# 2d Ising model in CFT and statistical mechanics

When I recently started to read about conformal field theory, one of the basic examples there is the so called Ising model. It is characterized by certain specific collection of fields on the plane acted by the Virasoro algebra with certain central charge, and by a specific operator product expansion. In the conformal fields literature I read it is claimed that this model comes from the statistical mechanics.

In the literature on statistical mechanics what is called the Ising model. It is something completely different: one fixes a discrete lattice on the plane, and there is just one field which attaches numbers $\pm 1$ to each vertex of the lattice.

As far as I heard there is a notion of scaling limit when the lattice spacing tends to zero. At this limit (at the critical temperature?) some important quantities converge to a limit. My guess is that this scaling limit should be somehow relevant to connect the two Ising models I mentioned above.

Question. Is there a good place to read about explicit relation between the two Ising models? In particular I would be interested to understand how to obtain the operator product expansion and the central charge starting from the statistical mechanics description.

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The standard reference is "Conformal Field Theory" by Philippe Francesco, Pierre Mathieu and David Senechal. –  Hunter May 26 at 14:10
@Hunter: I am familiar with this book a little bit. It does not discuss the statistical mechanics approach at all, and rather takes the central charge and OPE for granted. –  MKO May 26 at 14:12
Oh ok, I thought it did, but I'm might be wrong. Sorry –  Hunter May 26 at 14:14
@Hunter: I have to apologize. I checked again, in the book you mentioned indeed there is a discussion of the relation between the two subjects (Section 12.2). It is somewhat brief, but still might be useful. Thank you. –  MKO May 27 at 8:32
See physicsoverflow.org/22034 for answers –  Arnold Neumaier Sep 17 at 19:09