String compactifications on Calabi-Yau stacks

In this paper we study string compactifications on Deligne-Mumford stacks. The basic idea is that all such stacks have presentations to which one can associate gauged sigma models, where the group gauged need be neither finite nor effectively-acting. Such presentations are not unique, and lead to physically distinct gauged sigma models; stacks classify universality classes of gauged sigma models, not gauged sigma models themselves. We begin by defining and justifying a notion of ``Calabi-Yau stack,'' recall how one defines sigma models on (presentations of) stacks, and calculate of physical properties of such sigma models, such as closed and open string spectra. We describe how the boundary states in the open string B model on a Calabi-Yau stack are counted by derived categories of coherent sheaves on the stack. Along the way, we describe numerous tests that IR physics is presentation-independent, justifying the claim that stacks classify universality classes. String orbifolds are one special case of these compactifications, a subject which has proven controversial in the past; however we resolve the objections to this description of which we are aware. In particular, we discuss the apparent mismatch between stack moduli and physical moduli, and how that discrepancy is resolved.