  We propose a framework for treating F-theory directly, without resolving or
deforming its singularities. This allows us to explore new sectors of gauge
theories, including exotic bound states such as T-branes, in a global context.
We use the mathematical framework known as Eisenbud's matrix factorizations for
hypersurface singularities. We display the usefulness of this technique by way
of examples, including affine singularities of both conifold and orbifold type,
as well as a class of full-fledged compact elliptically fibered Calabi-Yau
fourfolds.
