Chamber Structure and Wallcrossing in The ADHM Theory of Curves I

TL;DR

This paper introduces a generalized stability condition in the ADHM theory of curves, revealing a chamber structure and setting the stage for wallcrossing analysis and applications to local stable pair invariants.

Contribution

It generalizes ADHM invariants by incorporating a real parameter-dependent stability condition, unveiling a chamber structure in the moduli space.

Findings

Establishment of a chamber structure in the ADHM theory of curves.

Definition of residual ADHM invariants via equivariant virtual integration.

Foundation for wallcrossing formulas and applications in subsequent work.

Abstract

ADHM invariants are equivariant virtual invariants of moduli spaces of twisted cyclic representations of the ADHM quiver in the abelian category of coherent sheaves of a smooth complex projective curve X. The goal of the present paper is to present a generalization of this construction employing a more general stability condition which depends on a real parameter. This yields a chamber structure in the ADHM theory of curves, residual ADHM invariants being defined by equivariant virtual integration in each chamber. Wallcrossing results and applications to local stable pair invariants will be presented in the second part of this work.