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Register a new userGeometry and supersymmetry of heterotic warped flux AdS backgrounds
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We classify the geometries of the most general warped, flux AdS backgrounds of heterotic supergravity up to two loop order in sigma model perturbation theory. We show under some mild assumptions that there are no $AdS_n$ backgrounds with $n ot=3$. Moreover the warp factor of AdS$_3$ backgrounds is constant, the geometry is a product $AdS_3times M^7$ and such solutions preserve, 2, 4, 6 and 8 supersymmetries. The geometry of $M^7$ has been specified in all cases. For 2 supersymmetries, it has been found that $M^7$ admits a suitably restricted $G_2$ structure. For 4 supersymmetries, $M^7$ has an $SU(3)$ structure and can be described locally as a circle fibration over a 6-dimensional KT manifold. For 6 and 8 supersymmetries, $M^7$ has an $SU(2)$ structure and can be described locally as a $S^3$ fibration over a 4-dimensional manifold which either has an anti-self dual Weyl tensor or a hyper-Kahler structure, respectively. We also demonstrate a new Lichnerowicz type theorem in the presence of $alpha$ corrections.
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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 $AdS_ntimes_w M^{10-n}$ preserve $2^{[{nover2}]} k$ for $nleq 4$ and $2^{[{nover2}]+1} k$ for $4<nleq 7$ supersymmetries, $kin bN_{>0}$. 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 $2^{[{nover2}]} k$ for $2<nleq 4$, $2^{[{n+1over2}]} k$ for $4<nleq 8$, and $2^{[{nover2}]} k$ for $n=9, 10$ supersymmetries.
According to the t Hooft-Susskind holography, the black hole entropy,$S_mathrm{BH}$, is carried by the chaotic microscopic degrees of freedom, which live in the near horizon region and have a Hilbert space of states of finite dimension $d=exp(S_mathrm{BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the AdS$_2[mathbb{Z}_N]$ discrete, finite and random geometry, where $Npropto S_mathrm{BH}$. It has been constructed by purely arithmetic and group theoretical methods in order to explain, in a direct way, the finiteness of the entropy, $S_mathrm{BH}$. What has been left as an open problem is how the smooth AdS$_2$ geometry can be recovered, in the limit when $Ntoinfty$. In the present article we solve this problem, by showing that the discrete and finite AdS$_2[mathbb{Z}_N]$ geometry can be embedded in a family of finite geometries, AdS$_2^M[mathbb{Z}_N]$, where $M$ is another integer. This family can be constructed by an appropriate toroidal compactification and discretization of the ambient $(2+1)$-dimensional Minkowski space-time. In this construction $N$ and $M$ can be understood as infrared and ultraviolet cutoffs respectively. The above construction enables us to obtain the continuum limit of the AdS$_2^M[mathbb{Z}_N]$ discrete and finite geometry, by taking both $N$ and $M$ to infinity in a specific correlated way, following a reverse process: Firstly, by recovering the continuous, toroidally compactified, AdS$_2[mathbb{Z}_N]$ geometry by removing the ultraviolet cutoff; secondly, by removing the infrared cutoff in a specific decompactification limit, while keeping the radius of AdS$_2$ finite. It is in this way that we recover the standard non-compact AdS$_2$ continuum space-time. This method can be applied directly to higher-dimensional AdS spacetimes.
In this paper we look for AdS solutions to generalised gravity theories in the bulk in various spacetime dimensions. The bulk gravity action includes the action of a non-minimally coupled scalar field with gravity, and a higher-derivative action of gravity. The usual Einstein-Hilbert gravity is induced when the scalar acquires a non-zero vacuum expectation value. The equation of motion in the bulk shows scenarios where AdS geometry emerges on-shell. We further obtain the action of the fluctuation fields on the background at quadratic and cubic orders.
We present all the symmetry superalgebras $mathfrak{g}$ of all warped AdS$_ktimes_w M^{d-k}$, $k>2$, flux backgrounds in $d=10, 11$ dimensions preserving any number of supersymmetries. First we give the conditions for $mathfrak{g}$ to decompose into a direct sum of the isometry algebra of AdS$_k$ and that of the internal space $M^{d-k}$. Assuming this decomposition, we identify all symmetry superalgebras of AdS$_3$ backgrounds by showing that the isometry groups of internal spaces act transitively on spheres. We demonstrate that in type II and $d=11$ theories the AdS$_3$ symmetry superalgebras may not be simple and also present all symmetry superalgebras of heterotic AdS$_3$ backgrounds. Furthermore, we explicitly give the symmetry superalgebras of AdS$_k$, $k>3$, backgrounds and prove that they are all classical.
We review the remarkable progress that has been made the last 15 years towards the classification of supersymmetric solutions with emphasis on the description of the bilinears and spinorial geometry methods. We describe in detail the geometry of backgrounds of key supergravity theories, which have applications in the context of black holes, string theory, M-theory and the AdS/CFT correspondence unveiling a plethora of existence and uniqueness theorems. Some other aspects of supersymmetric solutions like the Killing superalgebras and the homogeneity theorem are also presented, and the non-existence theorem for certain smooth supergravity flux compactifications is outlined. Amongst the applications described is the proof of the emergence of conformal symmetry near black hole horizons and the classification of warped AdS backgrounds that preserve more than 16 supersymmetries.
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