Abstract: B-branes on Calabi-Yau (CY) X manifolds are in 1-1 correspondence with objects in the derived category of coherent sheaves. B-branes on Gauged Linear Sigma Models (GLSM) can be roughly interpreted as objects of the derived category of a CY stack. This interpretation can be used to generate equivalences between various categories associated to geometric and nongeometric phases of a CY. I will introduce these concepts and present some results for the case when the stack is not CY (physically, this means the GLSM is anomalous), that can be derived from physics. In particular I will explain how the equivalences between categories associated to phases are modified and comment on some consequence on K-theory using as a guide the example of Hirzebruch-Jung resolutions of cyclic surface singularities.
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