The geometry of Nekrasov's gauge origami theory

TL;DR

This paper explores the geometric and algebraic structures of Nekrasov's gauge origami theory, providing a rigorous definition of its partition function, establishing key properties, and proposing a conjectural description of its moduli space.

Contribution

It offers an algebro-geometric formulation of the origami moduli space and partition function, computes signs for fixed points, and connects to framed sheaves on a fourfold.

Findings

Defined the origami moduli space as the zero locus of an isotropic section

Established an integrality result and dimensional reduction formulas

Proposed a conjectural description of the moduli space via 2D framed sheaves

Abstract

Nekrasov's gauge origami theory provides a (complex) 4-dimensional generalization of the ADHM quiver and its moduli spaces of representations. We describe the origami moduli space as the zero locus of an isotropic section of a quadratic vector bundle on a smooth space. This allows us to give an algebro-geometric definition of the origami partition function in terms of Oh--Thomas virtual cycles. The key input is the computation of a sign associated to each torus fixed point of the moduli space. Furthermore, we establish an integrality result and dimensional reduction formulae, and discuss an application to non-perturbative Dyson--Schwinger equations following Nekrasov's work. Finally, we conjecture a description of the origami moduli space in terms of certain 2-dimensional framed sheaves on $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, which we verify at the level of torus fixed points.