  In this short note we give an overview of recent work on string propagation
on stacks and applications to gauged linear sigma models. We begin by outlining
noneffective orbifolds (orbifolds in which a subgroup acts trivially) and
related phenomena in two-dimensional gauge theories, which realize string
propagation on gerbes. We then discuss the `decomposition conjecture,' equating
conformal field theories of strings on gerbes and strings on disjoint unions of
spaces. Finally, we apply these ideas to gauged linear sigma models for
complete intersections of quadrics, and use the decomposition conjecture to
show that the Landau-Ginzburg points of those models have a geometric
interpretation in terms of a (sometimes noncommutative resolution of a)
branched double cover, realized via nonperturbative effects, rather than as the
vanishing locus of a superpotential. These examples violate old unproven lore
on GLSM's (e.g., that geometric phases must be related by birational
transformations), and we conclude by observing that in these examples (and
conjecturing more generally in GLSM's), the phases are instead related by
Kuznetsov's `homological projective duality.'
