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.

💡 Research Summary

The paper develops a new approach to quantum K‑theoretic invariants of Grassmannians by exploiting the geometry of Quot schemes on curves. After recalling the definition of quantum K‑theory as introduced by Givental and Lee—where invariants are Euler characteristics of K‑theory classes on Kontsevich’s moduli space of stable maps—the authors replace the Kontsevich space with the Grothendieck Quot scheme Quot₍d₎(P¹,N,r). This replacement yields a more explicit and computable compactification of the space of degree‑d maps from P¹ to Gr(r,N).

A key technical device is the definition of K‑theory classes e O_λ on the Quot scheme, obtained by translating the Schur functor classes S_ν(Sₚ) and the Hubert cycle classes O_{X_ν} via the transition matrix between the Schur and Hubert bases. The virtual Euler characteristic of a product of such classes on Quot₍d₎(P¹,N,r) is shown (Theorem 1.2) to coincide exactly with the structure constants N^ν_{λ,μ} of the small quantum K‑ring QK(Gr(r,N)). Consequently, the S₃‑symmetry of these constants and the degree bound deg N^ν_{λ,μ} ≤ min{r,N−r} become immediate consequences of the vanishing of certain Euler characteristics on high‑degree Quot schemes (Theorem 1.12).

The authors then construct a (1+1)‑dimensional topological quantum field theory (TQFT) valued in the power‑series ring ℂ