Abstract
We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2, C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector technique and the Casimir operator approach. Furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. In this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method.
Highlights
The holographic interpretation of the conformal blocks is rather new result which certainly brings the AdS3/CFT2 correspondence to the new conceptual level of understanding.2 We recall that along with the characters of the symmetry algebra, conformal blocks represent the main kinematical ingredients of any CFT while the dynamical properties of the theory encoded in the structure constants of the operator algebra
The most simple regime is reduced to the global slp2, Cq conformal blocks while the most complicated one is known as the classical conformal blocks
Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block
Summary
The Virasoro algebra contains the maximal finite-dimensional subalgebra slp, Cq Ă V ir of projective conformal transformations which will be further referred to as the global conformal algebra. We consider correlation functions invariant under the global conformal algebra. A field φ “ φpzq is conformal if it satisfies the slp, Cq highest-weight relations. Let us consider 5-point correlation function of the primary fields φipziq with dimensions. Using the projective invariance one can fix its holomorphic dependence as follows. The exponents are chosen so that the 4-point function is directly obtained from (2.2) by taking ∆3 “ 0 along with BG{Bu “ 0. In this case the right-hand side of (2.2) does not depend on z3, while the projective invariance remains intact. The same is true when going further to the 3-point function: taking ∆3 “ ∆4 “ 0 along with BG{Bu “ BG{Bv “ 0 gives the standard Polyakov expression
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