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Gauge origami on broken lines

Sergej Monavari

TL;DR

This work develops a geometric framework for gauge origami on broken lines by modeling the moduli space as a Quot scheme on the union of coordinate axes and embedding it into a noncommutative Quot scheme. It constructs virtual fundamental classes and $K$-theoretic invariants, and derives a closed-form partition function ${\mathcal{Z}}_{\overline{r}}(q)$ that factorizes into rank-1 building blocks and matches Nekrasov-type counts in appropriate limits. A Nekrasov-Okounkov twist refines the invariants, yielding a compact plethystic-exponential expression, while the cohomological limit recovers a simple power-form generating function. The paper also relates the gauge origami partition function to the ADHM/quiver framework and to the Quot scheme on ${\mathbb{A}}^2$, highlighting connections to fundamental/antifundamental matter and to the broader landscape of instanton invariants in string theory. Together, these results provide a rigorous algebraic-geometry foundation for coupled vortex systems arising from intersecting brane configurations and offer exact, computable invariants across ranks.

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 $χ_{y}$-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.

Gauge origami on broken lines

TL;DR

This work develops a geometric framework for gauge origami on broken lines by modeling the moduli space as a Quot scheme on the union of coordinate axes and embedding it into a noncommutative Quot scheme. It constructs virtual fundamental classes and -theoretic invariants, and derives a closed-form partition function that factorizes into rank-1 building blocks and matches Nekrasov-type counts in appropriate limits. A Nekrasov-Okounkov twist refines the invariants, yielding a compact plethystic-exponential expression, while the cohomological limit recovers a simple power-form generating function. The paper also relates the gauge origami partition function to the ADHM/quiver framework and to the Quot scheme on , highlighting connections to fundamental/antifundamental matter and to the broader landscape of instanton invariants in string theory. Together, these results provide a rigorous algebraic-geometry foundation for coupled vortex systems arising from intersecting brane configurations and offer exact, computable invariants across ranks.

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 -theoretic invariants. We compute its associated partition function for all ranks, and show that it reproduces the generating series of equivariant -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.

Paper Structure

This paper contains 32 sections, 25 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. The moduli space
  4. Quiver model
  5. Partition function
  6. Strategy of the proof
  7. Nekrasov-Okounkov twist
  8. Cohomological limit
  9. Elliptic genus
  10. ADHM quiver
  11. Geometry and Physics
  12. Compact curves
  13. Moduli space
  14. Quot scheme
  15. Quiver model
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\overline{r}=(r_1, r_2)$ and $n\geq 0$. There exists a vector bundle ${\mathcal{V}}$ on ${\mathcal{M}}^{\rm nc}_{\overline{r}, n}$ together with a section $s$ such that ${\mathcal{M}}_{\overline r,n}$ is realised as the zero locus $Z(s)$\begin{tikzcd} & \CV\arrow[d]\\ \CM_{\overline{r}, n}\cong

Figures (3)

  • Figure 1: Framed quiver $\mathsf{L}$.
  • Figure 2: Framed quiver ${\tilde{\mathsf{L}}}$.
  • Figure 3: ADHM quiver.

Theorems & Definitions (45)

  • Theorem 1.1: \ref{['thm: isotropic construction']}
  • Theorem 1.2: \ref{['thm:factorization']}, \ref{['cor: explicit expression inv']}
  • Corollary 1.3: \ref{['cor: NO inv']}
  • Corollary 1.4: \ref{['cor: cohom']}
  • Example 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Remark 3.1
  • ...and 35 more