An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories

TL;DR

This work investigates a triad of interrelated theories—the $A_r$ quiver matrix models, $d=2$ $A_r$ Toda CFTs, and $d=4$ ${\\rm N}=2$ conformal $A_r$ quiver gauge theories—and explicates how the Dijkgraaf–Vafa matrix-model framework reproduces key quantities across these domains. It develops explicit calculations in the $A_1$ case (tests with Selberg integrals, hypergeometric structures, and spectral curves) and extends to general $A_r$ to map three- and higher-point functions to Toda blocks and Nekrasov partition functions, including the role of filling fractions and spectral curves. A $eta$-deformation is discussed for general $\epsilon_1,\epsilon_2$, and a speculative proposal for a $5d$ Nekrasov extension via $q$-deformed matrix models is offered, with partial checks against known 5d instanton structures. The results reinforce a unifying picture in which a single Riemann-surface geometry underlies gauge theory, Toda CFT, and matrix-model computations, and they point toward broader generalizations to higher ADE algebras and five-dimensional theories. The practical impact lies in providing concrete, cross-checked computational frameworks that connect seemingly disparate theories through spectral curves and conformal blocks, offering new tools for exact calculations in supersymmetric gauge theories and CFTs.

Abstract

We explore the connections between three classes of theories: A_r quiver matrix models, d=2 conformal A_r Toda field theories and d=4 N=2 supersymmetric conformal A_r quiver gauge theories. In particular, we analyse the quiver matrix models recently introduced by Dijkgraaf and Vafa and make detailed comparisons with the corresponding quantities in the Toda field theories and the N=2 quiver gauge theories. We also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions.