Quantum $K$-invariants via Quot schemes I

TL;DR

This work develops a Quot-scheme–driven framework for quantum $K$-theory of Grassmannians, showing that virtual Euler characteristics over Quot schemes for curves organize into a $1+1$-dimensional TQFT valued in $\mathbb{Z}[[q]]$ and yield a complete reconstruction of the small quantum $K$-ring of $\mathrm{Gr}(r,N)$. By introducing the quantum reduction map $\kappa$, the authors connect representation-theoretic data to the quantum $K$-ring, obtaining a Whitney-type presentation that recovers known relations and clarifies the finiteness and $S_3$-symmetry of structure constants. Wall-crossing arguments (epsilon-stability and light-to-heavy transitions) relate Quot invariants to quasimap invariants, enabling explicit computations of quantum $K$-invariants via degeneration formulas and torus localization, including a rigorous degeneration gluing framework for all genera. The results provide a versatile, geometry-driven route to explicit formulas for quantum $K$-invariants and have potential extensions to broader GIT quotients and quasimap theories in $K$-theory. Hence Quot schemes offer a powerful, unifying toolkit for computing and understanding quantum $K$-theoretic structures in Grassmannians and beyond.

Abstract

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 $\mathbb{Z}[[q]]$. 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.

Theorems & Definitions (88)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Remark 1.6
• Definition 1.7
• Definition 1.8
• Theorem 1.9
• Corollary 1.10