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

Abstract

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the q → 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 quivers. For a large class of symmetric quivers, we prove that quiver A-polynomials, that is, 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, reminiscent of the permutohedron. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We work out all diagonal quivers with adjacency matrix C = diag(α, α, . . . , α), α ≥ 2, and give a sketch when the diagonal entries are all distinct. We also conjecture that it holds for all symmetric quivers.