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

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arxiv 1910.02462 v3 pith:4VTMQ67D submitted 2019-10-06 hep-th

classification hep-th
keywords dimensionsimaginarylargemodelpartscftscomplexeffects
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In $d$ dimensions and for large $N$, we find that they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $N$, the $O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional $O(N)$ models may be studied using the techniques of numerical bootstrap.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Surprises in the Ordinary: $O(N)$ Invariant Surface Defect in the $\epsilon$-expansion

    hep-th 2024-11 conditional novelty 7.0 of 10

    The authors compute defect CFT data to third order in the epsilon expansion and discover approximate 'shadow' relations between surface and bulk scaling dimensions.

  2. Regge trajectories, detectors, and distributions in the critical ${\rm O}(N)$ model

    hep-th 2025-06 conditional novelty 6.0 of 10

    In the critical O(N) model, renormalizing detector and distribution light-ray operators at leading order in 1/N yields Regge intercepts, the leading-twist splitting function, and a BFKL-type anomalous spin.

  3. The large $N$ vector model on $S^1\times S^2$

    hep-th 2024-11 conditional novelty 6.0 of 10

    The authors compute the leading finite-size corrections to thermal observables of the large-N critical O(N) model on S^1 x S^2 using a perturbative expansion in beta/r.

  4. Complex CFTs: Holography and Interfaces

    hep-th 2026-08 conditional novelty 4.0 of 10

    Holographic and CFT constructions of interfaces between complex conjugate CFTs, with a leading-order holographic check of the Im-flip relation and super-transmission signatures of non-unitarity.

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