| 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).
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