The Quiver Matrix Model and 2d-4d Conformal Connection

TL;DR

This work clarifies how the quiver matrix model (ITEP) reproduces 2d–4d conformal structures and links to Seiberg–Witten theory. It introduces a quantum spectral curve $\langle\langle \det(x - i g_s \partial \phi(z)) \rangle\rangle = 0$ at finite $N$ and general $\beta$, and derives planar loop equations from $W_n$ constraints. Through residue analyses of both the matrix-model curves and SW curves, it establishes an explicit isomorphism between the spectral curves for ADE quivers in the planar limit, with detailed mass-parameter matching in the $A_2$ (SU(3)) case and a general $n$ extension. The multi-Penner approach of Dijkgraaf–Vafa provides a concrete bridge to Nekrasov–Ok Nekrasov partition functions and Toda/CFT via AGT, yielding a unified description of 4d ${\mathcal N}=2$ quiver gauge theories and their 2d Toda representations. The results supply finite-$N$ spectral curves, illuminate the role of non-commutative geometry in the beta-deformed setting, and generalize the matrix-model/SW correspondence to arbitrary ADE types and $n$, reinforcing the deep 2d–4d conformal connection.

Abstract

We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories. On the basis of the CFT representation of the beta deformation of the model, a quantum spectral curve is introduced as << det (x- i g_s \partial φ(z)) >>=0 at finite N and for beta \neq 1. The planar loop equation in the large N limit follows with the aid of W_n constraints. Residue analysis is provided both for the curve of the matrix model with the "multi-log" potential and for the Seiberg-Witten curve in the case of SU(n) with 2n flavors, leading to the matching of the mass parameters. The isomorphism of the two curves is made manifest.