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

Hiraku Nakajima|arXiv (Cornell University)|Mar 12, 2015

Black Holes and Theoretical Physics16 citations

TL;DR

This paper proposes a mathematical definition of the Coulomb branch in 3D $σ$-model gauge theories using vanishing cycle cohomology of a moduli space on the 2-sphere. It verifies that the cohomology group's graded dimensions match the monopole formula from physics, establishing a rigorous foundation for the Coulomb branch's coordinate ring, with a ring structure to be introduced in the sequel.

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.

Motivation & Objective

• To provide a mathematically rigorous definition of the Coulomb branch for 3D $σ$-model gauge theories with $σ$-model on $S^2$.
• To establish the graded dimensions of the Coulomb branch's coordinate ring using vanishing cycle cohomology.
• To verify consistency with the monopole formula from physics, particularly for $σ$-model with $(G,\mathbf{M})$.
• To lay the groundwork for a ring structure on the cohomology, to be completed in the sequel.
• To bridge physical intuition in $σ$-model gauge theories with rigorous mathematical constructions in algebraic geometry and representation theory.

Proposed method

• Define the Coulomb branch via the vanishing cycle cohomology group of a moduli space associated with a gauged $σ$-model on $S^2$ for $(G,\mathbf{M})$.
• Use the monopole formula from Cremonesi, Hanany, and Zaffaroni as a benchmark for expected graded dimensions.
• Apply the $q$-binomial theorem and infinite product identities to compute generating functions of the cohomology dimensions.
• Employ a bijection between partitions and weight data to relate the cohomology to the monopole formula's terms.
• Use the Mathai-Quillen formalism as a heuristic guide for infinite-dimensional geometry, though not rigorously applied.
• Work with the moduli space of flat connections with Higgs fields, focusing on the zero locus of a section in an infinite-dimensional setting.

Experimental results

Research questions

• RQ1 Does the vanishing cycle cohomology of the moduli space on $S^2$ yield a coordinate ring for the Coulomb branch with the correct graded dimensions predicted by the monopole formula?
• RQ2 Can the mathematical construction of the Coulomb branch be validated against known physical invariants in 3D $σ$-model gauge theories?
• RQ3 How does the cohomology of the moduli space reflect the representation-theoretic data of the gauge group $G$ and its quaternionic representation $\mathbf{M}$?
• RQ4 Is there a consistent way to define the Coulomb branch as a hyper-Kähler manifold via algebraic geometry, even when the underlying QFT is not rigorously defined?
• RQ5 Can the structure of the Coulomb branch be extended to include a ring structure, as expected from physical and mathematical consistency?

Key findings

• The graded dimensions of the vanishing cycle cohomology group match exactly the monopole formula predictions from Cremonesi, Hanany, and Zaffaroni.
• The generating function for the cohomology dimensions is expressed as a product of $q$-Pochhammer symbols, confirming the match with the monopole formula.
• The computation uses a bijection between partitions and weight data, allowing the sum over weights to be rewritten in terms of $k_\alpha$, the number of times a weight $\alpha$ appears.
• The sum over $m \in \mathbb{Z}$ is split into $m \geq 0$ and $m < 0$, and the divergent parts are canceled by subtracting 1, leading to a convergent expression.
• The final generating function is $\prod_{\alpha=0}^\infty \frac{1}{(t^{N|\alpha|}z^{\alpha}\Lambda; t^2)_\infty} \cdot \prod_{\alpha=1}^\infty \frac{1}{(t^{N|\alpha|}z^{-\alpha}\Lambda; t^2)_\infty}$, which matches the monopole formula.
• The result confirms that the proposed cohomological construction correctly encodes the Coulomb branch's coordinate ring in terms of physical expectations.