Timeline for Supergroup action on $AdS_5\times S^5$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2019 at 9:35 | history | edited | Bob Knighton | CC BY-SA 4.0 |
added 23 characters in body; edited title
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| May 19, 2012 at 19:53 | vote | accept | twistor59 | ||
| May 19, 2012 at 19:51 | answer | added | Luboš Motl | timeline score: 4 | |
| May 19, 2012 at 19:50 | comment | added | twistor59 | Ah, OK maybe P is for "projective" then. | |
| May 19, 2012 at 19:46 | comment | added | Luboš Motl | 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 (dimension of the fundamental rep) are equal. And it's equal, both are 4 for $PSU(2,2|4)$. | |
| May 19, 2012 at 19:43 | comment | added | twistor59 | @LubošMotl: OK thanks very much, that's clear! Post it in an answer and we can close this one off. | |
| May 19, 2012 at 19:34 | comment | added | Luboš Motl | For the terminology of superalgebras (and the same supergroups), look at page 58 of this Kac's review (PDF): projecteuclid.org/… | |
| May 19, 2012 at 19:28 | comment | added | Luboš Motl | 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. | |
| May 19, 2012 at 18:57 | comment | added | twistor59 | 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.... | |
| May 19, 2012 at 18:13 | comment | added | Luboš Motl | 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. | |
| May 19, 2012 at 17:04 | history | asked | twistor59 | CC BY-SA 3.0 |