  F-theory compactified on singular, elliptically fibered Calabi-Yau five-folds
gives rise to two-dimensional gauge theories preserving N=(0,2) supersymmetry.
In this paper we initiate the study of such compactifications and determine the
dictionary between the geometric data of the elliptic fibration and the 2d
gauge theory such as the matter content in terms of (0,2) superfields and their
supersymmetric couplings. We study this setup both from a gauge-theoretic point
of view, in terms of the partially twisted 7-brane theory, and provide a global
geometric description based on the structure of the elliptic fibration and its
singularities. Global consistency conditions are determined and checked against
the dual M-theory compactification to one dimension. This includes a discussion
of gauge anomalies, the structure of the Green-Schwarz terms and the
Chern-Simons couplings in the dual M-theory supersymmetric quantum mechanics.
Furthermore, by interpreting the resulting 2d (0,2) theories as heterotic
worldsheet theories, we propose a correspondence between the geometric data of
elliptically fibered Calabi-Yau five-folds and the target space of a heterotic
gauged linear sigma-model (GLSM). In particular the correspondence between the
Landau-Ginsburg and sigma-model phase of a 2d (0,2) GLSM is realized via
different T-branes or gluing data in F-theory.
