In the context of the AdS/CFT correspondence I was trying to understand how the symmetry group of the underlying space $AdS_5 \times S^5$ comes out to be the supergroup $SU(2,2|4)$. I can see how the bosonic subrgoup $SU(2,2)\times SU(4)_R$ crops up as the group of isometries of $AdS_5\times S^5$, since $SU(2,2)$ is a double cover of $SO(2,4)$ and this preserves the signature (+ + - - - -)space from which $AdS_5$ arose. Moreover the group of R symmetries $SU(4)_R$ bears an astonishing resemblance to $SO(6)$ which preserves the $S^5$ metric. So far so good in bosonic land. However, what about the fermionic parts of $SU(2,2|4)$ ? How do they act on $AdS_5\times S^5$ ? Something to do with the branes I guess, but I'm not sure....
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1$\begingroup$ Dear Twistor, this is a confusing question. Fermionic generators of course don't act just geometrically on a bosonic space. At most, you could consider a superspace extension of $AdS_5\times S^5$ but superspaces are not too useful if there are too many supercharges (they have too many components). So it's a kind of misguided approach to ask about the action of the supercharges on the spacetime only; one should learn what is the action of the supergroup at the Hilbert space - the whole actual theory - and it is pretty straightforward if you define the $N=4$ gauge theory. $\endgroup$– Luboš MotlCommented May 19, 2012 at 18:13
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$\begingroup$ Hi Lubos, the reason for the question was Witten's statement in slide number 11 in his talk in this link. He says "$AdS_5XS^5$ has symmetry group $PSU(2,2|4)$". I couldn't make sense of the geometric interpretation of the fermionic generators. Maybe he is alluding to something quantum mechanical as you say, but it wasn't clear in the slide. Also I'm not sure what the "P" means either.... $\endgroup$– twistor59Commented May 19, 2012 at 18:57
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$\begingroup$ Dear Twistor, OK, Witten really wants to say that $PSU(2,2|4)$ is the (or "a") maximal supergroup of symmetries that a theory defined on $AdS_5\times S^5$ may have. But he surely doesn't mean that all the generators - the fermionic ones in particular - may be defined as differential operators acting on the 10 bosonic coordinates of this spacetime only. $\endgroup$– Luboš MotlCommented May 19, 2012 at 19:28
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$\begingroup$ For the terminology of superalgebras (and the same supergroups), look at page 58 of this Kac's review (PDF): projecteuclid.org/… $\endgroup$– Luboš MotlCommented May 19, 2012 at 19:34
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$\begingroup$ @LubošMotl: OK thanks very much, that's clear! Post it in an answer and we can close this one off. $\endgroup$– twistor59Commented May 19, 2012 at 19:43
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Fermionic generators of course don't act just geometrically on a bosonic space; all differential operators acting on bosonic coordinates are bosonic.
At most, you could consider a superspace extension of $AdS_5 \times S^5$ but superspaces are not too useful if there are too many supercharges (they have too many components). So it's a kind of misguided approach to ask about the action of the supercharges on the spacetime only; one should learn what is the action of the supergroup at the Hilbert space – the whole actual theory – and it is pretty straightforward if you define the $N=4$ gauge theory.
Witten – when he mentions that the group acting on the AdS-space times the sphere is the supergroup – really wants to say that $PSU(2,2|4)$ is the (or "a") maximal supergroup of symmetries that a theory defined on $AdS_5\times S^5$ may have. But he surely doesn't mean that all the generators - the fermionic ones in particular - may be defined as differential operators acting on the 10 bosonic coordinates of this spacetime only.
For the terminology of superalgebras (and the same supergroups), look at page 58 of this Kac's review (PDF).
Finally, the extra $P$ in $PSU$ means that one eliminates a block-diagonal "hypercharge-like" generator from $SU$. It's similar to the embedding of $SU(3)\times SU(2)\times U(1)$ in $SU(5)$ in grand unification; in the superalgebra case, one can consistently eliminate the $U(1)$ here, at least if the number of bosonic entries and fermionic entries (dimensions of the fundamental representation) are equal. And it's equal, both are 4 for $PSU(2,2|4)$. If you check a SE question about an $SU(5)$ decomposition here
Introduction to Physical Content from Adjoint Representations
the equal dimensions allow us to set the "hypercharges" of the off-block-diagonal entries (all the fermionic generators) which were $\pm 5/6$ above to zero and eliminate the "hypercharge" $U(1)$ which was just shown to become a center (generator commuting with all others) completely.
Without the $P$ which stands for "projective", the bosonic subgroup of $SU(2,2|4)$ would really be $SU(2,2)\times SU(4)\times U(1)$ with the extra last factor that actually gets eliminated in $PSU$.
answered May 19, 2012 at 19:51