Exact solutions to complex Type IIB supergravity for complex superalgebra $F(4)$ and its real forms

We construct the general local solutions to complexified Type IIB supergravity which are invariant under the complexified Lie superalgebra $F(4)$. The geometry is a product of complexified maximally symmetric spaces ${\mathcal{M}}_{6\mathbb{C}}$ and ${\mathcal{M}}_{2\mathbb{C}}$ warped over a complexified surface ${\mathrm{\Sigma }}_{\mathbb{C}}$. We classify the reality conditions that may be imposed consistently to obtain real form solutions within real forms of complex Type IIB supergravity. The latter comprise standard Type IIB, Type IIB${}^{\star }$ and IIB${}^{\mathrm{\prime }}$, as well as theories with $3$, $5$, $7$ and $9$ time-like directions. Our classification of real solutions is consistent with and exhausts the real forms of $F(4)$, whose classification we confirm by elementary methods. The geometry of each real form solution is a product of real maximally symmetric spaces ${\mathcal{M}}_6$ and ${\mathcal{M}}_2$ warped over a Riemann surface $\mathrm{\Sigma }$, with various signatures. The real solutions include, among others, known $Ad{S}_6\times S^2\times \mathrm{\Sigma }$ and $Ad{S}_2\times S^6\times \mathrm{\Sigma }$ solutions to standard Type IIB as well as new solutions of the form $d{S}_{1,5}\times S^2\times \mathrm{\Sigma }$ in Type IIB${}^{\star }$. There are no real forms of the complex solutions with $\mathfrak{s}\mathfrak{o}(7;\mathbb{R})\oplus \mathfrak{s}\mathfrak{o}(3;\mathbb{R})$ symmetry. We discuss the relevance of the complex solutions, and of analytic continuations from $d{S}_{1,5}\times S^2\times \mathrm{\Sigma }$ to $S^6\times S^2\times \mathrm{\Sigma }$ within complex Type IIB, in connection with holography for the polarized IKKT model.