Gromov-Witten Invariants in Complex and Morava-Local K-Theories

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 $\mathbb{K}$ which is K_p(n)-local for some Morava K-theory K_p(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 $\mathbb{K}$ -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 of 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 K_p(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.