Lifting $I$-functions from the Grassmannians to their cotangent bundles

TL;DR

This paper introduces a geometric operator called balancing that lifts $I$-functions from Grassmannians to their cotangent bundles, connecting enumerative geometry, difference operators, and Bethe-Ansatz equations.

Contribution

It defines the balancing operator in $K$-theory, providing an explicit geometric method to relate $I$-functions of Grassmannians and their cotangent bundles, and links these to integrable systems.

Findings

Balancing operator lifts $I$-functions to cotangent bundles.

Compatibility of balancing with difference operators for certain functions.

Recovery of Bethe-Ansatz equations for $T^*G(r,n)$.

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 functions and the corresponding difference operators that annihilate them, including the $I$-functions of projective spaces $\mathbb{P}^n$, 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.