Abstract
We elaborate and extend the method of Wronskian differential equations for conformal blocks to compute four-point correlation functions on the plane for classes of primary fields in rational (and possibly more general) conformal field theories. This approach leads to universal differential equations for families of CFT’s and provides a very simple re-derivation of the BPZ results for the degenerate fields ϕ1,2 and ϕ2,1 in the c < 1 minimal models. We apply this technique to compute correlators for the WZW models corresponding to the Deligne-Cvitanović exceptional series of Lie algebras. The application turns out to be subtle in certain cases where there are multiple decoupled primaries. The power of this approach is demonstrated by applying it to compute four-point functions for the Baby Monster CFT, which does not belong to any minimal series.
Highlights
On the way we encounter a significant complication relative to the cases addressed in the previous section, which was noted in specific cases in [8, 9] and investigated in a somewhat more general context in [17]: when a primary has a multiplicity, the indices must be combined pairwise into definite representations that flow in the intermediate channel, and this creates “selection rules” which influence the singular behaviour of the conformal blocks, the Wronskian and the differential equation
We have seen that the Wronskian method, relying only on a knowledge of conformal dimensions and fusion rules, provides explicit expressions for the conformal blocks whenever a given correlator receives contributions from at most two conformal blocks
Families of RCFT’s are classified not by their chiral algebra but by the order of differential equation satisfied by their characters
Summary
We explain the Wronskian method to obtain differential equations for correlation functions of primaries in RCFT’s. Conformal invariance is used to rewrite the desired correlator as some standard factors times a function of the cross-ratio: In this way the original PDE can be converted to an ordinary differential equation in z. Eq (2.26) is a master equation that determines the correlators for all four-point functions having two conformal blocks, as long as the primaries have no degeneracy. This equation agrees perfectly with eq (5.19) of [1] once we take (z2, z3, z4) → (0, 1, ∞) in the latter. We have reproduced the differential equation for two-block 4-point correlators of identical fields in all minimal models!
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