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

Summary

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 manifolds and orbifolds (formulas, either previously known from works of Coates– Givental, Tseng, and Coates–Corti–Iritani–Tseng, or newly developed for this purpose 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 l₁; ....; l_k in a complex projective space of dimension.

0. Motivation

K-theoretic Gromov–Witten invariants of a compact complex algebraic manifold X are defined as holomorphic Euler characteristics of various interesting vector bundles over moduli spaces of stable maps of compact complex curves to X.