Manifolds with Group Actions and their Quotients

George Mason University, Fairfax VA

Investigators

Abstract

Abstract Award: DMS-0606869 Principal Investigator: Rebecca Goldin Orbifolds are among the simplest type of singular spaces. They arise in symplectic geometry as reductions, in algebraic geometry (where they are called Deligne-Mumford stacks) as certain moduli spaces, and in topology as quotient spaces. Algebraic invariants of these spaces, such as stringy Betti numbers, twisted Hodge numbers, and Chen-Ruan cohomology (also called orbifold cohomology) have recently gained interest because they are subtle enough to see the singularities, and in some cases describe the cohomology of a (crepant) resolution of singularities. This grant will explore new methods to compute the Chen-Ruan cohomology of an orbifold, following work done by the PI and coauthors in the case that the orbifold is a global quotient by a compact abelian Lie group. These methods should produce a new combinatorial fomula for the Chen-Ruan cohomology of a hypertoric variety (a hyperkahler reduction of quaternionic n-space by a compact torus), a formula for the K-theory of global abelian quotients, and the Chen-Ruan cohomology of non-abelian (Lie group) quotients. We hope to use these methods to discount or prove certain conjectures about or bifolds and crepant resolutions. In a separate undertaking, the PI is pursuing a general combinatorial formula for questions about the equivariant cohomology of manifolds with circle actions and isolated fixed points. We have found a topological formula for the restriction of certain canonical classes to any other fixed point, which has led to a kind of classification in 6-dimensions. In higher dimensions, we have found a "positive" restriction formula in the case that the manifold is Kahler and carries an invariant Palais-Smale metric (or has other rather rigid structure, such as being a GKM space). This work should lead to combinatorial formulas for the product structure in the equivariant cohomology, a topic of broad interest because it applies to flag manifolds and toric varieties, among other varieties. In a third project, we propose to investigate different formulae for the structure constants in Schubert calculus using methods developed on the calculus of Bott-Samelson manifolds. This should also generalize to fixed-point restriction formulas in the case of a symmetric group (the fixed points of an involution on G) acting on G/B. This research has roots in classical mechanics. Phase space, which consists of the position and momentum of a particle, is an example of a symplectic space, with an action by the symmetry group, also called a Hamiltonian action. Symmetry arises from the fact that the physics is the same for all observers. By "collapsing" a high-dimensional symplectic space by the symmetry, we can get a smaller space that is sometimes easier to work with. In a broad sense, this grant is concerned with describing these spaces with group actions, and their quotients. It will have impact on certain questions in physics that have arisen in string theory, as well as contribute to the discussion of how "rigid" our physical universe is by virtue of symmetry.