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Sinh Deformed Nakajima Operators

Boan Zhao, Paul Luis Roehl, Chunhao Li

TL;DR

This work constructs a sinh deformed version of Nakajima operators a_± that act on the equivariant K-theory of Hilbert schemes of points on C^2 by tensoring with square roots of canonical bundles along Nakajima correspondences. It proves a precise Heisenberg-like commutation relation [a_-, a_+] = (sqrt(q1) - 1/sqrt(q1))(sqrt(q2) - 1/sqrt(q2)), realized through both inverse tangent weight computations and contour integral methods, and interprets these operators physically as instanton line operators in 6d (1,1) SYM wrapped on a circle, with a 2d sigma-model perspective via topological interfaces. The paper further develops the correspondence between 2d amplitudes and 5d-6d instanton dynamics, including localization computations that reproduce the expected operator compositions even in nontransverse settings and connects to AGT via boundary states, thereby linking geometric representation theory with higher-dimensional gauge theory and duality frameworks. Overall, it provides a robust bridge between Nakajima type constructions in K-theory, instanton line operators in higher dimensions, and the AGT/Verma-module structures that appear in related correspondences.

Abstract

We prove a novel action of the (three-dimensional) Heisenberg algebra on the equivariant K-theory of the Hilbert scheme of points on C2. These operators are defined via pushforwards and pullbacks via the Nakajima correspondences while tensoring the square roots of the canonical line bundles of the correspondences. We show, using supersymmetric localisation in 6d (1, 1) Super Yang-Mills compactified on a circle, that these operators correspond to instanton line operators wrapping the extra circle.

Sinh Deformed Nakajima Operators

TL;DR

This work constructs a sinh deformed version of Nakajima operators a_± that act on the equivariant K-theory of Hilbert schemes of points on C^2 by tensoring with square roots of canonical bundles along Nakajima correspondences. It proves a precise Heisenberg-like commutation relation [a_-, a_+] = (sqrt(q1) - 1/sqrt(q1))(sqrt(q2) - 1/sqrt(q2)), realized through both inverse tangent weight computations and contour integral methods, and interprets these operators physically as instanton line operators in 6d (1,1) SYM wrapped on a circle, with a 2d sigma-model perspective via topological interfaces. The paper further develops the correspondence between 2d amplitudes and 5d-6d instanton dynamics, including localization computations that reproduce the expected operator compositions even in nontransverse settings and connects to AGT via boundary states, thereby linking geometric representation theory with higher-dimensional gauge theory and duality frameworks. Overall, it provides a robust bridge between Nakajima type constructions in K-theory, instanton line operators in higher dimensions, and the AGT/Verma-module structures that appear in related correspondences.

Abstract

We prove a novel action of the (three-dimensional) Heisenberg algebra on the equivariant K-theory of the Hilbert scheme of points on C2. These operators are defined via pushforwards and pullbacks via the Nakajima correspondences while tensoring the square roots of the canonical line bundles of the correspondences. We show, using supersymmetric localisation in 6d (1, 1) Super Yang-Mills compactified on a circle, that these operators correspond to instanton line operators wrapping the extra circle.
Paper Structure (19 sections, 135 equations, 4 figures)

This paper contains 19 sections, 135 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical Definition of the Operators $a_\pm$
  3. $a_\pm$ as Instanton Operators in 6d Super Yang-Mills
  4. Proof of the Commutator $[a_+, a_-]$
  5. Review of $U(1)$ Instanton Moduli Spaces
  6. Review of Nakajima Correspondences
  7. Matrix Elements of $a_\pm$
  8. Vanishing of Off-diagonal Components of $[a_-, a_+]$
  9. Diagonal Components of $[a_-, a_+]$
  10. Amplitudes of 2d $\sigma$-Models and Correspondences
  11. Norm of BPS states
  12. Amplitudes Associated with Correspondences
  13. Composition of Correspondences
  14. Instanton Line Operators in 6d $(1, 1)$ Super Yang-Mills
  15. Topological Twist
  16. ...and 4 more sections

Figures (4)

  • Figure 1: A closed string propagating in $M_k$.
  • Figure 2: A topological interface inserted at $x^0 = 0.5$ (a circle) which corresponds to $M_{k, k + 1}$ (a multivalued map $M_k\to M_{k + 1}$). The number $0.5$ can be replaced by any number between $0$ and $1$.
  • Figure 3: Two topological interfaces inserted at $x^0 = 0.3, 0.6$ which correspond to $M_{k, k + 1}, M_{k + 1, k +2 }$ respectively. The numbers $0.3, 0.6$ can be replaced by any two distinct numbers between $0$ and $1$.
  • Figure 4: 6d $(1, 1)$ super Yang-Mills on $\mathbb{C}^2$ times a cylinder