The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory

TL;DR

This paper characterizes genus-0 quantum K-theoretic Gromov-Witten invariants of complex manifolds using cohomological invariants, applying advanced Riemann-Roch formulas and symplectic geometry techniques, and demonstrates applications to tangent space structures and complete intersections.

Contribution

It introduces a novel framework combining adelic characterization and overruled Lagrangian cones to analyze genus-0 quantum K-theory invariants, extending previous methods with new formulas and structures.

Findings

Complete characterization of genus-0 K-theoretic Gromov-Witten invariants.

Tangent spaces of quantum K-theory cones form modules over finite-difference operator algebra.

Explicit computation for complete intersections in projective space.

Abstract

We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by applying (a virtual version of) the Kawasaki-Hirzebruch-Riemann-Roch formula for expressing holomorphic Euler characteristics of orbibundles on moduli spaces of genus-0 stable maps, analyzing the sophisticated combinatorial structure of inertia stacks of such moduli spaces, and employing various quantum Riemann--Roch formulas from "fake" (i.e. orbifold-ignorant) quantum K-theory of manifold and orbifolds (formulas, either previously known from works of Coates-Givental, Tseng, and Coates-Corti-Iritani-Tseng, or newly developed for this purpose in separate papers by Tonita). The ultimate formulation combines properties of overruled Lagrangian cones in symplectic loop spaces (the language, that has become traditional in description of generating functions of genus-0 Gromov-Witten theory) with a novel framework of "adelic characterization" of such cones. As an application, we prove that tangent spaces of the overruled Lagrangian cones of quantum K-theory carry a natural structure of modules over the algebra of finite-difference operators in Novikov's variables. As another application, we compute one of such tangent spaces for each of the complete intersections given by equations of degrees $l_1,...,l_k$ in a complex projective space of dimension $\geq l_1^2+...+l_k^2-1$.