# The $O(N)$ Model in $4<d<6$: Instantons and Complex CFTs

**Authors:** Simone Giombi, Richard Huang, Igor R. Klebanov, Silviu S. Pufu, Grigory Tarnopolsky  
**arXiv:** [1910.02462](https://arxiv.org/abs/1910.02462)  
**Date:** 2019-10-06  
**Summary page:** https://www.summarizepaper.com/en/paper/1910.02462v3/

## Key points

- Study of the scalar $O(N)$ model in the dimension range $4<d<6"
- Previous work shows a fixed point where operator scaling dimensions are real and above unitarity bound
- Scaling dimensions acquire small imaginary parts due to instanton effects
- Imaginary parts are of order $e^{-N f(d)}$ for large $N$, where $f(d)$ is the sphere free energy of a conformal scalar in $(d-2)$ dimensions
- Implies that at large enough values of $N$, $O(N)$ models can be considered as complex CFTs in the range of $4<d<6$
- Non-perturbatively small imaginary parts also appear in other observables such as sphere free energy and two/three-point function coefficients
- Calculations related to these observables have significant implications
- Numerically negligible imaginary parts allow for studying five-dimensional O(N) models using numerical bootstrap techniques
- Provides new insights into behavior of scalar O(N) models in higher dimensions and their connection to complex CFTs
- Opens up avenues for further research and exploration in this field.

## Summary

In the study of the scalar $O(N)$ model in the dimension range $4<d<6$, the effects caused by its metastability are explored. Previous work has shown that this model possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. However, further investigation reveals that these scaling dimensions acquire small imaginary parts due to instanton effects. The researchers find that in $d$ dimensions and for large $N$, these imaginary parts are of order $e^{-N f(d)}$, where the function $f(d)$ is equal to the sphere free energy of a conformal scalar in $(d-2)$ dimensions. This discovery is remarkable as it suggests that at sufficiently large values of $N$, the $O(N)$ models in the range of $4<d<6$ can be considered as complex conformal field theories (CFTs). Notably, these non-perturbatively small imaginary parts also appear in other observables such as the sphere free energy and two and three-point function coefficients. The researchers present some calculations related to these observables which have significant implications; when these imaginary parts become numerically negligible at large enough values of N, it becomes possible to study five-dimensional O(N) models using numerical bootstrap techniques. Overall, this study provides new insights into the behavior of scalar O(N) models in higher dimensions and highlights their connection to complex CFTs. The findings open up avenues for further research and exploration in this field.