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We construct a class of codimension-2 solutions in supergravity that realizeT-folds with arbitrary $O(2,2,\mathbb{Z})$ monodromy and we develop a geometricpoint of view in which the monodromy is identified with a product of Dehntwists of an auxiliary surface $\Sigma$ fibered on a base $\mathcal{B}$. Thesedefects, that we call T-fects, are identified by the monodromy of the mappingtorus obtained by fibering $\Sigma$ over the boundary of a small diskencircling a degeneration. We determine all possible local geometries bysolving the corresponding Cauchy-Riemann equations, that imply the equations ofmotion for a semi-flat metric ansatz. We discuss the relation with theF-theoretic approach and we consider a generalization to the T-duality group ofthe heterotic theory with a Wilson line.

The monodromy of T-folds and T-fects