We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFT’s.
"TQFT's and gerbes." Algebr. Geom. Topol. 4 (1) 243 - 272, 2004. https://doi.org/10.2140/agt.2004.4.243