Gaiotto–Witten superpotential and Whittaker D-modules on monopoles

Abstract

Abstract. Let G be an almost simple simply connected group over C. For a positive element α of the coroot lattice of G letα denote the space of maps from P 1 to the flag variety B of G sending ∞ ∈ P 1 to a fixed point in B of degree α. This space is known to be isomorphic to the space of framed G-monopoles on R 3 with maximal symmetry breaking at infinity of charge α.In [6] a system of (étale, rational) coordinates onα is introduced. In this note we compute various known structures on• Z α in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential studied in [9] and relate it to the theory of Whittaker D-modules discussed in [8].

Comultiplication for shifted Yangians and quantum open Toda lattice

“…The isomorphism β of Section 2.5 reduced modulo gives rise to the same named isomorphism β : [BFM,Theorem 2.12] and [T,Theorem 6.3]). We also have an isomorphism Ξ : C[ [BDF,1.4(3)]. We denote the composition of the above isomorphisms at the level of spectra by Υ = ι • Spec Ξ • Spec β : Z n ∼ −→ • Z n .…”

Why is Landau-Ginzburg link cohomology equivalent to Khovanov homology?

“…Afterwards in Section 4 we construct su n Landau-Ginzburg link cohomology n LGCoh(L) starting from the monopole moduli space arising in a (2,0) six-dimensional theory with a gauge algebra su n [10]. We argue that n LGCoh(L) is an invariant of link L.…”

“…The superpotential for what one could call a naive monopole model analog, or quasi-classical monopole model, when monopole centers are well-separated, see (3.3), is given in [10]. Here we use a low energy effective description with a su n Yang-Yang superpotential that in terms of integrals (2.3) gives a free field formulation for SU (n) WZW-model with N punctures correspondingly [34]:…”

Twisted zastava and q-Whittaker functions

“…In effect, let denote the above hypersurface in , and its projection to . Then the open subvarieties and are isomorphic by Lemma below and, for example, [, 5.6]. This isomorphism extends to due to normality of (Proposition below).…”

“…Here means this arrow belongs to the orientation chosen above and ; and means this arrow belongs to the orientation and or . This section extends regularly through the generic points of the irreducible components of the quasidiagonal divisor due to examples in Section 2.5, Section 2.6 (for a divisor ); due to [, 5.6] (for a divisor ); due to [, 5.5] (for a divisor ). Moreover, this section is ‐invariant, hence it descends to a rational section of that is regular nonvanishing at the generic points of the irreducible components of the quasidiagonal divisor (again due to [, 5.5]).…”