Scaling Behaviour of Quiver Quantum Mechanics

TL;DR

This paper investigates the vacuum degeneracy in large-rank Kronecker quiver quantum mechanics by computing the Witten index, revealing exponential growth for certain intersection numbers and discussing broader rank cases.

Contribution

It introduces a method to compute the Witten index for large-rank Kronecker quivers and uncovers exponential growth patterns in vacuum degeneracy.

Findings

Exponential growth of the Witten index for k ≥ 3

Counting index of mutation equivalent quivers

Speculation on general ranks including nonprimitive cases

Abstract

We explore vacuum degeneracy of Kronecker quiver with large ranks, by computing Witten index of corresponding 1d gauged linear sigma model. For $(d-1,d)_k$ quivers with the intersection number $k$, we actually counted index of its mutation equivalent, $(d,(k-1)d+1)_k$, and find exponentially large behaviour whenever $k\ge 3$. We close with speculation on more general ranks of Kronecker quiver including the nonprimitive cases.