Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
Hiroshi Itoyama, Takeshi Oota
TL;DR
This work develops a beta-deformed matrix-ensemble framework to generate and relate $q$-expansion coefficients for four-point conformal blocks and the $ ext{Nekrasov}$ partition function for $SU(2)$ with $N_f=4$. By treating the Dotsenko-Fateev integral as a perturbed double-Selberg system and exploiting the $q=0$ decoupling into two Selberg sectors, the authors construct two generating functions and derive exact finite-$N$ loop equations as well as planar $g_s$-corrected resolvents, linking conformal data to gauge-theory parameters through a precise 0d–4d dictionary. They obtain the first nontrivial conformal-block coefficient and derive the second Nekrasov coefficient, uncovering a natural pair of Young diagrams and a Jack-polynomial structure in the expansion. A free-field representation of the Nekrasov function is provided, further strengthening the AGT correspondence and offering a robust method to compute higher-order $q$-expansion terms in this Beta-deformed setting.
Abstract
We observe that, at beta-deformed matrix models for the four-point conformal block, the point q=0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of) two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko-Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q=0, it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q-expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU(2) with N_f =4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q=0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents. The planar free energy in the q-expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given.