In relativistic quantum field theory, physical quantities such as tensor and spinor can be considered as representations of the Poincaré group $\mathrm{ISO}(d,1)$, the isometry group of the given $(d+1)$-dimensional flat space-time.

With this perspective, it seems reasonable to consider physical quantities on a curved space-time (i.e. a pseudo-Riemanninan manifold) as representations of the isometry group of this manifold.

In short,

  1. Why don't we consider the representation theory of isometry group of pseudo-Riemannian manifolds?

In addition, I heard that in string theory (anti-) dS/CFT duality corresponds to flat space time conformal group and $dS$, $AdS$ isometry group, both of which we know quite well already. However,

  1. Why is this duality discussed only on maximally symmetric spaces, but not on general pseudo-Riemannian manifolds?

First, general curved space doesn't have isometries at all (moreover, manifold doesn't have to have a metric, which is of course necessary to define isometries), while it has all the geometrical objects you mentioned -- scalar fields, vectors, tensors of arbitrary rank, and spinors if the manifold is spin. There is no reason to restrict ourselves to representations of isometry group, and usually one doesn't do it. Say, it is really uncommon to think of various objects living on $S^2$ in terms of representations of $SO(3)$ group which is the isometry group of this space.

Second, the canonical example of AdS/CFT correspondence deals with $AdS_5 \times S^5$ space which is not maximally symmetric (while the factors are). Isometry group of this space is $SO(4,2) \times SO(6)$ which corresponds to the spacetime conformal group $SO(4,2)$ of SYM, and its R-symmetry group $SO(6)$. Working with this correspondence, one extensively uses the representations of both groups. The holographic correspondence may be formulated for less symmetric spaces in order to consider less symmetric boundary QFTs. To be more precise, one has to deform $AdS_5$ in order to break the conformal invariance, i.e. to allow the theory to flow under the RG group, and to deform $S^5$ in order to break ${\cal N}=4$ SUSY to the smaller one. Detailed discussion of all these topics (and many more) can be found in the classical review of AdS/CFT.


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