Gromov-Witten invariants in complex and Morava-local $K$-theories

TL;DR

This paper develops Gromov-Witten invariants valued in complex K-theory and Morava-local K-theories for symplectic manifolds, extending quantum K-theory with new axioms and a unified algebraic framework.

Contribution

It introduces Gromov-Witten invariants in Morava-local K-theories, extending quantum K-theory axioms and establishing a formalism for counting theories in this context.

Findings

Constructed Gromov-Witten invariants in complex and Morava-local K-theories.

Proved axioms including a splitting axiom for these invariants.

Defined quantum K-theory and quantum theory as deformations of cohomology rings.

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 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 $K_p(n)$-local theories, we introduce a formalism of `counting theories' for enumerative invariants on a category of global Kuranishi charts.