The geometry of Nekrasov's gauge origami theory

TL;DR

This work establishes a rigorous algebro-geometric framework for Nekrasov's gauge origami by realizing the origami moduli $M_{Q_4}( vec{r},n)$ as the zero locus of an isotropic section of a quadratic vector bundle on an ambient smooth moduli, yielding an Oh–Thomas virtual cycle and partition functions that reproduce Nekrasov’s Boltzmann weights via fixed-point signs. It proves dimensional reduction to 2D/3D ADHM theories, analyzes the torus-fixed loci (labeled by collections of partitions) to enable localization, and proves connectedness of the origami moduli. The work proposes a framed-sheaves description on $(P^1)^4$ and proves a conjectural isomorphism with the quiver moduli at fixed points, supported by a derived, $(-2)$-shifted symplectic structure and explicit quotient-Quot scheme correspondences. It also develops non-perturbative Dyson–Schwinger identities in this setting and explores wall-crossing phenomena, integrality properties, and potential modular structure of the resulting generating functions, providing a robust geometric bridge between 4D gauge origami and framed-sheaf theories.

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.

Figures (4)

• Figure 1: The quiver $Q_2$.
• Figure 2: The quiver $Q_4$. The arrows from $r_A$ to $n$ are labelled by $I_A$ and the arrows from $n$ to $r_A$ are labelled by $J_A$.
• Figure 3: The moduli space of Example \ref{['ex:butterfly n=1']}.
• Figure 4: The moduli space of Example \ref{['ex:book n=1']}.

Theorems & Definitions (116)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Example 2.2
• Example 2.3
• Proposition 2.4