Abstract
We study the classical c\to \infty limit of the Virasoro conformal blocks. We point out that the classical limit of the simplest nontrivial null-vector decoupling equation on a sphere leads to the Painleve VI equation. This gives the explicit representation of generic four-point classical conformal block in terms of the regularized action evaluated on certain solution of the Painleve VI equation. As a simple consequence, the monodromy problem of the Heun equation is related to the connection problem for the Painleve VI.
Highlights
The conformal blocks are associated with the moduli spaces Mg,n of genus g Riemann surfaces with n punctures
Suppressing again the dependence on ∆i, i = 1, . . . , 4, we denote by FP (x) the 4-point conformal block associated with the dual diagram in figure 2
The classical limits of the conformal blocks appear when the Virasoro central charge c goes to infinity along with all the dimensions, so that the ratios ∆i/c and ∆(Pα)/c remain fixed
Summary
The classical conformal blocks are closely related to the monodromy problem for ordinary linear differential equations. It is well known that for the n-punctured sphere the complex dimension of this space is exactly 2 (n − 3) This means that the differential equation (2.1) generally does not admit continuous isomonodromic deformations, and 2 (n− 3) parameters ci, zi The consistency with the null-vector decoupling equation (2.7) requires that the accessory parameters are determined in terms of the classical conformal block as follows. In our case fν(z) is the classical conformal block, and the accessory parameters given by (2.9) solve another monodromy problem, which is holomorphic in z, and involves n−3 additional parameters ν, as follows.. Of the accessory parameter C in (2.13) fixes the conjugacy class of the monodromy along the path γ12 = γ1 ◦ γ2 (see figure 4), Tr (M (γ12)) = −2 cos(πν).
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