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Tetrahedron instantons in Donaldson-Thomas theory

Nadir Fasola, Sergej Monavari

Abstract

Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define K-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.

Tetrahedron instantons in Donaldson-Thomas theory

Abstract

Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define K-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.
Paper Structure (28 sections, 22 theorems, 175 equations, 2 figures)

This paper contains 28 sections, 22 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tetrahedron instantons
  3. Mathematical formulation
  4. The moduli space
  5. Partition function
  6. Strategy of the proof
  7. Cohomological limit
  8. Witten genus
  9. Relation to Donaldson-Thomas Theory
  10. String Theory
  11. Moduli space of tetrahedron instantons
  12. Virtual fundamental classes
  13. Quot scheme
  14. Non-commutative Quot scheme
  15. Zero-locus construction
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\overline{r}=(r_1, r_2, r_3, r_4)$ and $n$ be non-negative integers. Then \begin{tikzcd} & \CL\arrow[d]\\ \CM_{\overline{r}, n}\cong Z(s)\arrow[r, hook, "\iota"] &\CM^{\nc}_{\overline{r}, n},\arrow[u, bend right, swap, "s"] \end{tikzcd}where ${\mathcal{M}}^{\rm nc}_{\overline{r}, n}$ is a smoot

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (49)

  • Theorem 1.1: Theorem \ref{['thm: isotropic construction']}
  • Theorem 1.2: Theorem \ref{['thm:factorization']}, \ref{['thm: explicit expression inv']}
  • Corollary 1.3: Corollary \ref{['cor: cohom']}
  • Example 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • Remark 3.1
  • ...and 39 more