SU(3)xSU(2)xU(1) is special!
I was always puzzled by the fact that nature chooses this particular combination of symmetry groups. Today Victor Kac showed an incredible way of realizing this symmetry in his talk at CTP. Let me try to summarize my humble understanding.
He managed to extend Cartan's classification of simple Lie algebras to superalgebras some years ago. Here instead of 4 (parametrized) series of algebras one gets 10 series, plus only 5 exceptional ones. You can understand these exceptional ones in terms of their maximal subgroups. Two of them have SU(3)xSU(2)xU(1) and one has SU(5). And if you believe in SU(5) unification you are left with a unique choice since only one of the preceding ones can be embedded in the latter. We can name them now: E(3|6) and E(5|10).
He and Rudakov studied the representations of E(3|6) and the result resembles nature surprisingly much. There are three families of fermions, five families of quarks and the same gauge sectors but no superpartners. Though there are important problems like anomaly cancellation issues due to the mismatch of lepton and quark families, weird neutrino sector, charged gluons and so forth.
Despite its immature physics right now, I found it extremely compelling, because this is the first place standard model appears as a special structure in mathematics. Also there are naturally occurring families. Most important question now is how to write Lagrangians with these symmetries.
Results are new even for mathematics community and I don't think that they find their way into physics literature. Most accessible introductions by Kac are math.QA/9912235 and math-ph/0302016. Also there is a rather diverse discussion in sci.physics.research. And finally since it is my specialty, I can tell you a place where you can watch a very similar talk by Kac along with some lecture notes.
Does anybody know any reasonable introduction for physicists? Feel free to leave comments and references.
He managed to extend Cartan's classification of simple Lie algebras to superalgebras some years ago. Here instead of 4 (parametrized) series of algebras one gets 10 series, plus only 5 exceptional ones. You can understand these exceptional ones in terms of their maximal subgroups. Two of them have SU(3)xSU(2)xU(1) and one has SU(5). And if you believe in SU(5) unification you are left with a unique choice since only one of the preceding ones can be embedded in the latter. We can name them now: E(3|6) and E(5|10).
He and Rudakov studied the representations of E(3|6) and the result resembles nature surprisingly much. There are three families of fermions, five families of quarks and the same gauge sectors but no superpartners. Though there are important problems like anomaly cancellation issues due to the mismatch of lepton and quark families, weird neutrino sector, charged gluons and so forth.
Despite its immature physics right now, I found it extremely compelling, because this is the first place standard model appears as a special structure in mathematics. Also there are naturally occurring families. Most important question now is how to write Lagrangians with these symmetries.
Results are new even for mathematics community and I don't think that they find their way into physics literature. Most accessible introductions by Kac are math.QA/9912235 and math-ph/0302016. Also there is a rather diverse discussion in sci.physics.research. And finally since it is my specialty, I can tell you a place where you can watch a very similar talk by Kac along with some lecture notes.
Does anybody know any reasonable introduction for physicists? Feel free to leave comments and references.
