On Supergravity and Noncritical Strings

Motivated by the long-term goal of finding holographic descriptions for ${\mathcal N}=1$ and ${\mathcal N}=2$ super QCD, we revisit the subject of ''noncritical'' superstring theory. Focusing on string models with 6d super Poincar\'e invariance, we provide a careful worldsheet derivation of the leading-order effective theories for the lowest modes. We identify them as $\textit{seven}$-dimensional, $\textit{maximally supersymmetric}$ gauged supergravities: the $SO(4)$ theory for noncritical IIA and the $ISO(4)$ theory for noncritical IIB. The same theories also arise as consistent truncations on $S^3$ of the 10d IIB and IIA supergravities, respectively, where the chirality flip is as expected from T-duality. These effective supergravities should be interpreted in the following sense. The noncritical string can be viewed as a special case of a discrete series of backgrounds labelled by an integer $k$ (which counts the number of NS5 branes in a certain duality frame); the ''noncritical'' value is $k=2$, while for $k \to \infty$ one recovers a weakly-curved 10d target space. The effective supergravities described here give an accurate description of the interactions among the lowest modes for $k \to \infty$, with higher derivative corrections suppressed by powers of $1/k$. We discuss BPS solutions of the 7d gauged supergravities and their uplift to 10d solutions. In particular, we find a novel class of solutions with RR flux, parametrized by a function of three variables that solves an elegant PDE. While we cannot solve the PDE in closed form except in trivial cases, we confirm that our solutions correspond to a 10d IIA Hanany-Witten setup with continuous distributions of both ''color'' D4 branes and ''flavor'' D6 branes.