Mutation, Witten Index, and Quiver Invariant

TL;DR

This paper investigates how Seiberg-like dualities, or mutations, affect the Witten index and quiver invariants in 1d ${ m N}=4$ quiver quantum mechanics, revealing nuanced mutation rules that preserve certain supersymmetric spectra.

Contribution

It introduces a detailed analysis of mutations in 1d ${ m N}=4$ quiver quantum mechanics, distinguishing between Witten index preservation and quiver invariant mutation rules, with numerical confirmation.

Findings

Witten indices are preserved under specific mutations with fixed FI parameters.

Quiver invariants mutate more straightforwardly than Witten indices.

A new mutation rule for quiver invariants is proposed and numerically validated.

Abstract

We explore Seiberg-like dualities, or mutations, for ${\cal N}=4$ quiver quantum mechanics in the context of wall-crossing. In contrast to higher dimensions, the 1d Seiberg-duality must be performed with much care. With fixed Fayet-Iliopoulos constants, at most two nodes can be mutated, one left and the other right, mapping a chamber of a quiver into a chamber of a mutated quiver. We delineate this complex pattern for triangle quivers and show how the Witten indices are preserved under such finely chosen mutations. On the other hand, the quiver invariants, or wall-crossing-safe part of supersymmetric spectra, mutate more straightforwardly, whereby a quiver is mapped to a quiver. The mutation rule that preserves the quiver invariant is different from the usual one, however, which we explore and confirm numerically.