# Einstein manifolds as the asymptotic background spacetimes in Type IIB string theory

On page 30 of the notes by Veronika Hubeny on The AdS/CFT correspondence, we find the following:

So far, we have been describing just one particular case of the AdS/CFT duality, namely (3.3). There are however many ways in which the correspondence can be extended and generalized. The earliest and most immediate one was to consider different number of dimensions, i.e. to describe a dual of a gravitational theory on $AdS_{d+1}$ (times a compact manifold) in terms of a $d$-dimensional CFT, on which we elaborate below. Another straightforward type of generalization constitutes starting with (3.3) and deforming both sides in a controlled way. If we add extra terms to the Lagrangian, the gauge theory is no longer conformal; it will undergo renormalization group flow (which gives an effective description at a given energy by integrating out the higher-energy degrees of freedom). This is directly mimicked by the behavior of the bulk geometry in the radial direction. Depending on the type of deformation, we can get quite a rich set of possibilities, including ones where the low-energy physics is confining, massive, chiral symmetry breaking, etc. One can also replace the $S^5$ by any other Einstein manifold or a quotient of the $S^5$, which gives rise to more complicated gauge theories. More interestingly, one can even consider different asymptotics.

Are $AdS_{5}\times S^{5}$ and $S^{5}$ Einstein manifolds?

Do only Einstein manifolds and quotients of Einstein manifolds qualify as the asymptotic background spacetimes on which you can define a Type IIB string theory?

• An Einstein manifold is a manifold where $R_{ij} = kg_{ij}$ for some constant k. Yes, $AdS_{5}\times S^{5}$ is an Einstein manifold, anti-deSitter space is a Lorentzian (specific case of an Einstein) manifold and of course $S^{5}$ is an Einstein manifold as well because it is a Calabi-Yau manifold. I'm not sure enough regarding your second question to write this as an answer but I am pretty sure the answer is 'yes'. – Not_Here Jun 6 '17 at 23:23