Two-dimensional critical Ising model

The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge formula_1. 
Correlation functions of the spin and energy operators are described by the formula_2 minimal model. While the minimal model has been exactly solved, the solution does not cover other observables such as connectivities of clusters.
The minimal model.
Space of states and conformal dimensions.
The Kac table of the formula_2 minimal model is:
This means that the space of states is generated by three primary states, which correspond to three primary fields or operators:
The decomposition of the space of states into irreducible representations of the product of the left- and right-moving Virasoro algebras is 
where formula_7 is the irreducible highest-weight representation of the Virasoro algebra with the conformal dimension formula_8.
In particular, the Ising model is diagonal and unitary.
Characters and partition function.
The characters of the three representations of the Virasoro algebra that appear in the space of states are 
where formula_10 is the Dedekind eta function, and formula_11 are theta functions of the nome formula_12, for example formula_13.
The modular S-matrix, i.e. the matrix formula_14 such that formula_15, is 
where the fields are ordered as formula_17.
The modular invariant partition function is 
Fusion rules and operator product expansions.
The fusion rules of the model are 
The fusion rules are invariant under the formula_20 symmetry formula_21.
The three-point structure constants are 
Knowing the fusion rules and three-point structure constants, it is possible to write operator product expansions, for example
where formula_24 are the conformal dimensions of the primary fields, and the omitted terms formula_25 are contributions of descendant fields.
Correlation functions on the sphere.
Any one-, two- and three-point function of primary fields is determined by conformal symmetry up to a multiplicative constant. This constant is set to be one for one- and two-point functions by a choice of field normalizations. The only non-trivial dynamical quantities are the three-point structure constants, which were given above in the context of operator product expansions. 
with formula_28.
The three non-trivial four-point functions are of the type formula_33. For a four-point function formula_34, let formula_35 and formula_36 be the s- and t-channel Virasoro conformal blocks, which respectively correspond to the contributions of formula_37 (and its descendants) in the operator product expansion formula_38, and of formula_39 (and its descendants) in the operator product expansion formula_40. Let formula_41 be the cross-ratio. 
In the case of formula_42, fusion rules allow only one primary field in all channels, namely the identity field.
In the case of formula_44, fusion rules allow only the identity field in the s-channel, and the spin field in the t-channel.
In the case of formula_46, fusion rules allow two primary fields in all channels: the identity field and the energy field. In this case we write the conformal blocks in the case formula_47 only: the general case is obtained by inserting the prefactor formula_48, and identifying formula_49 with the cross-ratio.
Connectivities of clusters.
The Ising model has a description as a random cluster model due to Fortuin and Kasteleyn. In this description, the natural observables are connectivities of clusters, i.e. probabilities that a number of points belong to the same cluster. 
The Ising model can then be viewed as the case formula_51 of the formula_52-state Potts model, whose parameter formula_52 can vary continuously, and is related to the central charge of the Virasoro algebra. 
In the critical limit, connectivities of clusters have the same behaviour under conformal transformations as correlation functions of the spin operator. Nevertheless, connectivities do not coincide with spin correlation functions: for example, the three-point connectivity does not vanish, while formula_54. There are four independent four-point connectivities, and their sum coincides with formula_55. Other combinations of four-point connectivities are not known analytically. In particular they are not related to correlation functions of the minimal model, although they are related to the formula_56 limit of spin correlators in the formula_52-state Potts model.