Re: Lie algebra cohomology

[Jacques Distler <distler@golem.ph.utexas.edu>]

In article <970v75$lgv$1@news.state.mn.us>,
 Thomas Larsson <Thomas.Larsson@hdd.se> wrote:

>> But you still haven't explained why you think these algebraic structures 
>> might be RELEVANT to string theory, M-theory, or the pricing of pork 
>> belly futures. "They were unknown in 1984" is not much of an argument.
>
>I do not think that they are relevant to string or M-theory, because I
>do not believe that string and M-theory are relevant to physics. But I do
>claim that these structures are deeper and more general than anything
>seen in string theory.

Deeper than Elliptic Cohomology?
Deeper than the Arithmetical properties of the "Attractor Mechanism"?  
Deeper than the Batyrev-Borisov/Strominger-Yau-Zaslow conjectures on 
"Mirror Symmetry"?
Deeper than the Seiberg-Witten solution of Donaldson theory?
Deeper than . . . ?

Not that any of these are necessarily relevant to physics but they are 
mathematically deep, in anyone's definition of "deep".

[Now, the AdS/CFT correspondence, *that's* something which is incredibly 
deep *and* definitely relevant to physics. How does it stack up relative 
to E(3,8) ?]

>The grade zero subalgebra of E(3,8), which largely governs the 
>representations,
>is sl(3)+sl(2)+gl(1). I think it is quite remarkable that the sole requirement 
>of maximal depth immediately leads to the symmetries of the standard model, 
>without any arbitrary symmetry breaking, compactification, magic, or mystery.

Come back when you have derived the entries in the Kobayashi-Maskawa 
matrix from E(3,8). 

The Standard Model, even to those of us who know and love her, is an 
ugly hack. 

Moreover, the unification of coupling and the prediction of sin^2 of the 
Weinberg angle is striking evidence that there is more going on than 
SU(3)xSU(2)xU(1).

So I think I speak for a great many theorist who would be surprised and 
disappointed if the Standard Model were the end of the story and if, 
moreover, there was a mathematical "proof" of that fact.

>Kac has appearently been talking about the connection between the exceptional 
>Lie superalgebras and the standard model for over a year. Now, Viktor Kac 
>has some rather solid scientific credentials, including a Wolf prize in 
>physics, and you might expect that if the world's strongest algebraist 
>says the words "exceptional algebraic structure" and "standard model" 
>in the same sentence, people would notice. Nonetheless, he told me that 
>I was the first physicist to take him seriously (Sept. 2000). If the physics 
>community doesn't listen to such a person, who would they listen to?

Unfortunately, Alain Connes (the Fields Medalist) got there first, and 
have been claiming to be able to "derive" the Standard Model for over a 
decade now. I think he even claims to understand the value of sin^2 of 
the Weinberg angle (though I'm not sure anyone has told him that it's 
*not* equal to 3/8).

So if Viktor Kac wants to get attention for *his* "derivation" of the 
Standard Model, he will have to do Connes one better and compute the KM 
matrix (or, at least, he could make a prediction for the Higgs mass -- 
it's still open season on that number).

Anyway, for the reasons stated above, I am not waiting with bated breath 
for such a derivation.

>2. In quantum theory, symmetries are only represented projectively.
>On the Lie algebra level, this means that the symmetry algebra acquires
>an extension (provided that the algebra is big enough).

Projective representations are usually associated to the symmetry 
suffering an anomaly. I would hope that the diffeomorphisms do not 
suffer an anomaly.

>I guess that nobody will be really surprised that I agree with everything 
>said in the Woit paper (http://www.arxiv.org/abs/physics/0102051).

I found Woit's paper hilariously funny.  But, as any joke is diminished 
an attempt at explication, I will refrain from trying to explain *why* 
it was so funny. Paul Shockley makes some relevant comments elsewhere in 
this thread. You can probably figure it out from there.

Jacques

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