N\to\infty$.