Bootstrapping $O(N)$ Vector Models with Four Supercharges in $3 \leq d \leq4$

TL;DR

This work uses the conformal bootstrap to constrain 3d–4d ${\cal N}=2$ SCFTs with four supercharges and an $O(N)\times U(1)$ flavor symmetry, focusing on the 4-point function of $O(N)$-fundamental chiral operators $Z_i$ with no chiral primary in the $O(N)$-singlet OPE. By combining bootstrap constraints with localization data, non-integer-dimension extrapolations, and the $4-\epsilon$ expansion, the authors identify features near the IR fixed point of the theory with $W = \frac{g}{2} X\sum_i Z_i^2$ and with general bounds under the chiral ring relation $\sum_i Z_i^2 \sim 0$. They obtain robust lower bounds on central charges $c_J^{O(N)}$, $c_J^{U(1)}$, $c_T$, and upper bounds on the lowest unprotected $O(N)$-singlet scalar, observing kinks and peaks that align with localization predictions, especially for larger $N$ and in non-integer dimensions. The results also reveal a family of kinks in $d\in[3,4]$ tied to the chiral ring constraint and show how the spectrum tightens when fixing central charges, offering precision checks for epsilon-expansion approximations and guidance for large-$N$ behavior in these SCFTs.

Abstract

We analyze the conformal bootstrap constraints in theories with four supercharges and a global $O(N) \times U(1)$ flavor symmetry in $3 \leq d \leq 4$ dimensions. In particular, we consider the 4-point function of $O(N)$-fundamental chiral operators $Z_i$ that have no chiral primary in the $O(N)$-singlet sector of their OPE. We find features in our numerical bounds that nearly coincide with the theory of $N+1$ chiral super-fields with superpotential $W = X \sum_{i=1}^N Z_i^2$, as well as general bounds on SCFTs where $\sum_{i=1}^N Z_i^2$ vanishes in the chiral ring.

Figures (8)

• Figure 1: Lower bounds on the $O(N)$ flavor current central charge $c_J^{O(N)}$ as function of the scaling dimension $\Delta_{Z_i}$ of the chiral $O(N)$-fundamental primary in dimensions $d=3$ and $d=3.5$, for $N = 2, 3, 4, 10, 20$. Different shadings of orange denote the allowed regions for each $N$. The black vertical lines denote localization values of $\Delta_{Z_i}$ for each $N$, while the asterisks denote the localization values of $c_J^{O(N)}$ for each $N$ (see Table \ref{['val3']}). These bounds were computed using $\ell_\text{max}=25$ and $\Lambda=19$.
• Figure 2: Lower bounds on the $U(1)$ central charge $c_J^{U(1)}$ in terms of the scaling dimension $\Delta_{Z_i}$ of the chiral $O(N)$-fundamental primary in dimensions $d=3$ and $d=3.5$, for $N = 2, 3, 4, 10, 20$. Different shadings of orange denote the allowed regions for each $N$. The black vertical lines denote localization values of $\Delta_{Z_i}$ for each $N$, while the asterisks denote the localization values of $c_J^{U(1)}$ for each $N$ (see Table \ref{['val3']}). These bounds were computed using $\ell_\text{max}=25$ and $\Lambda=19$.
• Figure 3: Lower bounds on central charge $c_T$ in terms of the scaling dimension $\Delta_{Z_i}$ of the chiral $O(N)$-fundamental primary in dimension $d=3$ and $d=3.5$, for $N = 2, 3, 4, 10, 20$. Different shadings of orange denote the allowed regions for each $N$. The black vertical lines denote localization values of $\Delta_{Z_i}$ for each $N$, while the asterisks denote the localization values of $c_T$ for each $N$ (see Table \ref{['val3']}). These bounds were computed using $\ell_\text{max}=25$ and $\Lambda=19$.
• Figure 4: The ratio between the bootstrap lower bound on the coefficient $c_J^{O(N)}$ of the $O(N)$ current 2-point function and the value predicted using the generalization of the $F$-value of the 3d SCFT in the IR limit of \ref{['superpot']} with $g_2=0$ to non-integer dimensions Giombi:2014xxa. The curves, from bottom to top, correspond to $N = 2, \, 3, \, 4, \, 10$.
• Figure 5: Upper bound on the scaling dimension $\Delta'_{Ss, 0}$ of the lowest-lying $O(N)$-singlet scalar in the $Z_i\times\bar{Z}_j$ OPE, as a function of $\Delta_{Z_i}$ in dimensions $d=3$ and $d=3.5$, for $N = 2, 3, 4, 10, 20$. Different shadings of orange denote the allowed regions for each $N$. The black vertical lines denote localization values (Table \ref{['val3']}) of $\Delta_{Z_i}$ for each $N$. The red dots denote the stronger bounds once the localization values of $c_J^{U(1)}$ and $c_J^{O(N)}$ (Table \ref{['val3']}) are imposed on the spectrum. The asterisks indicate the Padé approximation to the 3-loop $\epsilon$-expansion values of $\Delta'_{Ss, 0}$ for each $N$. Note that for $N=2$ there is a gap in the allowed region for $.75<\Delta_{Z_i}<.875$, and that the range of $\Delta_{Z_i}$ is smaller in this plot than it is in the $d=3$ central charge plots. These bounds were computed using $\ell_\text{max}=25$ and $\Lambda=19$.