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Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches

Yaolong Shen, Changjian Su, Rui Xiong

TL;DR

The paper establishes a concrete link between shifted twisted Yangians and the quantized Coulomb branch of a 3d ${\mathcal N}=4$ quiver gauge theory with involution, via a GKLO-type representation and a central algebra homomorphism. It proves that the shifted twisted Yangian $\mathbf{Y}^{\tau}_{\mu}(\mathfrak{g}_Q)$ embeds into the Coulomb-branch algebra ${\mathcal A}_{\hbar}$ by sending Cartan generators to the Gelfand–Tsetlin subalgebra and $b$-generators to minuscule monopole operators, with the construction compatible with 3D mirror symmetry and recovering the BFN framework in the symmetric-pair limit. The work relies on equivariant localization, difference-operator realizations, and detailed verifications of ($h$, $b$) commutators and Serre relations in a twisted, shifted setting, producing a robust geometric realization of twisted shifted Yangians. These results provide a foundation for further connections between $(\imath)$quantum groups, GKLO-type representations, and Coulomb branch geometry, and they pave the way to K-theoretic extensions and broader types via 3D mirror symmetry.

Abstract

To a quiver with involution, we study the Coulomb branch of the 3d $\mathcal{N} = 4$ involution-fixed part of the quiver gauge theory. We show that there is an algebra homomorphism from the corresponding shifted twisted Yangian to the quantized Coulomb branch algebra. This gives a new instance of 3D mirror symmetries.

Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches

TL;DR

The paper establishes a concrete link between shifted twisted Yangians and the quantized Coulomb branch of a 3d quiver gauge theory with involution, via a GKLO-type representation and a central algebra homomorphism. It proves that the shifted twisted Yangian embeds into the Coulomb-branch algebra by sending Cartan generators to the Gelfand–Tsetlin subalgebra and -generators to minuscule monopole operators, with the construction compatible with 3D mirror symmetry and recovering the BFN framework in the symmetric-pair limit. The work relies on equivariant localization, difference-operator realizations, and detailed verifications of (, ) commutators and Serre relations in a twisted, shifted setting, producing a robust geometric realization of twisted shifted Yangians. These results provide a foundation for further connections between quantum groups, GKLO-type representations, and Coulomb branch geometry, and they pave the way to K-theoretic extensions and broader types via 3D mirror symmetry.

Abstract

To a quiver with involution, we study the Coulomb branch of the 3d involution-fixed part of the quiver gauge theory. We show that there is an algebra homomorphism from the corresponding shifted twisted Yangian to the quantized Coulomb branch algebra. This gives a new instance of 3D mirror symmetries.
Paper Structure (22 sections, 11 theorems, 150 equations)

This paper contains 22 sections, 11 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Coulomb branches of cotangent type
  3. BFN Coulomb Branch
  4. Quiver with involution
  5. Monopole operators
  6. Shifted twisted Yangians
  7. Main Results
  8. Relations (I)
  9. Relation \ref{['eq:hbcomm']}: commutativity of $h_{i,r}$ and $b_{j,s}$
  10. The case $j\in \{i,\tau i\}$
  11. The case $j\notin \{i,\tau i\}$
  12. Some identites
  13. Relation \ref{['eq:bcomm']}: commutativity of $b_{i,r}$'s
  14. The case $j\neq \tau i$, $i=j$
  15. The case $j\neq \tau i$, $c_{ij}=-1$
  16. ...and 7 more sections

Key Result

Theorem 1

There exists an algebra homomorphism from shifted twisted Yangian $\mathbf{Y}^\imath_\mu$ to the quantized Coulomb branch algebra $\mathcal{A}_\hbar$ of $(G,{\mathbf{N}})$.

Theorems & Definitions (29)

  • Theorem : \ref{['thm:main']}
  • Example 2.1: Diagonal Type
  • Example 2.2: Type AIII
  • Example 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 19 more