The monodromy of T-folds and T-fects

We construct a class of codimension-2 solutions in supergravity that realize T-folds with arbitrary $O(2,2,\mathbb{Z})$ monodromy and we develop a geometric point of view in which the monodromy is identified with a product of Dehn twists of an auxiliary surface $\Sigma$ fibered on a base $\mathcal{B}$. These defects, that we call T-fects, are identified by the monodromy of the mapping torus obtained by fibering $\Sigma$ over the boundary of a small disk encircling a degeneration. We determine all possible local geometries by solving the corresponding Cauchy-Riemann equations, that imply the equations of motion for a semi-flat metric ansatz. We discuss the relation with the F-theoretic approach and we consider a generalization to the T-duality group of the heterotic theory with a Wilson line.