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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?

Connor Behan
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Andrew Hardy
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  • Regarding the relation between minimal models and interacting scalar field theory, see the following reference, arguing that the $\mathcal{M}_p$ minimal models (with central charge $c=1-6/p(p+1)$) are described by critical points in scalar theories with interaction terms of the form $\phi^{2(p-1)}$: https://inis.iaea.org/search/search.aspx?orig_q=RN:18062595 – Seth Whitsitt Oct 20 '22 at 13:58
  • @SethWhitsitt Thanks for the reference. The boson mapping I am familiar with. I'm not able to find a pdf of it through my university library though, so I'm not sure if it mentions any connection to the free fermion description. Is there a good reference on how these are both in the same universality class? – Andrew Hardy Oct 20 '22 at 15:11
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    Do you have access to Di Francesco, Mathieu, and Senechal's textbook "Conformal Field Theory" (often called "the big yellow book")? This certainly discusses the relation between free fermions and the Ising model. In some sense, the fact that both models have $c=1/2$, which only has one possible set of primary operators, already shows that they must be equivalent (though showing how observables map to each other requires more work). There are also various transformations/limits starting from an Ising model going to scalar field theory of fermionic field theory. – Seth Whitsitt Oct 23 '22 at 18:24

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