Gerbes, Quantum Mechanics and Gravity

We prove that invariance of a quantum theory under the semiclassical vs. strong– quantum duality S/~ ←→ ~/S, where S is the classical action, is equivalent to noncommutativity (of the Heisenberg–algebra type) of the coordinates of the space on which S is defined. We place these facts in correspondence with gerbes and Neveu– Schwarz B–fields and discuss their implications for a quantum theory of gravity. Feynman’s propagator turns out to be closely related to the trivialisation of a flat gerbe on configuration space. The 2–cocycle defining this flat gerbe is given by the exponential of the surface integral of the Neveu–Schwarz field.