Check of AGT Relation for Conformal Blocks on Sphere

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.