We compare two different constructions of higher-dimensional parallel transport. On the one hand, there is the two-dimensional parallel transport associated with 2-connections on 2-bundles studied by Baez–Schreiber , Faria Martins–Picken  and Schreiber–Waldorf . On the other hand, there are the higher holonomies associated with flat superconnections as studied by Igusa , Block–Smith  and Arias Abad–Schätz . We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the two-dimensional holonomies provided by the two approaches coincide.
Differential Geometry and Its Applications, 2015, Vol 40, p. 14-42