Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type

TL;DR

This paper explores the massive scaling limits of a beta-deformed matrix model of Selberg type, deriving integral representations of irregular conformal blocks and verifying Nekrasov coefficients through instanton expansions.

Contribution

It introduces a new scaling limit process for the beta-deformed matrix model, connecting it to irregular conformal blocks and providing explicit integral representations.

Findings

Derived integral representations of irregular conformal blocks.

Verified Nekrasov coefficients for N_f=3,2 cases.

Established connections between matrix models and conformal field theory.

Abstract

We consider a series of massive scaling limits m_1 -> infty, q -> 0, lim m_1 q = Lambda_{3} followed by m_4 -> infty, Lambda_{3} -> 0, lim m_4 Lambda_{3} = (Lambda_2)^2 of the beta-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, epsilon_{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 lim frac{1}{C(q)} : e^{(1/2) \alpha_1 \phi(0)}: (int_0^q dz : e^{b_E phi(z)}:)^n : e^{(1/2) alpha_2 phi(q)}: and is subsequently analytically continued.