[Paper Review] $K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

Gus Schrader, Alexander Shapiro|arXiv (Cornell University)|Oct 7, 2019

Algebraic structures and combinatorial models4 citations

TL;DR

This paper constructs a cluster algebra quiver $Q_{\Gamma}$ associated to a 3d $\mathcal{N}=4$ quiver gauge theory and proves that the $K$-theoretic Coulomb branch subalgebra $\mathscr{A}'_q$, generated by equivariant $K$-theory of a point and dressed minuscule monopole operators, embeds injectively into the quantized algebra of regular functions on the corresponding cluster variety. The result provides strong evidence for Gaiotto's conjecture that $K$-theoretic Coulomb branches carry a quantum cluster structure.

ABSTRACT

Let $\\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $\\mathcal{N}=4$ quiver gauge theory with quiver $\\Gamma$, and $\\mathscr{A}'_q \\subseteq \\mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{\\varpi_{i,1},f}$ and $M_{\\varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $\\mathcal{Q}_\\Gamma$ and provide an embedding of the subalgebra $\\mathscr{A}'_q$ into the quantized algebra of regular functions on the corresponding cluster variety.

Motivation & Objective

• To investigate the quantum cluster structure of $K$-theoretic Coulomb branches in 3d $\mathcal{N}=4$ quiver gauge theories.
• To test Gaiotto's conjecture that these Coulomb branches carry a quantum cluster variety structure.
• To construct a quiver $Q_\Gamma$ from the gauge quiver $\Gamma$ such that $\mathscr{A}'_q$ embeds into the quantized cluster algebra of the associated cluster variety.
• To show that generators of $\mathscr{A}'_q$—equivariant $K$-theory classes and dressed monopole operators—map to Laurent polynomials in any cluster, ensuring global regularity.

Proposed method

• Construct a quiver $Q_\Gamma$ by gluing cluster quivers from relativistic open Coxeter-Toda systems for each node of $\Gamma$, using the Toda lattice's relation to $K$-theory of the affine Grassmannian.
• Use the $b$-Whittaker transform from prior work [SS18] to diagonalize quantum Toda Hamiltonians and realize eigenfunctions as symmetric functions in spectral parameters.
• Embed the $K$-theoretic Coulomb branch $\mathscr{A}_q$ into an algebra of $q$-difference operators acting on variables $w_{i,n}$, where $i$ indexes nodes and $n$ runs over $\dim V_i$.
• Define cluster coordinates via $Y(\lambda_0; \lambda_1, \dots, \lambda_M)$ and $\mathscr{E}_k(\lambda, J)$ to express monopole operators as Laurent polynomials in cluster variables.
• Establish an injective algebra homomorphism from $\mathscr{A}'_q$ to the quantum cluster chart of the cluster variety labeled by $Q_\Gamma$, showing that all generators land in the global regular functions.
• Verify the embedding by explicit computation in examples: $\Gamma = (4), (3,1), (2,2), (2,1,1)$, mapping monopole operators to $q$-deformed $Y$-functions and checking Laurentness across clusters.

Experimental results

Research questions

• RQ1 Does the $K$-theoretic Coulomb branch of a 3d $\mathcal{N}=4$ quiver gauge theory admit a quantum cluster structure?
• RQ2 Can the subalgebra $\mathscr{A}'_q$ generated by equivariant $K$-theory of a point and dressed minuscule monopole operators be embedded into a quantized cluster algebra?
• RQ3 Is the image of $\mathscr{A}'_q$ under the embedding contained in the algebra of global regular functions on the cluster variety?
• RQ4 How does the cluster structure of the relativistic Toda system relate to the $K$-theoretic Coulomb branch of quiver gauge theories?
• RQ5 What is the precise quiver $Q_\Gamma$ that realizes the cluster structure for a given framed quiver $\Gamma$?

Key findings

• The subalgebra $\mathscr{A}'_q$ of the $K$-theoretic Coulomb branch embeds injectively into the quantized algebra of regular functions on a cluster variety associated to $Q_\Gamma$.
• Generators of $\mathscr{A}'_q$, including equivariant $K$-theory classes and dressed monopole operators $M_{\varpi_{i,1},f}$, $M_{\varpi^*_{i,1},f}$, map to Laurent polynomials in any cluster, confirming their global regularity.
• The quiver $Q_\Gamma$ is constructed by gluing Toda system quivers for each node of $\Gamma$, reflecting the Toda lattice's role in $K$-theory of the affine Grassmannian.
• Explicit computations for partitions $(4)$, $(3,1)$, $(2,2)$, and $(2,1,1)$ confirm that monopole operators map to $q$-deformed $Y$-functions with Laurent structure across clusters.
• The $b$-Whittaker transform from [SS18] provides a key technical tool, enabling the diagonalization of quantum Toda Hamiltonians and realizing eigenfunctions as symmetric functions in spectral parameters.
• The result supports Gaiotto's conjecture that $K$-theoretic Coulomb branches of 3d $\mathcal{N}=4$ theories are quantum cluster varieties, offering a partial but concrete realization of the structure.