Massive scaling limit of theβ-deformed matrix model of Selberg type

Abstract

We consider a series of massive scaling limits m 1 → ∞, q → 0, lim m 1 q = Λ 3 followed by2 of the β-deformed matrix model of Selberg type (N c = 2, N f = 4) which reduce the number of flavours to N f = 3 and subsequently to N f = 2. This keeps the other parameters of the model finite, which include n = N L and N = n+N R , namely, the size of the matrix and the "filling fraction". Exploiting the method developed before, we generate instanton expansion with finite g s , ǫ 1,2 to check the Nekrasov coefficients (N f = 3, 2 cases) to the lowest order. The limiting expressions provide integral representation of irregular conformal blocks which contains a 2d operator limn : e (1/2)α 2 φ(q) : and is subsequently analytically continued. *

Seiberg-Witten curve via generalized matrix model

“…Thus, such an extension seems to be quite non-trivial. In a sphere case, it is known that the matrix expression for the conformal block is valid even for finite N [42]- [49]. So, it would be quite interesting to study whether such an expression is possible also for the torus case.…”

“…(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].…”

Discrete Painlevé system for the partition function of N f = 2 SU(2) supersymmetric gauge theory and its double scaling limit

“…It is also known that the GWW model has connection with the Painlevé III equation [46,47]. 5 Another type of generalization of the GWW model can be found, for example, in [48]. The phase transition of the asymmetric GWW model is studied in [49].…”

“…where ω ≡ exp(2πi/3). At η = ±1, ζ = 0, the quartic equation (4.3) turns into 5) and the three roots out of four degenerate to ξ = 0. Therefore, we choose the critical values of (ξ, η, ζ) as…”

“…Taking the double scaling limit [9,10,11] to the critical point of this system, we succeeded in deriving the Painlevé II differential equation containing the parameter M ≡ α 1+2 + N = (m 2 − m 1 )/g s . See [12,8,5,1] and Appendix F of this paper for detail of these transcriptions or matrix model/4d gauge theory dictionary. This critical point is the simplest example of the Argyres-Douglas type points [13,14,15,1] well studied by using the Seiberg-Witten curve of cubic or quartic type [16,17].…”