Quantum $K$-invariants via Quot schemes I

We study the virtual Euler characteristics of sheaves over Quot schemes of curves, establishing that these invariants fit into a topological quantum field theory (TQFT) valued in $\mathbb{Z}[[q]]$. We show that the three-pointed genus-zero $K$-theoretic stable map invariants of the Grassmannian coincide with the genus-zero $K$-theoretic invariants defined via the Quot scheme. Utilizing Quot scheme compactifications alongside the TQFT framework, we derive presentations of the small quantum $K$-ring of the Grassmannian. Our approach offers a new method for finding explicit formulas for quantum $K$-invariants.