UC Berkeley

We prove a twisting theorem for nodal classes in permutation-equivariant quantum K-theory, and combine it with existing theorems of Givental to obtain a twisting a theorem for general characteristic classes of the virtual tangent bundle of the moduli space of stable maps. Using this result, we develop complex cobordism-valued 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. 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 y-genus.

We then introduce Euler-theoretic Gromov-Witten invariants, which are based on the ordinary topological Euler characteristics of loci of curves. These invariants have useful enumerative properties. They are integer valued, there is a reliable way to remove boundary contributions, and when the moduli spaces are smooth orbifolds, have a concrete geometric intepretation. In this case, the invariants also encompass the ordinary (i.e. integer) topological Euler characteristic of the moduli space of stable maps and its compactification.

We give a Wick-type formula computing these invariants in terms of twisted cohomological Gromov-Witten invariants, and we also show that they can be regarded as the limit as y approaches 1 of the Hirzebruch-theoretic invariants introduced earlier in this work.

Specializing to the case of a point target space, our formulas yield similar results to the computation of the Euler characteristics of the moduli spaces of n-pointed curves, given by Bini-Harer.