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

By Alexander Givental, Yuan-Pin Lee. 2001

173 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f9f6386941b59fe62afdee0676999cd256e702c

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Citations in the corpus

Quantum $K$-theory, I: Foundations (2001, methodology): “The K-theoretic push-forward χ will be defined in Section 2.1 and Section 4.1 for algebraic and topological K-theory respectively.”
Geometry of canonical bases and mirror symmetry (2013, methodology): “Therefore it provides, generalising Givental’s work [37] for G = GLm in non-equivariant setting, the integral presentation of the special solution of equivariant quantum cohomology Dh̄-module for the flag variety BL studied in [30–33,36,38,56,58,64,69,70].”
Elliptic stable envelopes (2016, background): “Givental and his collaborators studied difference equations in quantum K-theory [34,35], and their theory is very general — X can be an arbitrary nonsingular algebraic variety, or even an orbifold, as long as rational curves in X satisfy certain properness assumptions.”
On the WDVV equation in quantum K-theory. (2000, background): “This result and its conjectural generalization [ 7 ] to the flag manifolds X = G/B of complex simple Lie algebras links quantum K-theory to representation theory and quantum groups.”
Vortex Counting and Lagrangian 3-Manifolds (2010, background): “CPN−1, and the K-theoretic version of the equivariant J -function has the form [29]:”
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory (2011, background): “This structure, although manifest in some examples (see [14]) and predictable on heuristic grounds of S(1)equivariant Floer theory [8, 9], has been missing so far in the realm of K-theoretic GW-invariants.”
Weyl modules and $$q$$q-Whittaker functions (2012, background): “As usually, we have to add that [16] deals with G = SL(N ), however, the proof goes through word for word for arbitrary semisimple G.”
QUANTUM K-THEORY OF GRASSMANNIANS (2008, background): “This ring was introduced by Givental and Lee, motivated by a study of the relationship between Gromov-Witten theory and integrable systems [22, 24, 39].”