General proof of the sequential-integration formula for arbitrary quivers

Establish the closed-form large-N superconformal-index formula for arbitrary n-node SU quiver gauge theories by proving that sequential large-N integration over gauge nodes reproduces the determinant and inverse-kernel structure beyond the explicit n=2 and n=3 checks.

Background

The paper studies the large-N superconformal index of product-group SU quiver gauge theories with adjoint, bifundamental, and vector-like fundamental matter. One derivation treats all gauge nodes simultaneously using a fixed-mode complex Gaussian and yields a matrix determinant-and-inverse-kernel formula.

A complementary derivation integrates gauge nodes one at a time, regarding the remaining gauge factors as flavor symmetries at each step. The authors report explicit agreement with the matrix structure for two- and three-node quivers, but characterize the extension of this sequential procedure to arbitrary numbers of nodes as conjectural. Establishing that arbitrary-n sequential integration always reproduces the proposed formula would turn this partial check into a general proof.