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The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue

Taro Kimura, Go Noshita

Abstract

We present a contour integral formalism for computing the K-theoretic equivariant Pandharipande--Thomas (PT) 4-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that the PT 4-vertex can be obtained from the same integrand as the Donaldson--Thomas (DT) 4-vertex by choosing a different reference vector. We illustrate the formalism through examples involving curves and surfaces on the 4-fold. Furthermore, we investigate the DT/PT correspondence for the 4-fold setting together with its higher rank and supergroup-like generalizations.

The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue

Abstract

We present a contour integral formalism for computing the K-theoretic equivariant Pandharipande--Thomas (PT) 4-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that the PT 4-vertex can be obtained from the same integrand as the Donaldson--Thomas (DT) 4-vertex by choosing a different reference vector. We illustrate the formalism through examples involving curves and surfaces on the 4-fold. Furthermore, we investigate the DT/PT correspondence for the 4-fold setting together with its higher rank and supergroup-like generalizations.

Paper Structure

This paper contains 55 sections, 6 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Magnificent four and JK-residue
  3. Magnificent four
  4. Jeffrey--Kirwan (JK) residue
  5. Magnificent four and solid partitions
  6. Plethystic exponential (PE) formula
  7. DT4 counting and JK-residue
  8. DT4 counting with boundary conditions
  9. Leg boundary conditions
  10. Surface boundary conditions
  11. Leg and surface boundary conditions
  12. Framing node and JK-residue
  13. Leg boundary conditions
  14. Surface boundary conditions
  15. Choice of contour
  16. ...and 40 more sections

Key Result

Theorem 2.1

The poles of the rank one magnificent four giving non-zero JK-residue are classified as where we defined For later use, we also introduce

Theorems & Definitions (16)

  • Theorem 2.1: Nekrasov:2017cihNekrasov:2018xsb
  • Definition 3.1
  • Definition 3.2
  • Claim 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.4
  • Conjecture 6.1
  • Conjecture 6.2
  • Conjecture 6.3
  • ...and 6 more