# $O(N)$ Models with Boundary Interactions and their Long Range Generalizations

**Authors:** Simone Giombi, Himanshu Khanchandani  
**arXiv:** [1912.08169](https://arxiv.org/abs/1912.08169)  
**Date:** 2019-12-17  
**Summary page:** https://www.summarizepaper.com/en/paper/1912.08169v3/

## Key points

- Study investigates critical properties of scalar field theories in $d+1$ dimensions with $O(N)$ invariant interactions localized on a $d$-dimensional boundary
- Evidence for existence of non-trivial $O(N)$ Boundary Conformal Field Theories (BCFTs) in range $1<d<4$
- Models have free fields in the bulk, resulting in presence of bulk higher-spin currents conserved except for terms localized on the boundary
- Suggestion of set of protected spinning operators on the boundary with vanishing anomalous dimensions
- Discussion of long-range $O(N)$ models in $d$ dimensions, closely related to main focus
- Examination of weakly coupled description of long-range $O(N)$ model in one dimension near upper critical value using non-local non-linear sigma model
- Combination of known perturbative descriptions and estimates for critical exponents in one dimension ($d=1)
- Comprehensive study spanning 64 pages with figures included

## Summary

In their study, Simone Giombi and Himanshu Khanchandani investigate the critical properties of scalar field theories in $d+1$ dimensions with $O(N)$ invariant interactions localized on a $d$-dimensional boundary. They employ a combination of large $N$ and epsilon expansions to provide evidence for the existence of non-trivial $O(N)$ Boundary Conformal Field Theories (BCFTs) in the range $1<d<4$. The authors highlight that these models have free fields in the bulk, which results in the presence of bulk higher-spin currents that are conserved except for terms localized on the boundary. This leads them to suggest that there should be a set of protected spinning operators on the boundary, and they provide evidence that these operators have vanishing anomalous dimensions. Additionally, Giombi and Khanchandani discuss long-range $O(N)$ models in $d$ dimensions, which are closely related to their main focus. Specifically, they examine a weakly coupled description of the long-range $O(N)$ model in one dimension near its upper critical value. This description is given by a non-local non-linear sigma model. To further support their findings, the authors combine known perturbative descriptions and provide estimates for critical exponents in one dimension ($d=1$). Their comprehensive study spans 64 pages and includes several figures. It is worth noting that this version is an updated version from previous versions with minor changes and added references; it was published in JHEP (Journal of High Energy Physics) with DOI: 10.1007/JHEP08(2020)010.