Quantum $K$-invariants via Quot schemes II

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 our previous work 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.