Lifting $I$-functions from the Grassmannians to their cotangent bundles
Kamyar Amini
TL;DR
The paper builds a purely algebraic balancing framework that lifts the $I$-function of the Grassmannian $G(r,n)$ to the quasimap vertex function of its cotangent bundle $T^*G(r,n)$. By pairing a $\lambda_y$-balanced $K$-theoretic class with a corresponding $q$-difference operator, it shows that the vertex function can be recovered from the $I$-function via the balancing operator, after the substitution $y=-q^{-1}\hbar$. This yields a direct link between the quantum $K$-theory of a base variety and that of its cotangent bundle, compatible with abelianization and leading to Bethe-Ansatz equations in the general Grassmannian case. The results are illustrated by a detailed P^1 example, clarifying the localization and balancing procedures at fixed points. Overall, the work provides a new, concrete mechanism to connect enumerative invariants of a space with those of its cotangent bundle through algebraic twisting and abelian/nonabelian correspondence, with potential applications to integrable systems via Bethe equations.
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.