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 𝕂 which is Kp(n)-local for some Morava K-theory Kp(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 Kp(n)-local theories, we introduce a formalism of `counting theories' for enumerative invariants on a category of global Kuranishi charts.