Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups

Abstract

Let (x : y) be homogeneous coordinates on CP . A degree d holomorphic map CP 1 → CP is uniquely determined, up to a constant scalar factor, by N + 1 relatively prime degree d binary forms (f0(x : y) : ... : fN(x : y)). Omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps CP 1 → CP to a complex projective space of dimension (N +1)(d+1)− 1. We denote this compactification of the space of maps by CP d . This construction defines the compactification Πd = Π r i=1CP ni−1 di of the space of degree d = (d1, ..., dr) maps from CP 1 to Π. Composing degree d holomorphic maps from CP 1 to the flag manifold X with the Plucker embedding, we embed the space of such maps into Πd. The closure QMd of this space in Πd is often referred to as the Drinfeld’s compactification of the space of degree d maps from CP 1 to X and will be called the space of quasimaps (following [14]). It is a (generally speaking — singular) irreducible projective variety of complex dimension dimX + 2d1 + ...+ 2dr. The flag manifold is a homogeneous space of the group SLr+1(C) and of its maximal compact subgroup SUr+1. The action of these groups on X lifts naturally to the spaces Π, Πd, and QMd. In addition to this action the spaces Πd and the subspaces QMd carry the circle action induced by the rotation of CP 1 defined by (x : y) 7→ (x : ey). Thus the product group G = S × SUr+1 and its complex version GC = C ∗ × SLr+1(C) act on the quasimap spaces. We will see later that QMd have G-equivariant desingularizations QMd.