| Résume | 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-Kähler manifold, such as instanton moduli spaces on $R^4$, $SU(2)$-monopole moduli spaces on $R^3$, etc.
In this talk, 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 $(G,M)$. |
