Given a closed symplectic manifold X, we construct Gromov-Witten-type
invariants valued both in (complex) K-theory and in any complex-oriented
cohomology theory K which is Kpβ(n)-local for some Morava
K-theory Kpβ(n). We show that these invariants satisfy a version of the
Kontsevich-Manin axioms, extending Givental and Lee's work for the quantum
K-theory of complex projective algebraic varieties. In particular, we prove a
Gromov-Witten type splitting axiom, and hence define quantum K-theory and
quantum K-theory as commutative deformations of the corresponding
(generalised) cohomology rings of X; the definition of the quantum product
involves the formal group of the underlying cohomology theory. The key
geometric input to these results is a construction of global Kuranishi charts
for moduli spaces of stable maps of arbitrary genus to X. On the algebraic
side, in order to establish a common framework covering both ordinary
K-theory and Kpβ(n)-local theories, we introduce a formalism of `counting
theories' for enumerative invariants on a category of global Kuranishi charts.Comment: 63 pages, 2 figure