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Marco Mackaay, Universidade do Algarve

TQFT's and Gerbes

The general setup for a categorical construction of state-sum invariants of PL $4$-manifolds given by the speaker contains all known categorical/combinatorial constructions as especial cases and has the potential of being more general. In order to find more examples of this construction, one would like to understand whether there is a link with the famous smooth invariants of $4$-manifolds defined by Donaldson and by Seiberg and Witten. For this reason one first ought to understand the differential geometry behind the categorical construction. As a first tiny step in this direction Roger Picken and the speaker studied the holonomy of gerbes with connections and proved some interesting structural theorems about such holonomies. This talk will first sketch the results about the contruction above in order to make clear the motivation for the interest in gerbes with connections, and then will explain the results about the aforementioned holonomies.