Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology

Research Article

The full orbifold K-theory of abelian symplectic quotients

Rebecca Goldina1, Megumi Haradaa2, Tara S. Holma3 and Takashi Kimuraa4

a1 Mathematical Sciences MS 3F2, George Mason University, 4400 University Drive, Fairfax, Virginia 22030, USA, rgoldin@math.gmu.edu

a2 Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S4K1, Canada, Megumi.Harada@math.mcmaster.ca

a3 Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201, USA, tsh@math.cornell.edu

a4 Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215, USA, kimura@math.bu.edu

Abstract

In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold K-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

(Received April 25 2009)

Key Words

  • full orbifold K-theory;
  • inertial K-theory;
  • Hamiltonian T -space;
  • symplectic quotient