Trace identities for quiver representations

TL;DR

This paper derives a trace-based formula for the determinant of the twisted Laplacian in quiver representations, connecting algebraic identities with combinatorial interpretations involving cycle-rooted forests.

Contribution

It introduces a novel expression for the twisted Laplacian determinant in terms of cycle holonomies and proves a general block matrix determinant identity with two different proofs.

Findings

Determinant expressed via traces of holonomies for quiver representations

Block matrix determinant identity proven through enumerative and generating series methods

Combinatorial interpretation as weighted counts of cycle-rooted forests

Abstract

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for the determinant of a block matrix in terms of traces of products of its blocks. We give two proofs, one purely enumerative and one using generating series. In the special case of a finite graph equipped with a vector bundle and a connection, the twisted Laplacian determinant admits a combinatorial interpretation as a weighted count of tuples of oriented cycle-rooted spanning forests, where the weights involve traces of holonomies along cycles formed by combining the edges of the forests.