Multiplicative Quantum Cobordism Theory

Abstract

We prove a twisting theorem for nodal classes in permutation-equivariant quantum K-theory, and combine it with existing theorems of Givental [5] to obtain a twisting a theorem for general characteristic classes of the virtual tangent bundle. Using this result, we develop complex cobordismvalued Gromov-Witten invariants defined via K-theory, and relate those invariants to K-theoretic ones via the quantization of suitable symplectic transformations. This procedure is a K-theoretic analogue of the quantum cobordism theory developed by Givental and Coates in [3]. Using the universality of cobordism theory, we give an example of these results in the context of "Hirzebruch K-theory", which is the cohomology theory determined by the Hirzebruch χ−y-genus.