Exact Results in Quantum Theory & Gravity

In these lectures, I shall discuss various geometric and field-theoretic aspects of the Gauge Principle for multi-phase sigma-models with a topological (Wess-Zumino) term, as formulated by K.Gawędzki, K.Waldorf and myself in the framework of gerbe theory and generalised geometry. The focus shall be on universal structures and constructions rather than peculiarities of the 2d setting in hand. Thus, upon recalling some field-theoretic, geometric and categorial prerequisites (the world-sheet description of multi-phase CFT, the 2-category of bundle gerbes with connection, and basic theory of (Lie) groupoids and algebroids), I shall elaborate on the following issues:- a groupoidal/algebroidal description of rigid symmetries, its generalised-geometric interpretation and a nice picture of the local gauge anomaly;- rendering rigid symmetries local - the minimal coupling recipe and beyond;- the Principle of (2-)Categorial Descent, and equivariant structures (bundles, gerbes etc.);- the role of (topologically) non-trivial gauge bundles in the construction of a consistent model on the symmetry coset;- (equivariant-)cohomological classification of obstructions to gauging and inequivalent variants thereof.Time permitting, I shall also outline possible applications of the framework described in an attempt at understanding T-duality (resp. mirror symmetry) for toroidally fibred target spaces, and the (non-)geometry of T-folds.