Perturbative and Nonperturbative Studies of CFTs with MN Global Symmetry

TL;DR

This paper explores the fixed points of 3D conformal field theories with $MN_{m,n}$ symmetry, using perturbative and nonperturbative methods, and compares findings with numerical bootstrap and experimental data.

Contribution

It introduces a large $m$ expansion for certain fixed points and analyzes their behavior across different parameters, connecting perturbative and nonperturbative regimes.

Findings

One family of fixed points approaches a perturbative limit with increasing $m$.

Another family persists only for $n=2$, approaching a non-perturbative limit.

The large $m$ expansion aligns well with numerical bootstrap data.

Abstract

Fixed points in three dimensions described by conformal field theories with $MN_{m,n}= O(m)^n\rtimes S_n$ global symmetry have extensive applications in critical phenomena. Associated experimental data for $m=n=2$ suggest the existence of two non-trivial fixed points, while the $\varepsilon$ expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters $m$ and $n$, with critical exponents in good agreement with experimental determinations in the $m=n=2$ case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters $m$ and $n$. We find that one family of kinks approaches a perturbative limit as $m$ increases, and using large spin perturbation theory we construct a large $m$ expansion that fits well with the numerical data. This new expansion, akin to the large $N$ expansion of critical $O(N)$ models, is compatible with the fixed point found in the $\varepsilon$ expansion. For the other family of kinks, we find that it persists only for $n=2$, where for large $m$ it approaches a non-perturbative limit with $\Delta_\phi\approx 0.75$. We investigate the spectrum in the case $MN_{100,2}$ and find consistency with expectations from the lightcone bootstrap.