Five-brane webs, 3d $\mathcal{N}=2$ theories and quantum curves

TL;DR

This paper explores the connection between 3d $ ext{N}=2$ supersymmetric theories realized by 5-brane webs, their $S^3$ partition functions, and quantum curves, proposing a conjecture linking Newton polygons to toric diagrams.

Contribution

It introduces a conjecture relating quantum curves and brane web diagrams, providing explicit derivations for Lagrangian theories and evidence for non-Lagrangian cases.

Findings

Conjecture that Newton polygons of quantum curves match toric diagrams of 5-brane webs.

Explicit derivation of the relation for Lagrangian theories using localization and Fermi gas formalism.

New matrix models for 5-brane webs, including arbitrary $(p,q)$ 5-branes, linking brane configurations to quantum curves.

Abstract

We propose a relation between the brane configurations consisting of D3-branes and 5-brane webs which realize 3d $\mathcal{N}=2$ supersymmetric Chern-Simons theories and quantum curves by focusing on the $S^{3}$ partition functions. In particular, we conjecture that the Newton polygons of the quantum curves are equal to the toric diagrams which are dual to the 5-brane webs. For brane configurations whose worldvolume theories have Lagrangian descriptions, we show an explicit derivation of the relation by using the supersymmetric localization and the Fermi gas formalism. We also provide some evidence of the conjecture for non-Lagrangian theories. We see that our conjecture gives us new matrix models for 5-brane webs including a $\left(p,q\right)$5-brane with arbitrary $p$. This leads to explicit relations between the brane configurations, the matrix models and genus one quantum curves.