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Gauge origami and quiver W-algebras II: Vertex function and beyond quantum $q$-Langlands correspondence

Taro Kimura, Go Noshita

TL;DR

This work advances gauge origami by coupling D2-brane systems to screening currents of W-algebras, identifying the vertex function for quasimaps to Nakajima varieties with the gauge origami partition function. It establishes that the magnetic W-algebra block directly solves the $q$-KZ equation, enabling a direct electric–magnetic equivalence via chamber/radial ordering, and extends the quantum $q$-Langlands correspondence to double affine settings using the four-parameter W$_{q_{1,2,3,4}}$ algebra. The origami vertex function is proposed as a higher-rank PT vertex for $\mathbb{C}^3$ and $\mathbb{C}^4$, with contour-integral constructions and pole analyses aligning with PT counts; this yields a unified algebraic and geometric framework linking stable envelopes, Higgsing, and quiver varieties to enumerative invariants. Through explicit elliptic and trigonometrically deformed blocks, R-matrices, and Higgsing identifications, the paper connects 3d/5d/4d gauge theories to Hilbert/Quot scheme vertex functions, enriching the geometric Langlands program in a double-affine context.

Abstract

We continue the study of generalized gauge theory called gauge origami, based on the quantum algebraic approach initiated in [arXiv:2310.08545]. In this article, we in particular explore the D2 brane system realized by the screened vertex operators of the corresponding W-algebra. The partition function of this system given by the corresponding conformal block is identified with the vertex function associated with quasimaps to Nakajima quiver varieties and generalizations, that plays a central role in the quantum $q$-Langlands correspondence. Based on the quantum algebraic perspective, we address three new aspects of the correspondence: (i) Direct equivalence between the electric and magnetic blocks by constructing stable envelopes from the chamber structure of the vertex operators, (ii) Double affine generalization of quantum $q$-Langlands correspondence, and (iii) Conformal block realization of the origami vertex function associated with intersection of quasimaps, that realizes the higher-rank multi-leg Pandharipande-Thomas vertices of 3-fold and 4-fold.

Gauge origami and quiver W-algebras II: Vertex function and beyond quantum $q$-Langlands correspondence

TL;DR

This work advances gauge origami by coupling D2-brane systems to screening currents of W-algebras, identifying the vertex function for quasimaps to Nakajima varieties with the gauge origami partition function. It establishes that the magnetic W-algebra block directly solves the -KZ equation, enabling a direct electric–magnetic equivalence via chamber/radial ordering, and extends the quantum -Langlands correspondence to double affine settings using the four-parameter W algebra. The origami vertex function is proposed as a higher-rank PT vertex for and , with contour-integral constructions and pole analyses aligning with PT counts; this yields a unified algebraic and geometric framework linking stable envelopes, Higgsing, and quiver varieties to enumerative invariants. Through explicit elliptic and trigonometrically deformed blocks, R-matrices, and Higgsing identifications, the paper connects 3d/5d/4d gauge theories to Hilbert/Quot scheme vertex functions, enriching the geometric Langlands program in a double-affine context.

Abstract

We continue the study of generalized gauge theory called gauge origami, based on the quantum algebraic approach initiated in [arXiv:2310.08545]. In this article, we in particular explore the D2 brane system realized by the screened vertex operators of the corresponding W-algebra. The partition function of this system given by the corresponding conformal block is identified with the vertex function associated with quasimaps to Nakajima quiver varieties and generalizations, that plays a central role in the quantum -Langlands correspondence. Based on the quantum algebraic perspective, we address three new aspects of the correspondence: (i) Direct equivalence between the electric and magnetic blocks by constructing stable envelopes from the chamber structure of the vertex operators, (ii) Double affine generalization of quantum -Langlands correspondence, and (iii) Conformal block realization of the origami vertex function associated with intersection of quasimaps, that realizes the higher-rank multi-leg Pandharipande-Thomas vertices of 3-fold and 4-fold.
Paper Structure (30 sections, 26 theorems, 112 equations, 2 figures)

This paper contains 30 sections, 26 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Direct equivalence between electric and magnetic blocks
  3. Double affine quantum $q$-Langlands correspondence
  4. Origami vertex function, PT vertex, and W-algebra
  5. W$_{q_{1,2}}(A_1)$ conformal block
  6. Single screening current
  7. Pole structure
  8. Elliptic $R$-matrix
  9. Mirror symmetry
  10. Vertex function
  11. 5d gauge theory partition function
  12. Elliptic conformal block
  13. Elliptic vertex function
  14. Elliptic $q$-KZ equation
  15. Trigonometric conformal block
  16. ...and 15 more sections

Key Result

Theorem 1.1

The vertex function is the deformed conformal block of the $q$-deformed W-algebra.

Figures (2)

  • Figure 1: Quivers for $\mathrm{W}_{q_{1,2}}(A_1)$ block. The red edge describes the interaction of two nodes (the so-called 0d theory contribution).
  • Figure 2: Quivers for $\mathrm{W}_{q_{1,2,3,4}}(\widehat{A}_0)$ block. (F) and (M) stand for the Fock and the MacMahon formula, respectively. The solid (dashed) lines indicate the chiral (Fermi) multiplets, and the red ones are multiple 0d theory contributions.

Theorems & Definitions (51)

  • Theorem 1.1: Aganagic:2017smx
  • Theorem 1.2: Theorem \ref{['thm:e_qKZ']}
  • Theorem 1.3: Theorems \ref{['thm:vertex_Fock']}, \ref{['thm:vertex_MacMahon']}
  • Conjecture 1.4: Conjecture \ref{['conj:origami_vertex']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 41 more