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Quiver Yangians as Coulomb branch algebras

Tiantai Chen, Wei Li

TL;DR

The work develops a unified framework for the quantum Coulomb branch of 3D $\\mathcal{N}=4$ quiver gauge theories in the $\\Omega$-background, proposing that the algebra is captured by a truncated shifted quiver Yangian $Y_{\\hbar}(\\widehat{Q},\\widehat{W})$ built from the triple quiver of the original theory. By explicit analysis of tree-type quivers, the authors connect monopole and vector-multiplet data to quiver-Yangian generators, and realize the vortex Hilbert space as Verma modules of the truncated algebra. They further relate the Coulomb branch algebra to the CoHA of triple quivers and to the spherical shuffle algebra, arguing for a universal structure that encompasses non-simply-laced quivers and the Jordan quiver. The results provide concrete representations, examples (A-, D-, E-type theories, and high-valency $K$-star quivers), and a general conjecture governing general quivers, with potential implications for geometric representation theory and quantum algebras in gauge theories. Overall, the paper offers a robust algebraic framework that links physical BPS sectors to modern quiver Yangian theory and CoHA constructions, illuminating the quantization of Coulomb branches and their representation-theoretic content.

Abstract

For a 3D N=4 gauge theory, turning on the $Ω$-background in RxR$^2_ε$ deforms the Coulomb branch chiral ring into the quantum Coulomb branch algebra, generated by the 1/2-BPS monopoles together with the complex scalar in the vector-multiplet. We conjecture that for a 3D N=4 quiver gauge theory with unitary gauge group, the quantum Coulomb branch algebra can be formulated as the truncated shifted quiver Yangian Y$(\widehat{Q},\widehat{W})$ based on the triple quiver $\widehat{Q}$ of the original quiver Q with canonical potential $\widehat{W}$. We check this conjecture explicitly for general tree-type quivers Q by considering the action of monopoles on the 1/2-BPS vortex configurations. The Hilbert spaces of vortices approaching different vacua at spatial infinity furnish different representations of the shifted quiver Yangian, and all the charge functions have only simple poles. For quivers beyond tree-type, our conjecture is consistent with known results on special examples.

Quiver Yangians as Coulomb branch algebras

TL;DR

The work develops a unified framework for the quantum Coulomb branch of 3D quiver gauge theories in the -background, proposing that the algebra is captured by a truncated shifted quiver Yangian built from the triple quiver of the original theory. By explicit analysis of tree-type quivers, the authors connect monopole and vector-multiplet data to quiver-Yangian generators, and realize the vortex Hilbert space as Verma modules of the truncated algebra. They further relate the Coulomb branch algebra to the CoHA of triple quivers and to the spherical shuffle algebra, arguing for a universal structure that encompasses non-simply-laced quivers and the Jordan quiver. The results provide concrete representations, examples (A-, D-, E-type theories, and high-valency -star quivers), and a general conjecture governing general quivers, with potential implications for geometric representation theory and quantum algebras in gauge theories. Overall, the paper offers a robust algebraic framework that links physical BPS sectors to modern quiver Yangian theory and CoHA constructions, illuminating the quantization of Coulomb branches and their representation-theoretic content.

Abstract

For a 3D N=4 gauge theory, turning on the -background in RxR deforms the Coulomb branch chiral ring into the quantum Coulomb branch algebra, generated by the 1/2-BPS monopoles together with the complex scalar in the vector-multiplet. We conjecture that for a 3D N=4 quiver gauge theory with unitary gauge group, the quantum Coulomb branch algebra can be formulated as the truncated shifted quiver Yangian Y based on the triple quiver of the original quiver Q with canonical potential . We check this conjecture explicitly for general tree-type quivers Q by considering the action of monopoles on the 1/2-BPS vortex configurations. The Hilbert spaces of vortices approaching different vacua at spatial infinity furnish different representations of the shifted quiver Yangian, and all the charge functions have only simple poles. For quivers beyond tree-type, our conjecture is consistent with known results on special examples.

Paper Structure

This paper contains 78 sections, 317 equations, 12 figures.

Table of Contents

  1. Introduction and Summary
  2. Review
  3. 3D N=4 quiver gauge theories
  4. Quiver and framed quiver
  5. Decomposing into N=2 multiplets
  6. Masses and FI-parameters
  7. BPS states in the 3D N=4 quiver gauge theories
  8. The vacua
  9. Vortices and the vortex moduli space
  10. Monopoles
  11. Ω-deformation and N=4 quantum mechanics
  12. Ω-deformation
  13. N=4 quantum mechanics
  14. Hilbert space of the vortex states
  15. Quantum Coulomb branch algebra
  16. ...and 63 more sections

Figures (12)

  • Figure 1: A tree-type quiver, with the flavor node $\mathtt{f}$ and the root $\mathtt{r}=7$. All the arrows in $\mathrm{Q}$ point away from the root, and there is a unique path from $\mathtt{f}$ to any $a\in \mathrm{Q}_0$.
  • Figure 2: The handsaw quiver.
  • Figure 3: The quiver quantum mechanics $\textrm{QM}(\mathfrak{n},\nu)$ of D type.
  • Figure 4: Triple quiver for the D-type shifted quiver Yangian.
  • Figure 5: $D_4$ quiver as a $3$-star.
  • ...and 7 more figures