  We present a class of solvable models that resemble string theories in many
respects but have a strikingly different non-perturbative sector. In
particular, there are no exponentially small contributions to perturbation
theory in the string coupling, which normally are associated with branes and
related objects. Perturbation theory is no longer an asymptotic expansion, and
so can be completely re-summed to yield all the non-perturbative physics. We
examine a number of other properties of the theories, for example constructing
and examining the physics of loop operators, which can be computed exactly, and
gain considerable understanding of the difference between these new theories
and the more familiar ones, including the possibility of how to interpolate
between the two types. Interestingly, the models we exhibit contain a family of
zeros of the partition function which suggest a novel phase structure. The
theories are defined naturally by starting with models that yield
well-understood string theories and allowing a flux-like background parameter
to take half-integer rather than integer values. The family of models thus
obtained are seeded by functions that are intimately related to the classic
rational solutions of the Painleve II equation, and a family of
generalisations.
