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 $d=2+ε$ expansion starting from exact two-dimensional data.