  We study gerbes with connection over an etale stack via noncommutative
algebras of differential forms on a groupoid presenting the stack. We then
describe a dg-category of modules over any such algebra, which we claim
represents a dg-enhancement of the derived category of coherent analytic
sheaves on the gerbe in question.
  This category can be used to phrase and prove Fourier-Mukai type dualities
between gerbes and other noncommutative spaces. As an application of the
theory, we show that a gerbe with flat connection on a torus is dual (in a
sense analogous to Fourier-Mukai duality or T-duality) to a noncommutative
holomorphic dual torus.
