Einstein manifolds as asymptotic background spacetimes in Type IIB string theory

[Afonso Campos]

On page 30 of the notes (https://arxiv.org/abs/1501.00007) 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?

 

[AndyW]

Hello,

Thank you for bringing up this interesting topic on the AdS/CFT correspondence. To answer your first question, yes, both $AdS_{5}\times S^{5}$ and $S^{5}$ are considered Einstein manifolds. An Einstein manifold is defined as a Riemannian manifold with constant scalar curvature, and both of these spaces have this property. In fact, the $S^{5}$ is a maximally symmetric space with positive scalar curvature, while $AdS_{5}\times S^{5}$ has negative scalar curvature.

To address your second question, it is true that only Einstein manifolds and their quotients are considered as suitable asymptotic backgrounds for defining a Type IIB string theory. This is because the AdS/CFT correspondence relies on the fact that the spacetime on which the gravitational theory is defined (in this case, $AdS_{d+1}$) has a conformal boundary. This boundary is necessary for the dual CFT to exist and for the correspondence to hold. Einstein manifolds and their quotients are the only known spacetimes that have this property, making them the most suitable for defining a Type IIB string theory.

I hope this helps clarify any confusion. If you have any further questions, please feel free to ask.