Gauge origami on broken lines

TL;DR

This paper introduces a new gauge origami moduli space on broken lines, realized via Quot schemes and quiver representations, and computes related $K$-theoretic invariants, connecting to Nekrasov's partition function.

Contribution

It defines the gauge origami moduli space on broken lines, constructs its virtual classes, and relates its invariants to known partition functions and generating series.

Findings

Computed the partition function for all ranks.

Reproduces the generating series of equivariant $hi_{y}$-genus for smooth cases.

Links the partition function to Nekrasov's partition function.

Abstract

In analogy to Nekrasov's theory of gauge origami on intersecting branes, we introduce the gauge origami moduli space on broken lines. We realize this moduli space as a Quot scheme parametrising zero-dimensional quotients of a torsion sheaf on two intersecting affine lines, and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf, by which we define $K$-theoretic invariants. We compute its associated partition function for all ranks, and show that it reproduces the generating series of equivariant $\chi_{y}$-genus when the moduli space is smooth. Finally, we relate our partition function with the virtual invariants of the Quot schemes of the affine plane and Nekrasov's partition function.