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


Let M be a quaternionic representation of a compact Lie group G. Physicists study the Coulomb branch of the 3-dimensional gauge theory associated with (G,M), which is a hyper-Kaehler manifold, but have no rigorous mathematical definition. When M is of a form N + N∗, we introduce a variant of the affine Grassmannian Steinberg variety, define convolution product on its equivariant Borel-Moore homology group, and show that it is commutative. We propose that it gives a mathematical definition of the coordinate ring of the Coulomb branch. (Joint work by Braverman, Finkelberg and Nakajima)

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