Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact Lie group G c and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an SU(2)-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with C × -action when M is of a form N ⊕ N * , as the second step of the proposal given in [Nak16]. Temkin for the useful discussions. We also thank J. Hilburn and B. Webster for pointing out mistakes in an earlier version of the proof of Theorem 4.1, and the formula in (4.7) respectively.

This is a companion paper of [Part II]. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based rational maps from P 1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.

Let g be a simple complex Lie algebra, G -the corresponding simply connected group; let also g aff be the corresponding untwisted affine Lie algebra. For a parabolic subgroup P ⊂ G we introduce a generating function Z aff G,P which roughly speaking counts framed G-bundles on P 2 endowed with a P -structure on the horizontal line (the formal definition uses the corresponding Uhlenbeck type compactifications studied in [3]). In the case P = G the function Z aff G,P coincides with Nekrasov's partition function introduced in [23] and studied thoroughly in [24] and [22] for G = SL(n). In the "opposite case" when P is a Borel subgroup of G we show that Z aff G,P is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebraǧ aff -the Langlands dual Lie algebra of g aff . This clarifies somewhat the connection between certain asymptotic of Z aff G,P (studied in loc. cit. for P = G) and the classical affine Toda system. We also explain why the above result gives rise to a calculation of (not yet rigorously defined) equivariant quantum cohomology ring of the affine flag manifold associated with G. In particular, we reprove the results of [13] and [18] about quantum cohomology of ordinary flag manifolds using methods which are totally different from loc. cit.We shall show in a subsequent publication how this allows one to connect certain asymptotic of the function Z aff G,P with the Seiberg-Witten prepotential (cf.[2], thus proving the main conjecture of [23] for an arbitrary gauge group G (for G = SL(n) it has been proved in [24] and [22] by other methods.