Gaiotto-Witten superpotential and Whittaker D-modules on monopoles

Coulomb branches of quiver gauge theories with symmetrizers

“…Recall that there is an étale rational coordinate system (y i,r , w i,r ) i∈I,1≤r≤v i on the open zastava space Zα for finite type [FKMM99,BDF16]. We claim that it is compatible with the above folding, namely the coordinate system for B 2 , G 2 is the restriction of the coordinate system for A 3 , D 4 to the Z/m-fixed point (m = 2, 3) respectively.…”

“…d i in[BDF16] is the square length of the simple coroot for i, while it is of the simple root here. Therefore i (resp.…”

Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian

“…Recall the factorization map π α : Z α → A α and its section s α : A α → Z α , see e.g. [BDF16]: the restriction of π α to Zα ⊂ Z α takes a based map φ : P 1 → B to the pullback φ * S of the Q 0 -colored Schubert divisor (the boundary of the open B-orbit in B). Recall that A α = A |α| /S α where A |α| = i∈Q 0 A a i , and S α is the product of the symmetric groups i∈Q 0 S a i .…”

“…Recall that π α : Z α → A α is flat (since A α is smooth, Z α has rational singularities and hence it is Cohen-Macaulay [BF14b, Proposition 5.2], and all the fibers of π α have the same dimension |α| [BFGM02, Propositions 2.6, 6.4, and the line right after 6.4]). Recall the regular functions (w i,r , y i,r ) i∈Q 0 , 1≤r≤a i on Z α , see [BDF16,2.2]. Note that π α (w i,r , y i,r ) = (w i,r ).…”

“…The factorization property is well-known in the context of zastava space Z α , which is a natural partial compactification of Zα . (See [BDF16] and references therein. )…”