Lifting I-Functions from the Flag Varieties to Their Cotangent Bundles

Abstract

We relate two fundamental enumerative functions, namely the I-functions in the quantum K-ring of G(r,n) and of its cotangent bundle, by defining a K-theoretic operator on classes, called balancing. This operator lifts the I-function of G(r,n) to that of T^*G(r,n), providing an explicit geometric interpretation. We also define an operator acting on difference operators and show that, for certain K-theoretic classes and the corresponding difference operators that annihilate them—including the I-functions of projective spaces P^n—the balancing operation on difference operators and on classes is compatible. Moreover, for general G(r,n), we recover the Bethe-Ansatz equations for T^*G(r,n) via a procedure inspired by both balancing and the abelian/non-abelian correspondence.