On the Wilson-Fisher fixed point in the limit of integer spacetime 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 $d=2+\epsilon$ expansion starting from exact two-dimensional data.

arXiv commentjournal version

References in corpus (41)

• Conformal Four Point Functions and the Operator Product Expansion
• Precision Islands in the Ising and $O(N)$ Models
• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• A Mellin space approach to the conformal bootstrap
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• Conformal Bootstrap in Mellin Space
• Conformal Field Theories in Fractional Dimensions
• A Cheap Alternative to the Lattice?
• A scaling theory for the long-range to short-range crossover and an infrared duality
• Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions
• Long-range critical exponents near the short-range crossover
• The critical O(N) CFT: Methods and conformal data
• The Random-Bond Ising Model in 2.01 and 3 Dimensions
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• On the Three-point Function in Minimal Liouville Gravity
• Bootstrapping the long-range Ising model in three dimensions
• Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• Bootstrapping the 3d Ising Stress Tensor
• On Calculation of $2+\veps$ RG Functions in the Gross Neveu Model from Large $N$ Expansions of Critical Exponents
• Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model
• The structure of the spectrum of anomalous dimensions in the N-vector model in 4-epsilon dimensions
• Two-dimensional O(n) models and logarithmic CFTs
• Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• Critical Ising Model in Varying Dimension by Conformal Bootstrap
• Two Dimensional Renormalization Group Flows in Next to Leading Order
• Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite $\mathbf{\epsilon }$
• Ising model close to $d=2$
• Critical loop models are exactly solvable
• (No) Bootstrap for the Fractal Ising Model
• The tricritical Ising CFT and conformal bootstrap
• Slightly broken higher-spin current in bosonic and fermionic QED in the large-$N$ limit
• Coupled minimal models revisited II: Constraints from permutation symmetry
• Quantum Groups as Global Symmetries
• Disturbing news about the $d=2+\epsilon$ expansion
• Quantum Groups as Global Symmetries II. Coulomb Gas Construction
• Long-range minimal models
• Seven loops ${\varphi }^4$
• Phases of 2d Gauge Theories and Symmetric Mass Generation
• Constraints on the spectrum of field theories with non-integer $O(N)$ symmetry from quantum evanescence