Non-archimedean quantum K-invariants

TL;DR

This paper develops a new framework for quantum K-invariants in non-archimedean geometry using derived methods, enabling more flexible enumerative invariants and providing foundational tools for non-archimedean enumerative geometry and mirror symmetry.

Contribution

It introduces a derived approach to quantum K-invariants in non-archimedean geometry, establishing geometric relations and foundational tools for the field.

Findings

Derived geometric relations between stacks of stable maps

Imposition of incidence conditions with multiplicities

Development of foundational aspects of derived non-archimedean geometry

Abstract

We construct quantum K-invariants in non-archimedean analytic geometry. Contrary to the classical approach in algebraic geometry via perfect obstruction theory, we build on our previous works on the foundations of derived non-archimedean geometry, the representability theorem and Gromov compactness. We obtain a list of natural geometric relations between the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of quantum K-invariants. The derived approach produces highly intuitive statements and functorial proofs. The flexibility of our derived approach to quantum K-invariants allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities. This leads to a new set of enumerative invariants. For the proofs, we further develop the foundations of derived non-archimedean geometry in this paper: we study derived lci morphisms, relative analytification, and deformation to the normal bundle. Our motivations come from non-archimedean enumerative geometry and mirror symmetry.