Are there exactly solvable CFTs?

I am wondering if there are CFTs such that n-point correlation functions in them of the fields (may be the primaries or of some notion of twist fields) is exactly known.

• Are there such?

• Aren't minimal models supposed to have this property?

• Even if not exactly known, can the n-point functions (may be when written as a conformal block expansion) be written order-by-order as a power series in the central charge? (..I guess there is some theorem about the asymptotic exponential dependence of the conformal blocks on the central charge (reference help? ) and I want to know if 1/c corrections to that are known..)

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Are your $n$-point correlation functions in genus zero? – S. Carnahan Oct 13 '13 at 14:15
@S.Carnahan That would be the simplest scenario I would like to know of. (..if you can say something about higher genus then that is more exciting ;P..) – user6818 Oct 13 '13 at 18:15
Did you look at the work of Dotsenko and Fateev from mid 1980s. They computed correlation functions of a number of 2d conformal field theories. – José Figueroa-O'Farrill Oct 14 '13 at 19:33
@JoséFigueroa-O'Farrill Can you kindly refer to any paper that you have in mind? Is it known by some name that I can may be trace in the Francesco et. al's book? – user6818 Oct 15 '13 at 17:05
@user6818, I have lent my copy of the book by di Francesco et al., I'm afraid, but the papers I have in mind are the following: dx.doi.org/10.1016%2F0550-3213%2884%2990269-4 and dx.doi.org/10.1016%2FS0550-3213%2885%2980004-3 . – José Figueroa-O'Farrill Oct 15 '13 at 23:24

1 Answer

Indeed, as José said, one can get exact formulas for correlations using the Coulomb gas method introduced by Feigin, Fuchs, Dotsenko and Fateev. I don't know how explicit or simple the formulas are say for the general unitary minimal models $\mathcal{M}(m+1,m)$, but for the Ising model on the plane, i.e., $m=3$, the formulas are quite cute. One finds for the spin field $\phi$ $$\langle\phi(x_1)\cdots\phi(x_n)\rangle= \sqrt{\sum_{e_1,\ldots,e_n} \prod_{1\le i<j\le n} |x_i-x_j|^{\frac{e_i e_j}{2}}}$$ where the sum is over all neutral charge configurations $e_i=\pm 1$ with $\sum_i e_i=0$.

More remarkably, this is now a rigorous theorem proved in this article by Chelkak, Hongler, Izyurov (see page 1093) and this other one by Dubédat.

A physical literature derivation of this formula (also including the energy field and the disorder operator as well as the genus one case) can be found in this article by Di Francesco, Saleur and Zuber.

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There is a similar closed formula for compactifications of the free boson or vertex operators, which is eg derived in Buchholz-Mack-Todorov. This includes the free complex Fermion which can also be represent by the Wick formula – Marcel Bischoff Feb 13 at 7:00