Algebra and Geometry Seminar

Quantum K-theory is a K-theoretic generalization of Gromov-Witten theory, and genus-zero quantum K-invariants of a target manifold X are defined as the Euler characteristics of coherent sheaves over the Kontsevich moduli space, parametrizing stable maps from (possibly nodal) rational curves to X. 

In this talk, we consider the target space to be the Grassmannian. The space of maps from the Riemann sphere to the Grassmanninan has a simpler compactification -- the Quot scheme. We define K-theoretic invariants using the Quot scheme and provide explicit bialternant-type formulas for the one-pointed K-theoretic Quot scheme invariants. We prove that these invariants determine quantum K-invariants of the Grassmannian with up to three marked points. Our approach is "stacky" in the sense that the Quot scheme parametrizes maps not to the Grassmannian but to the GIT quotient stack containing it. I will explain the advantages of our stacky approach by providing the following applications:

1. We present simple, closed formulas for the inverse of the quantized metric and the structure constants in quantum K-theory.
2. We re-prove the finiteness property of the quantum K-product using a vanishing result for the K-theoretic Quot scheme invariants.
3. We derive relations in the quantum K-ring using a reduction map.
4. We obtain a complete set of explicit formulas (for one-pointed invariants, quantized metric, structure constants, etc.) in the rank 2 case. 

The talk is based on joint work with Shubham Sinha.