Method of generating q-expansion coefficients for conformal block and N=2 Nekrasov function by β-deformed matrix model

Abstract

We observe that, at β-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. *

Seiberg-Witten curve via generalized matrix model

“…(See also [42] for 1/N correction.) It has been discussed that direct integral calculation leads to the instanton (q-)expansion of the Nekrasov partition function (and the corresponding expansion of the conformal block) [43][44][45][46][47][48][49].…”

Classical Virasoro irregular conformal block

“…We will fix the position z 0 = 0 as the reference point and put z n+1 → ∞. The β-deformed Penner-type matrix model [15,16] is defined from the conformal block by factoring out the prefactor 0≤k<I≤n+1 (z k −z l ) −2α k α l as well as the terms containing z n+1 :…”

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

“…the DotsenkoFateev-like β-ensemble) representations are already known both for B (0) [18][19][20][21][22][23][24]70] and for B (1) [64,65]. The first one is represented [53] as an AMM decomposition [90,91,98] into two spherical Selberg integrals…”