Tabuada and Van den Bergh have argued in a recent paper that Hochschild (co)homology of a smooth variety is not sensitive to the presence of a twisting by a Brauer class.

I will argue that the situation is fundamentally different for orbifolds. Specifically, I will describe an HKR decomoposition of the Hochschild (co)homology of an orbifold endowed with a Brauer class, and I will explain how the invariants change when the Brauer class is turned on.

This gives mathematical justification for certain calculations in mirror symmetry of Vafa and Witten. (Joint work with Dima Arinkin.)