Supersymmetry of IIA warped flux AdS and flat backgrounds

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We identify the fractions of supersymmetry preserved by the most general warped flux AdS and flat backgrounds in both massive and standard IIA supergravities.

• We find that $AdS_n\times_w M^{10-n}$ preserve $2^{[{n\over2}]} k$ for $n\leq 4$ and $2^{[{n\over2}]+1} k$ for $4<n\leq 7$ supersymmetries, $k\in \bN_{>0}$.
• In addition we show that, for suitably restricted fields and $M^{10-n}$, the killing spinors of AdS backgrounds are given in terms of the zero modes of Dirac like operators on $M^{10-n}$.
• This generalizes the Lichnerowicz theorem for connections whose holonomy is included in a general linear group.