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Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, I

Hiraku Nakajima

TL;DR

This work proposes a rigorous algebraic model for the Coulomb branch $\mathcal{M}_C$ of 3d $\mathcal{N}=4$ gauge theories by defining its coordinate ring via vanishing-cycle cohomology of a gauged sigma-model moduli space on $S^2$, and verifies compatibility with the monopole formula. It situates the construction within a broader framework linking topological twists, 2+1D TQFT intuition, and 3d mirror symmetry, while outlining how line bundles and flavor symmetries act on $\mathcal{M}_C$ and how the Higgs branch data inform the Coulomb side. The paper develops a detailed scaffold—through hyper-Kähler quotient theory, generalized Seiberg–Witten reductions, and motivic Donaldson–Thomas-type invariants—that supports a mathematically precise definition of $\mathcal{M}_C$ and sets the stage for the noncommutative quantization and convolution structures to be treated in the sequel (BFN). The results provide concrete checks against the monopole formula in key abelian and quiver-type examples and establish a route to analyze complex dualities in non-Lagrangian settings. Overall, the work advances a rigorous, geometry-grounded approach to Coulomb branches with broad implications for 3d dualities and mathematical invariants in gauge theory.

Abstract

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on $\mathbb R^4$, $SU(2)$-monopole moduli spaces on $\mathbb R^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $σ$-model on the $2$-sphere associated with $(G,\mathbf M)$. In this first part, we check that the cohomology group has the correct graded dimensions expected from the monopole formula proposed by Cremonesi, Hanany and Zaffaroni arXiv:1309.2657. A ring structure (on the cohomology of a modified moduli space) will be introduced in the sequel of this paper.

Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, I

TL;DR

This work proposes a rigorous algebraic model for the Coulomb branch of 3d gauge theories by defining its coordinate ring via vanishing-cycle cohomology of a gauged sigma-model moduli space on , and verifies compatibility with the monopole formula. It situates the construction within a broader framework linking topological twists, 2+1D TQFT intuition, and 3d mirror symmetry, while outlining how line bundles and flavor symmetries act on and how the Higgs branch data inform the Coulomb side. The paper develops a detailed scaffold—through hyper-Kähler quotient theory, generalized Seiberg–Witten reductions, and motivic Donaldson–Thomas-type invariants—that supports a mathematically precise definition of and sets the stage for the noncommutative quantization and convolution structures to be treated in the sequel (BFN). The results provide concrete checks against the monopole formula in key abelian and quiver-type examples and establish a route to analyze complex dualities in non-Lagrangian settings. Overall, the work advances a rigorous, geometry-grounded approach to Coulomb branches with broad implications for 3d dualities and mathematical invariants in gauge theory.

Abstract

Consider the -dimensional supersymmetric gauge theory associated with a compact Lie group and its quaternionic representation . Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on , -monopole moduli spaces on , etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged -model on the -sphere associated with . In this first part, we check that the cohomology group has the correct graded dimensions expected from the monopole formula proposed by Cremonesi, Hanany and Zaffaroni arXiv:1309.2657. A ring structure (on the cohomology of a modified moduli space) will be introduced in the sequel of this paper.

Paper Structure

This paper contains 59 sections, 3 theorems, 178 equations, 7 figures.

Table of Contents

  1. Introduction
  2. 3-dimensional N=4 SUSY gauge theories
  3. Topological twist
  4. Coulomb branch
  5. A relation to topological invariants
  6. 2+1-dimensional TQFT
  7. Monopole formula
  8. Higgs branch, 3d mirror symmetry and symplectic duality
  9. Hikita conjecture
  10. A proposal of a definition of MC
  11. Examples of hyper-Kähler quotients
  12. Hyper-Kähler quotients of linear spaces
  13. Cotangent type
  14. Complete intersection
  15. Quiver gauge theory
  16. ...and 44 more sections

Key Result

Theorem 6.6

A solution of the generalized Seiberg-Witten equation eq:24 on a compact Kähler surface $X$ consists of a holomorphic $G_{\mathbb C}$-bundle $\mathscr P$, a holomorphic section $\overline{\Phi}$ of $K_X^{1/2}\otimes (\mathscr P\times_{G_{\mathbb C}}\mathbf M)$ satisfying

Figures (7)

  • Figure 1: An example of a mirror pair
  • Figure 2: McKay quiver for $\operatorname{\rm SU}(2)$
  • Figure 3: McKay quiver for $S^1$
  • Figure 4: $\mathcal{M}_C$ : $E_8$$k$-instantons on ${\mathbb R}^4$.
  • Figure 5: Mirror of a $3d$ Sicilian theory of type $A_\ell$
  • ...and 2 more figures

Theorems & Definitions (12)

  • Remark 1.3
  • Remark 4.2
  • Remark 4.5
  • Remark 6.2
  • Theorem 6.6
  • Remark 6.7
  • Theorem 6.9
  • Remark 7.4
  • Remark 7.6
  • Remark 7.10
  • ...and 2 more