Abstract
We uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non-perturbatively. We are able to embed the type 0A and 0B (A, A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We observe that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A, D) minimal string backgrounds. We explain how these and several string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painlevé IV equation plays a key role in organizing the string theory physics, joining its siblings, Painlevé I and II, whose roles have previously been identified in this minimal string context.
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