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.

给定一个闭辛流形X ，我们构造在（复） K理论和任何面向复数的上同调理论\(\mathbb{K}\)中均有价值的 Gromov-Witten 型不变量，即K _p ( n )-local对于一些 Morava K理论K _p ( n )。我们证明这些不变量满足 Kontsevich-Manin 公理的一个版本，扩展了 Givental 和 Lee 在复射影代数簇的量子K理论方面的工作。特别是，我们证明了 Gromov-Witten 型分裂公理，因此将量子K理论和量子\(\mathbb{K}\) -理论定义为X的相应（广义）上同调环的交换变形；量子乘积的定义涉及基础上同调理论的形式群。这些结果的关键几何输入是构建任意属到X的稳定映射的模空间的全局 Kuranishi 图。在代数方面，为了建立一个涵盖普通K理论和K _p ( n ) 局部理论的通用框架，我们引入了一种用于全局 Kuranishi 图类别上的枚举不变量的“计数理论”形式。