Perturbative and Nonperturbative Studies of CFTs with MN Global Symmetry

Abstract

Fixed points in three dimensions described by conformal field theories with $M{N}_{m,n}=O(m{)}^n⋊S_n$ 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 $\epsilon$ 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 $\epsilon$ 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 ${\Delta }_{\varphi }\approx 0.75$. We investigate the spectrum in the case $M{N}_{100,2}$ and find consistency with expectations from the lightcone bootstrap.

arXiv comment23 pages, 8 figures. v2: Minor emendations, additions and clarifications

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