We obtain the complete local solutions with 16 supersymmetries
to Type IIB supergravity on a space-time of the form

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0$. ]]>

0 \end{align} With this assumption $\rho^2$, given in~(\ref{eq:5_2c}), is real and can be made positive by appropriately choosing the sign of the constant $c$. To verify $|B| \leq 1$ we calculate $f^{2}$ using~(\ref{eq:4_4_1b}), \begin{equation} \label{eq:5_4c} f^{2} = \frac{1}{1 - |B|^{2}} = 1 + \frac{|\lambda - Z^{2}|^{2}}{(1 - |\lambda|^{2})(1 - |Z|^{4})} \end{equation} Since $\kappa^{2}(1 - R) = |\kappa_{-}|^{2}(1 - |\lambda|^{2})(1 - |Z|^{2})$,~(\ref{kR}) implies $f^{2} \geq 1$. The reality and positivity conditions are therefore given by~(\ref{kG}) and~(\ref{kR}). ]]>

1$ or $R<1$. We shall refer to these branches as $\mB_\pm$, defined by, \begin{align} \label{eq:5_4h} \mB_+ & = \left \{ \kappa^{2} > 0, \qquad \cG < 0, \qquad R < 1 \right \} \no \\ \mB_- & = \left \{ \kappa^{2} < 0, \qquad \cG > 0, \qquad R > 1 \right \} \end{align} These two branches are mapped into one another by an involution, which is a combination of complex conjugation, reversal of the complex structure on $\Sigma$, and reversal of the indices~$\pm$ on the functions $\cA_\pm$, given by, \begin{align} \label{eq:5_4e} \mathcal{A}_{\pm}(w) &\to \mathcal{A}'_{\pm}(w) = \bar{\mathcal{A}}_{\mp}(w) = \overline{\mathcal{A}_{\mp}(\bar{w})} \end{align} combined with $R\rightarrow R^{-1}$. This transformation leaves eq.~(\ref{eq:5_2b}) invariant and reverses the sign of $\kappa^2$ and $\cG$. It leaves the metric functions $f_2^2$, $f_6^2$ and $\rho^2$ invariant and complex conjugates the fields $B$ and $\cC$, \begin{align} \label{eq:5_4g} B(w,\bar{w}) \to B'(w,\bar{w}) & = \bar{B}(w,\bar{w}) = \overline{B(\bar{w},w)} \nonumber \\ \mathcal{C}(w,\bar{w}) \to \mathcal{C}'(w,\bar{w}) & = \bar{\mathcal{C}}(w,\bar{w}) = \overline{\mathcal{C}(\bar{w},w)} \end{align} ]]>

0$ and strictly negative for the branch $\mB_-$ with $\kappa^2<0$. Thus, no regular supergravity solutions exist when $\mathcal{G}$ vanishes on $\p \Sigma$. ]]>

0 \hskip 0.1in\text{on ${\rm int}(\Sigma)$} & \kappa^2&=\cG=0 \hskip 0.1in\text{on $\partial\Sigma$} \end{align} We can assume the same choice of holomorphic data as input for the ${\rm AdS}_2\times S^6\times\Sigma$ solutions. The expressions for $\kappa^2$ and $\cG$ in terms of the locally holomorphic functions are the same for ${\rm AdS}_2\times S^6\times\Sigma$ and ${\rm AdS}_6\times S^2\times\Sigma$, such that we realize~(\ref{eq:wick2}) in both cases. Eq.~(\ref{eq:5_2b}) then implies $\Lambda R>0$ in the interior of the upper half plane and $\Lambda R\rightarrow 1$ on the boundary, and we choose the branch $0<\Lambda R\leq 1$. This implies that $R$ is positive for ${\rm AdS}_6\times S^2\times\Sigma$ and negative for ${\rm AdS}_2\times S^6\times\Sigma$. We note that negative $R$ was not acceptable for solving the BPS equations, where $R$ was positive by construction. But, as noted at the end of section~\ref{app:EOM}, neither $R$ nor the constant $c$ are constrained by the equations of motion. So at the level of the equations of motion these ${\rm AdS}_2\times S^6$ configurations are acceptable, and we have, \begin{align} (\Lambda R)_{{\rm AdS}_2\times S^6\times\Sigma}&=(\Lambda R)_{{\rm AdS}_6\times S^2\times\Sigma} \end{align} \looseness=-1 The expressions for the supergravity fields in~(\ref{eqn:metric}),~(\ref{eq:5_2d}) and~(\ref{eq:5_2e}) depend on $R$ only through this combination $\Lambda R$. The form of the axion-dilaton $B$ in~(\ref{eq:5_2d}) is, in fact, exactly the same for ${\rm AdS}_2\times S^6\times\Sigma$ and ${\rm AdS}_6\times S^2\times\Sigma$. The metric functions in~(\ref{eqn:metric}) are real provided that $c$ is chosen real for ${\rm AdS}_6\times S^2\times\Sigma$ and imaginary for ${\rm AdS}_2\times S^6\times\Sigma$, to compensate for the phase in $\sqrt{\Lambda\cG}$. They only differ between ${\rm AdS}_2\times S^6\times\Sigma$ and ${\rm AdS}_6\times S^2\times\Sigma$ through their signs: while $f_2^2$, $f_6^2$ and $\rho^2$ all have the same sign for ${\rm AdS}_6\times S^2\times\Sigma$, the sign of $f_2^2$ and $f_6^2$ is opposite to that of $\rho^2$ for ${\rm AdS}_2\times S^6\times\Sigma$. This is precisely as expected for solutions connected by the analytic continuation in~(\ref{eq:acont}). The gauge potential $\cC$ in~(\ref{eq:5_2e}) differs only by an overall factor of $i$ between ${\rm AdS}_2\times S^6\times\Sigma$ and ${\rm AdS}_6\times S^2\times\Sigma$. This produces the expected behavior under a Wick rotation for the three-form field strength, where one of the components along $S^2$ becomes timelike and picks up a factor of $i$. We have thus recovered the analytic continuation of the global ${\rm AdS}_6\times S^2\times\Sigma$ solutions to ${\rm AdS}_2\times S^6\times\Sigma$ via~(\ref{eq:acont}), which is simply realized by the same choice of locally holomorphic functions. This naive analytic continuation does, however, not lead to physically regular solutions. Aside from the inappropriate signs for the metric functions, it is still the two-dimensional space that collapses on the boundary of $\Sigma$. This was the desired behavior for the ${\rm AdS}_6\times S^2\times\Sigma$ case, with the collapsing $S^2$ smoothly closing off spacetime. But it is not desirable for the ${\rm AdS}_2\times S^6\times\Sigma$ solutions to have the ${\rm AdS}_2$ cap off on $\partial\Sigma$. Moreover, the solutions are not supersymmetric, since we do not recover them from the BPS equations where $R\geq 0$ was required by construction. The loss of supersymmetry under Wick rotation may be understood from the change in the Clifford algebra due to the changed signature in the two- and six-dimensional spaces. The construction of physically regular supersymmetric solutions will therefore have to be revisited, with the regularity conditions and constraints spelled out in section~\ref{sec:regularity}. ]]>