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In article <412c53c6\$1@news.sentex.net>, Urs Schreiber wrote: >John Baez wrote: >> When we think about connections in this context, we see the natural >> equation is not >> >> $B = -F$ >> >> (which makes no sense!) >> >> but instead >> >> t(B) $= -F$ >> >> where F is a g-valued 2-form and B is an h-valued 2-form. >> >> We can then consider various special cases. >> >> At one extreme we can have G trivial and H abelian; then the >> above equation is vacuous. This is what happens in 2-form >> electromagnetism. >> >> At another extreme we can have $G = H$ and t the identity; then >> you get $B = -F$. That's the case you seem to like best. >True, I like that best at the moment, because it seems to me that a very >interesting class of physical applications corresponds to $G = H = U(N)$. By the way, there are closely related ways to build a 2-group, that are easy to mix up. It would be unfortunate to think you were working with one when you were really working with the other! I'm not accusing you of doing this, but I just think I should warn you of the danger, since I'm gotten confused myself plenty of times. Here they are: 1) Take $G = H,$ let t: $H -> G$ be the identity homomorphism, and let $\alpha$ be the action of G on H be via conjugation: $$\alpha(g)(h) = ghg^{-1}$$ 2) Take G = Aut(H) be the group of all automorphisms of H. Let $t: H -> G$ be the map sending any element h to the corresponding "inner automorphism" of H, that is, the automorphism given by conjugating by h: t(h)(h') $= hh'h^{-1}$ Let $\alpha$ be the obvious action of G = Aut(H) as automorphisms of H. These two constructions agree when the map t in construction 2) is one-to-one and onto. This is the case for SO(3) but not for SU(n) or U(n), since these have a nontrivial center. >From a pure mathematical viewpoint, construction 2) is actually a lot simpler and more important than construction 1), even though it doesn't look simpler in the lowbrow way I just described it. >But I do understand that 2-group theory covers the more general case where G $>\neq H$. I think pretty much everything I wrote so far with $G = H$ in mind >directly translates to this more general case when you think of all my Bs as >t(B)s. But I will make that more precise and explicit in the future. Okay... it may be no big deal for your applications, but it can be... >Meanwhile, since I have the feeling that I didn't express myself well >enough, let me try to rephrase the question that I am currently concerned with: > >Alvarez, Ferreira&Sanchez-Guillen in http://www.arxiv.org/abs/hep-th/9710147 found two classes of >consistent surface holonomies which happen to have $G = H$ but $B + F \neq$ . Of course it's possible to have $G = H$ but t: $H -> G$ not equal to the identity! That would be one possible explanation of their work, since this can give $B + dt(F) =$ but $B + F =/=$ . However, instead of guessing, I should just read their paper. >1) It contains a flaw and 2-group theory is right that only $t(B)+F =$ gives >well defined surface holonomy. I should note that it's not "2-group theory" which makes this claim, but a paper by Girelli and Pfeiffer. And I'm not even sure they claim this in an ironclad way. Personally I'm very confused about all this stuff, so more confusion is actually a good thing $- it$ might disentangle something. >2) It is correct. Then there must be a reason why 2-group theory cannot >obtain these surface holonomies with t(B) + F nonvanishing. >I argued that the latter is the case (namely that there are more solutions >to the 2-associativity condition than just those with $t(B)+F=0),$ but if I am >wrong about that please let me know where I went astray. I don't know what the "2-associativity" condition is. I guess I'll just have to read their stuff. Of course I wouldn't mind a wee summary in plain English....