Quantum K -theory, I: Foundations

Abstract

This work is devoted to the study of the foundations of quantum $K$-theory, a $K$-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary $K$-ring $K(X)$ of a smooth projective variety $X$, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.