Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition

TL;DR

This paper investigates algebraic Nahm equations related to symmetric quivers, proving that certain quiver A-polynomials are tempered and quantizable, revealing a combinatorial pattern in face polynomials, and conjecturing this holds for all symmetric quivers.

Contribution

It establishes the K-theoretic condition for a class of quiver A-polynomials and explores their combinatorial and polyhedral properties, advancing understanding of Nahm sums in quiver theory.

Findings

Quiver A-polynomials are tempered and quantizable for diagonal symmetric quivers.

Face polynomials follow a remarkable combinatorial pattern.

The K-theoretic condition holds for diagonal symmetric quivers and is conjectured for all symmetric quivers.

Abstract

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the $q\rightarrow 1$ limit of the $q$-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials -- specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that they are quantizable. Moreover, we find that their face polynomials obey a remarkable combinatorial pattern. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We show that this condition holds for the diagonal quivers with adjacency matrix $C = \mathrm{diag}(\alpha,\alpha,\dots,\alpha),\ \alpha\geq 2$, and provide several checks for non-diagonal quivers. Our conjecture is that the K-theoretic condition holds for all symmetric quivers.