  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-K\"ahler 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.
