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As far as I understand it, the $R$-symmetry group is just the largest subgroup of the automorphism group of the supersymmetry (SUSY) algebra which commutes with the Lorentz group. I know for $\mathcal{N}=1$ SUSY, the $R$-symmetry is $U(1)$, mainly due to there being only one supercharge. However, I was wondering: how does one find the $R$-symmetry group for an extended $\mathcal{N}>1$ supersymmetric theory?

Also, does the $R$-symmetry group depend on the dimension and/or geometry (e.g. if we had a compact spacetime manifold) of spacetime?

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Properties of R-symmetry group depends on spinor structure:

Here M means Maiorana spinors, MW - Maiorana-Weyl, S- symplectic.

Spinor structure depends on dimension and signature of space. For more details one can consult Tools for supersymmetry.

Put supersymmetric theory on curved space is not simple task. To do such procedure, one must find generalized Killing spinors on such manifold. Such spinors will depend on non trivial SUGRA background filelds. See for example An introduction to supersymmetric field theories in curved space.

Simpler task is to study SCFT, but in superconformal algebra R symmetry is not external symmetry. R symmetry appear in commutation relations. Such theories natural to put on conformal flat manifolds.

For spheres we have following R-symmetry groups:

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