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K-theory of stacks, often of geometric stacks such as differentiable stacks/Lie groupoids. Under mild assumptions equivalently the operator K-theory of their groupoid convolution algebras.
The twisted K-theory of geometric stacks $X$ assigns K-classes to maps $X \to \mathbf{B}^2 U(1)$ to the moduli 2-stack of circle 2-bundles.
We discuss two different but related definitions. The first
realizes the K-theory of a suitable Lie groupoid (a local quotient groupoid) as a class in the Grothendieck group of suitable equivariant vector bundles (Hilbert space bundles). The second
realizes it as the operator K-theory of the groupoid convolution algebra.
In the Properties-section below we discuss how both definitions are compatible.
For local quotient groupoids one can define their K-theory as, essentially, the Grothendieck group of equivariant vector bundles on the groupoid, or rather, in the twisted case, by equivalence classes of sections of Fredholm bundles. (…)
One can define the (twisted) K-theory of a Lie groupoid (equipped with a circle 2-bundle) to be the operator K-theory of its groupoid convolution algebra.
For $X$ a proper Lie groupoid, the two definition above agree. (TuXuLG 03, 4.6).
The twisted K-theory of local quotient differentiable stacks is defined in terms of Fredholm bundles in
The twisted K-theory of general differentiable stacks/Lie groupoids in terms of the operator K-theory of their twisted groupoid convolution algebras is discussed in
Functoriality of twisted K-theory of topological groupoids/Lie groupoids is discussed in
Alexander Alldridge, Jeffrey Giansiracusa, Contravariant functoriality for twisted K-theory of stacks, Oberwolfach Reports no. 46 (2006) 2796 – 2800 (pdf)
Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013
See also
(…)
Last revised on August 30, 2018 at 13:12:13. See the history of this page for a list of all contributions to it.