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

Abstract 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.