UC Berkeley

Given a projective smooth complex variety X, one way to associate to it numerical

invariants is by taking holomorphic Euler characteristics of interesting vector bundles on

the moduli spaces of genus 0, degree d stable maps with n marked points to X. We

call these numbers genuine quantum K-theoretic invariants of X. Their generating series is

called the genus 0 K-theoretic descendant potential of X and can be viewed as a function

on a suitable infinite dimensional vector space K+. Its graph is a uniruled Lagrangian cone

in the cotangent bundle of K+.

We give a complete characterisation of points on the cone, proving a Hirzebruch Riemann

Roch type theorem for the genuine K-theory of X. In particular, our result can be used

to recursively express all genus 0 K-theoretic invariants of X in terms of cohomological

ones (usually known as Gromov-Witten invariants). The main technical tool we use is the

Kawasaki Riemann Roch theorem of [Ka], which reduces the computation of holomorphic

Euler characteristic of a bundle on an orbifold to the computation of a cohomological integral

on the inertia orbifold.

In the process, we need to study more general cohomological Gromov-Witten invariants

of an orbifold X, which we call twisted invariants. These are obtained by capping the virtual

fundamental classes of the moduli spaces with certain multiplicative characteristic

classes. We twist the Gromov-Witten potential by three types of twisting classes and we

allow several twistings of each type. We use a Mumford's Grothendieck-Riemann-Roch

computation on the universal curve to give closed formulae which show the eect of each

type of twist on their generating series (the twisted potential). This generalizes earlier results

of [CG] and [TS].