We extend the construction of the Chen-Ruan cohomology in the setting of heterotic string theory. We show that it properly reduces to the Chen-Ruan cohomology in the case where the gauge bundle E is chosen to be the tangent bundle TX and examine its basic properties, followed by demonstrating nontrivial examples and computations. The second portion of this work examines the extension of the anomaly cancellation condition for gerbes through an extended example. Namely, we use Fourier-Mukai transforms and the methods of [Donagi-Pantev 04] to set up a construction of bundles over a gerbe which should be non-anomalous.