ADHM sheaf theory and wallcrossing

Abstract

In this article we survey the recent developments in ADHM sheaf theory on a smooth projective variety $X$. When $X$ is a curve the theory is an alternative construction of stable pair theory of Pandharipande and Thomas or Gromov–Witten theory on local curve geometries. The construction relies on relative Beilinson spectral sequence and Fourier–Mukai transformation. We will present some applications of the theory, including the derivations of the wallcrossing formulas, higher rank Donaldson–Thomas invariants on local curves, and the coholomogies of the moduli of stable Hitchin pairs.