Check of AGT Relation for Conformal Blocks on Sphere

TL;DR

This work tests the AGT conjecture beyond the four-point sphere blocks by computing explicit 5- and 6-point conformal blocks using a diagrammatic approach built from propagators and triple vertices, and matching them to Nekrasov partition functions up to the third order. It develops a systematic extraction of the $U(1)$ factor from free-field constructions and establishes precise parameter mappings between 2d CFT data and 4d $igotimes_{i=1}^n U(2)$ Nekrasov data, including mass-to-dimension relations and symmetry-induced solution branches. The results demonstrate consistency of AGT for the higher-point cases and identify how higher-point blocks reduce to lower-point ones at successive orders, while also outlining the challenges posed by non-comb (generalized quiver) configurations. Overall, the paper advances the verification of AGT for sphere blocks with more than four legs and clarifies the role and form of the universal $U(1)$ factor in this correspondence.

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.

Figures (10)

• Figure 1: This diagram defines the instanton partition function for the $\bigotimes\limits_{i=1}^{n}U(2)$ linear quiver theory. There is simple correspondence between the quiver theories and the diagrams.
• Figure 2: Ferrer-Young diagram $[14,12,9,8,8,7,6,2,2,1]$, hhere $s=(i,j)$ is a multiindex (coordinate on Ferrer-Young diagram) and $k^T_j(Y),k_i(Y)$ are the height of column and length of row in Ferrer-Young diagram correspondingly ($i=3,j=7$ in the picture).
• Figure 3: Diagram technique.
• Figure 4: This diagram defines six-point conformal block.
• Figure 5: This diagram defines the fusing rule for fields in the conformal block. We called this diagram type comb-type diagram. AGT conjecture speculates only about diagram of this type.