I will begin by reviewing some basic facts about vector bundles on the Grassmannian $Gr(r,n)$ and state the Borel–Weil–Bott theorem. The space of maps from a smooth projective curve $C$ to $Gr(r,n)$ is compactified by the Quot scheme. In this talk, we define $K$-theoretic invariants involving Euler characteristics of vector bundles over these Quot schemes. We show that these invariants naturally fit into a topological quantum field theory. Additionally, we demonstrate that the genus-zero invariants recover the quantum $K$-ring of $Gr(r,n)$, and provide a novel approach for deriving explicit formulas.