On the Wilson-Fisher Fixed Point in the Limit of Integer Spacetime Dimensions

The Wilson–Fisher (WF) fixed point, a scalar field theory with a quartic interaction in $d = 4 – epsilon$ dimensions, is the textbook example of a weakly coupled CFT. The epsilon expansion accurately reproduces the critical exponents of the $d = 3$ and the $d = 2$ Ising model. This has led to the common view that WF defines a continuous family of CFTs in non-integer dimensions, interpolating between the free theory at $d=4$ and the Ising model at $d = 3$ and $d = 2$. In this talk I will look at what happens to the WF CFT when the number of dimensions approaches an integer value. Identifying the $dto 2$ limit of WF as the $2d$ Ising CFT raises some issues, related to the fact that the $d=2$ CFT has infinitely many conserved currents that the $d>2$ theory does not have. I will propose a scenario, inspired by some exactly solved two-dimensional models, where the $dto 2$ limit of WF contains the $d=2$ Ising CFT as a subsector, but contains more operators which do not completely decouple from the Ising subsector. I will explain why this makes the task of obtaining a $d= 2 + epsilon$ expansion for the Ising model harder.
 
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