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On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions

Bernardo Zan

TL;DR

The work argues that the Wilson–Fisher fixed point in integer dimensions cannot be identically the 2D Ising CFT due to the emergence of Virasoro symmetry; instead, Ising data appear as a unitary subsector embedded within WF in the d → 2 limit. By analyzing the 2D O(n) model as a toy model and examining negative multiplicity representations and multiplet recombination, the authors show that non-Ising operators persist and cancel in Ising observables through precise dimension and O(d) representation matches. They identify candidate negative-multiplicity operators with Δ ≈ 5 and ℓ = 3 that can cancel the would-be spin-3 descendants, providing a consistent mechanism for the observed separation between full WF data and Ising subsector. The results challenge the feasibility of obtaining WF conformal data in d = 2 + ε purely from Ising data and highlight the need to understand the full d → 2 limit, with implications for bootstrap approaches and perturbative expansions in non-integer dimensions.

Abstract

The Wilson-Fisher fixed point defines a continuous family of interacting conformal field theories in non-integer dimensions. In integer dimensions, it is widely believed to lie in the same universality class as the critical Ising model. In this work, we revisit the identification between the Wilson-Fisher fixed point at integer dimensions and the Ising CFT. We argue that a literal equality between the two theories is incompatible with the emergence of Virasoro symmetry in two dimensions. Instead, we propose that the Ising model emerges only as a subsector of the Wilson-Fisher fixed point. We support this scenario through a detailed study of the two-dimensional $O(n)$ model and by examining operators transforming in irreducible representations of the orthogonal group whose multiplicities become negative for integer values of the spacetime dimension. Finally, we comment on the implications of these results for attempts to construct a $d=2+ε$ expansion starting from exact two-dimensional data.

On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions

TL;DR

The work argues that the Wilson–Fisher fixed point in integer dimensions cannot be identically the 2D Ising CFT due to the emergence of Virasoro symmetry; instead, Ising data appear as a unitary subsector embedded within WF in the d → 2 limit. By analyzing the 2D O(n) model as a toy model and examining negative multiplicity representations and multiplet recombination, the authors show that non-Ising operators persist and cancel in Ising observables through precise dimension and O(d) representation matches. They identify candidate negative-multiplicity operators with Δ ≈ 5 and ℓ = 3 that can cancel the would-be spin-3 descendants, providing a consistent mechanism for the observed separation between full WF data and Ising subsector. The results challenge the feasibility of obtaining WF conformal data in d = 2 + ε purely from Ising data and highlight the need to understand the full d → 2 limit, with implications for bootstrap approaches and perturbative expansions in non-integer dimensions.

Abstract

The Wilson-Fisher fixed point defines a continuous family of interacting conformal field theories in non-integer dimensions. In integer dimensions, it is widely believed to lie in the same universality class as the critical Ising model. In this work, we revisit the identification between the Wilson-Fisher fixed point at integer dimensions and the Ising CFT. We argue that a literal equality between the two theories is incompatible with the emergence of Virasoro symmetry in two dimensions. Instead, we propose that the Ising model emerges only as a subsector of the Wilson-Fisher fixed point. We support this scenario through a detailed study of the two-dimensional model and by examining operators transforming in irreducible representations of the orthogonal group whose multiplicities become negative for integer values of the spacetime dimension. Finally, we comment on the implications of these results for attempts to construct a expansion starting from exact two-dimensional data.
Paper Structure (15 sections, 49 equations, 1 table)

This paper contains 15 sections, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. The problem
  3. The two-dimensional $O(n)$ CFT as a toy model
  4. Correlation functions
  5. Correlation functions of non-Ising operators
  6. From $n=1$ to $n=1.01$
  7. The $d\to 2$ limit of WF
  8. Negative multiplicity operators
  9. Scaling dimensions
  10. $d=3$
  11. From $d=2$ to $d=2.01$
  12. Conclusions
  13. Details about the $O(n)$ model
  14. Partition function
  15. Correlation functions