# Supersymmetry and non-compact $R$-symmetry group?

The $R$-symmetry for $N$ supercharges is $U(N)$. Is it possible to generalize $R$-symmetry [let's take $U(4)$) to be something like $U(2,2)$ (maybe analogous to Wick rotation of $SO(3,1)$ to $SO(4)$?)]?

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Consider the $i$-th superpartner of a bosonic particle state, $|i\rangle$, where $i=1,2,\dots,N$. The inner product $\langle i|j\rangle$ of such 1-fermion states has to respect the symmetry. So for $U(M,N)$, it would be ${\rm diag}(+1,+1,\dots,-1,-1,\dots)$ with $M$ plus signs and $N$ minus signs. It would follow that the Hilbert space contains physical states with norms of both signs and the predicted probabilities could be negative, too.
Thanks Lubos! I'm OK with everything you say but what happens for example in bosonic string worldsheet theory, where you have inertial symmetry $SO(25,1)$ for scalar fields? There you seem to have noncompact inertial symmetry? –  jancore May 1 '13 at 22:08
Dear @jancore, I meant that internal noncompact symmetries that produce states transforming as a linear finite-dimensional representation of the symmetry are unacceptable (for the simple "sign of thenorm" reason I fully described). The internal symmetry you mention (spacetime Lorentz group) doesn't lead to any linear representation. Similarly, SUGRA theories have noncompact groups like $E_{7(7)}$ but they're realized nonlinearly. –  Luboš Motl May 11 '13 at 4:55