Supersymmetry of IIA warped flux AdS and flat backgrounds

TL;DR

This paper classifies the supersymmetry preserved by warped flux AdS and flat backgrounds in type IIA supergravity, generalizing the Lichnerowicz theorem and providing explicit relations between geometry and supersymmetry fractions.

Contribution

It determines the fractions of preserved supersymmetry in warped flux backgrounds and extends the Lichnerowicz theorem to these settings, linking Killing spinors to zero modes of Dirac-like operators.

Findings

Preservation of specific supersymmetry fractions in AdS and flat backgrounds.

Generalization of the Lichnerowicz theorem for these backgrounds.

Explicit relations between Killing spinors and Dirac operator zero modes.

Abstract

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. We also adapt our results to $\bR^{1,n-1}\times_w M^{10-n}$ backgrounds which underpin flux compactifications to $\bR^{1,n-1}$ and show that these preserve $2^{[{n\over2}]} k$ for $2<n\leq 4$, $2^{[{n+1\over2}]} k$ for $4<n\leq 8$, and $2^{[{n\over2}]} k$ for $n=9, 10$ supersymmetries.