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

Consider the $3$-dimensional N=4 supersymmetric gauge theory associated with a compact Lie group G and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-K\"ahler manifold, such as instanton moduli spaces on R⁴, $SU(2)$-monopole moduli spaces on R³, 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, 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.