The Principal Investigators will further develop and use their recent results in orbifold cohomology, orbifold K-theory and the orbifold Chern character. They will expand their work on the obstruction bundle to include orbifold Gromov-Witten theory which will have implications not only for calculations, but also important theoretical significance. Secondly, they will generalize their constructions of stringy and orbifold K-theory and cohomology from the case of a finite group to the case of a non-Abelian, infinite group with possibly infinite stabilizers. Third, they will study invariants arising from their stringy and orbifold Chern characters. In particular, they will investigate the Chern classes and similar structures in the stringy and orbifold settings. Finally, they will examine the relation of these invariants to their counterparts on various hyper-Kaehler and crepant resolutions of the underlying singular spaces.
Invariants of spaces are fundamental tools in topology and geometry. The development of new invariants is of great importance to these fields, as it provides new tools to identify and describe essential properties of geometric and topological spaces. Invariants also appear in theoretical physics as observables in topological quantum field theories, for example. In many physical and mathematical settings, the spaces of greatest importance also have symmetries, and it is important to understand how those symmetries interact with the geometric and topological properties of the space. Recently, the PIs have developed new invariants of spaces with symmetries (stringy K-theory) and have also made important progress in describing connections between their new invariants and previously known invariants, such as orbifold cohomology. They have also used their newly developed tools to refine and simplify many aspects of those previously known invariants. With the support of this grant, the PIs will use their theory of stringy K- theory as well as their improvements on orbifold cohomology to study spaces with symmetries. They will also further develop these tools to extend their applicability to more types of spaces, including spaces with continuous symmetries, which are common throughout mathematics and physics. They will also develop new invariants of such spaces, including enhancements of well-known classical invariants such as Chern classes, but accounting for symmetries. Such invariants are suggested by topological string theory and will provide powerful tools for understanding these spaces.
