Quantum $K$-invariants via Quot schemes II

TL;DR

This paper develops a $K$-theoretic analogue of the Vafa--Intriligator formula to compute Euler characteristics of vector bundles over Quot schemes, aiding the study of quantum $K$-rings of Grassmannians.

Contribution

It introduces a new $K$-theoretic formula for Quot schemes, enabling computation of quantum $K$-ring structure constants and providing vanishing results.

Findings

Derived a $K$-theoretic Vafa--Intriligator formula

Obtained vanishing results for quantum $K$-theory

Simplified formula involving Schur functions in genus-zero case

Abstract

We derive a $K$-theoretic analogue of the Vafa--Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree $d$ morphisms from a fixed projective curve to the Grassmannian $\mathrm{Gr}(r,N)$. As an application, we deduce interesting vanishing results, used in Part I (arXiv:2406.12191) to study the quantum $K$-ring of $\mathrm{Gr}(r,N)$. In the genus-zero case, we prove a simplified formula involving Schur functions, consistent with the Borel-Weil-Bott theorem in the degree-zero setting. These new formulas offer a novel approach for computing the structure constants of quantum $K$-products.