Conformal field theory (CFT) plays a crucial role in understanding phase transitions, whether quantum or statistical. Typically, CFTs are studied using perturbative methods, such as the 4-epsilon expansion and the large N expansion. In this talk, I will introduce an alternative approach: a fixed-dimensional perturbation theory that incorporates long-range, non-local interactions. I will demonstrate that by imposing the conformal Ward Identity, it is possible to recover the data of local CFTs. By re-summing the perturbative series using long-range solitons (solutions to the nonlinear equation of motion with fractional Laplacian), we obtain impressive results for the critical exponents.
