There are numerous Stack Exchange answers that explain how to construct a free fermion CFT ($c = 1/2$) which describes the critical point of a 2D Ising model.
However there are also sources that describe the theory as a $\mathcal{M}_3$ minimal Virasoro module. This is also detailed in Section 5 of Ginsparg's "Applied" CFT notes. I would like to better understand the relationship between these two. Does $\mathcal{M}_3$ also describe a free fermion theory? Adding to my confusion is that in Section 11.6 of Mussardo's Statistical Field Theory, he describes $\mathcal{M}_3$ as corresponding to a $\varphi^4$ scalar field theory.
Is there a way to understand how both an interacting scalar theory (which I don't even think $\varphi^4$ is conformally invariant) and a free fermion field can describe the same thing?
