Solving $\mathcal N=4$ SYM BCFT matrix models at large $N$

TL;DR

This paper analyzes large N matrix models arising from 4d N=4 SYM with boundary conditions, deriving saddle points using both localization and holography, and tests the dualities with precision.

Contribution

It introduces a method to find saddle points of boundary matrix models at large N using generalized Lambert W-functions, connecting localization results with holographic duals.

Findings

Derived explicit saddle points for boundary matrix models.

Connected matrix model solutions with holographic dual geometries.

Validated duality predictions through precision tests.

Abstract

Many observables in 4d $\mathcal N=4$ SYM with Gaiotto-Witten boundary conditions can be described exactly by matrix models via supersymmetric localization. The boundaries typically introduce new degrees of freedom, through a reduction of the gauge symmetry on the boundary or as explicit boundary degrees of freedom, leading to non-trivial matrix models. We derive the saddle points dominating these matrix models at large $N$, expressed in terms of generalized Lambert W-functions. In string theory the BCFTs are realized by D3-branes ending on D5 and NS5 branes. We independently derive the saddle points from the holographic duals with $\rm AdS_4\times S^2\times S^2\times\Sigma$ geometry and provide precision tests of the dualities.