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

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 $\sigma$-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.

54 pages, Videos of talks on this work are available at http://media.scgp.stonybrook.edu/video/video.php?f=20141028_2_qtp.mp4, https://www.youtube.com/watch?v=JyxPAV5qRWs and https://www.youtube.com/watch?v=HSA9tHM3rVQ; v2. Remark 4.5 and Added in Proof are added

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• Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface
• Seiberg-Witten monopoles with multiple spinors on a surface times a circle
• Some Open Questions in Quiver Gauge Theory