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

A bstract 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 $$ {\mathcal{M}}_{6\mathbb{C}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mn>6</mml:mn> <mml:mi>ℂ</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $$ {\mathcal{M}}_{2\mathbb{C}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>ℂ</mml:mi> </mml:mrow> </mml:msub> </mml:math> warped over a complexified surface Σ 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 ⋆ and IIB ′ , 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 $$ {\mathcal{M}}_6 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> and $$ {\mathcal{M}}_2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> warped over a Riemann surface Σ, with various signatures. The real solutions include, among others, known AdS 6 × S 2 × Σ and AdS 2 × S 6 × Σ solutions to standard Type IIB as well as new solutions of the form dS 1,5 × S 2 × Σ in Type IIB ⋆ . There are no real forms of the complex solutions with $$ \mathbf{\mathfrak{so}}\left(7;\mathbb{R}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> <mml:mfenced> <mml:mn>7</mml:mn> <mml:mi>ℝ</mml:mi> </mml:mfenced> </mml:math> ⨁ $$ \mathbf{\mathfrak{so}}\left(3;\mathbb{R}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> <mml:mfenced> <mml:mn>3</mml:mn> <mml:mi>ℝ</mml:mi> </mml:mfenced> </mml:math> symmetry. We discuss the relevance of the complex solutions, and of analytic continuations from dS 1,5 × S 2 × Σ to S 6 × S 2 × Σ within complex Type IIB, in connection with holography for the polarized IKKT model.