Check of AGT Relation for Conformal Blocks on Sphere

TL;DR

This paper investigates the AGT conjecture for spherical conformal blocks with more than four external legs, explicitly verifying the relation up to third order for 5- and 6-point cases.

Contribution

It extends the verification of the AGT relation to higher-point conformal blocks on the sphere and proposes a method to extract the U(1)-factor from free field vertex operators.

Findings

Confirmed AGT relation up to third order for 5- and 6-point blocks

Developed diagram technique involving propagators and vertices

Proposed extraction method for U(1)-factor from matrix elements

Abstract

The AGT conjecture identifying conformal blocks with the Nekrasov functions is investigated for the spherical conformal blocks with more than 4 external legs. The diagram technique which arises in conformal block calculation involves propagators and vertices. We evaluated vertices with two Virasoro algebra descendants and explicitly checked the AGT relation up to the third order of the expansion for the 5-point and 6-point conformal blocks on sphere confirming all the predictions of arXiv:0906.3219 relevant in this situation. We propose that U(1)-factor can be extracted from the matrix elements of the free field Vertex operators. We studied the n-point case, and found out that our results confirm the AGT conjecture up to the third order expansions.