Edward Witen is not researching LQG-like quantum gravity

[Mike2]

I think it's one of the deepest mysteries of our universe. It's easy to understand in certain cases, at least if the universe needs to obey the laws of logic at all. But, why should comprehensible mathematical laws govern so much of physical reality? Do we really need to be made of particles that are representations of Lie groups, for example? I see no obvious reason why this needs to be true: it's just an empirical observation of a remarkable fact. One might go so far as to say it's the only real miracle in nature - apart from the fact that the universe exists at all, which is an even bigger miracle.

If it turns out that physics can be derived from logic, then I think it would not be such a mystery why a language developed from logic (mathematics) would be useful in describing reality. Both are derived from logic.

 

[ccdantas]

Great. The review articles by Terry Gannon cited in Witten's paper are very good, but http://arxiv.org/abs/math/0109067" may be the most fun to start with.

Thanks! And thanks for your explanation. Quite interesting!

If 3d quantum gravity is related to the Monster group as Witten argues, it would be an incredible step forwards to understanding this puzzle.

Right. Now I see your point.

I again urge you to stop seeking short-term physics applications for Witten's new work.

A natural attitude for a non-mathematician, no? Being an astrophysicist in the first place drives me to think more in physical terms. In any case, I was not seeking short-terms applications of Witten's new work, but just trying to understand possible implications for future research.

Luckily I care about many other things... so I can enjoy what Witten did.

I think mathematics is, by far, less frustrating than physics. Actually I would love to turn myself into a mathematician, but the process would be too frustrating as well (!), so I limit myself to learn what it's possible to be learned in my lifetime.

BTW concerning the big mystery of why the physical world can be described by mathematics, that is certainly the most fundamental issue of all. And if physics is derived from logic (as Mike2 suggests, and would be great if he shows a proof that is so), one would have to end up with the same mystery: why the physical world can be described by logic. For me, the issue seems not to have progressed much and I would even say that it still revolves around a Kantian metaphysics on the basis of the human intellect. There is no obvious way to approach the question of the correspondence of the physical world (up to simplifying assumptions) to our mathematical internal formulations from a scientific point of view. There is still too much to be learned.

 

[ccdantas]

the space of all ways to make a torus look locally like the complex plane is a sphere.

I'm reading the paper you have mentioned, it's quite interesting, though I needed some help from Nakahara to get some basic points. Thanks!

 

[john baez]

Some basic questions.

I've tried to give a general sketchy introduction to Witten's paper in http://math.ucr.edu/home/baez/week254.html" - you might look at that.

Further, what is "holomorphic" factorization? (A pointer to the basic literature on this will suffice).

I don't really understand that term. It should be defined in http://arxiv.org/abs/hep-th/9205072" --- this paper speaks of "meromorphic conformal field theories" instead of "conformal field theories with holomorphic factorization", but they must be the same thing. However, I'm having a bit of trouble finding the precise definition! I just know a bunch of properties of these theories.

First, the central charge c is an integer multiple of 24.

Second, as a consequence, the partition function is really a well-defined number, not just defined up to (24/c)th root of unity. In other words, it's "modular invariant".

These two are very important in Witten's paper.

Third, as another consequence, the http://en.wikipedia.org/wiki/Schwinger_function" --- that is, holomorphic except for poles. This is not so important in Witten's paper, though.

Is it the only possible constraint?

Witten gives an argument that 3d quantum gravity has as its AdS/CFT dual a conformal field theory with c = 24k for some integer k = 1,2,3,... The main
nice thing is that - modulo a certain conjecture - Schellekens classified these conformal field theories for k = 1.

He argues that the (naive) partition function Z_0(q) differs from the "exact" Z(q) by terms of order O(q). Would this be correct for any k?

Yes, he argues this is true for any k. Then, around equation (3.13), he shows that this property, together with modular invariance of the exact partition function, completely determines the exact partition function! It's a certain explicit polynomial in the J function.

He finds that for k=1 the monster group is interpreted as the symmetry of 2+1-dimensional black holes. How sensitive is this result with respect to the value of k, and to respect to the other assuptions used in the derivation?

For k=1 he goes through Schelleken's list of 71 conformal field theories with c = 24 and picks the one that has the Monster group as its symmetries. He gives an argument for why this one is the right one, but it's not airtight.

He doesn't actually find the relevant conformal field theories with c = 24k for
higher values of k. He just figures out their supposed partition functions. Since the coefficients of their partition functions are - just as in the k = 1 case - dimensions of representations of the Monster group, it seems awfully plausible that these theories (if they really exist!) have the Monster group as symmetries.

However, this is something one would want to check. Nobody seems to know a c = 48 theory with Monster group symmetries, for example.

I will copy your questions and my answers to the http://golem.ph.utexas.edu/category/2007/07/this_weeks_finds_in_mathematic_15.html" , and hope some experts on conformal field theory (like Urs Schreiber and Jacques Distler) can help us out.

 

[ccdantas]

Dear John Baez,

Thanks a lot. I'll go in more detail into what you have written and of course I'll read with great interest your new TWF and blog entry.

Over at http://egregium.wordpress.com/2007/06/29/witten-on-3d-quantum-gravity-and-the-monster-group/" , in special, I was thinking about the section "Holomorphic functions can be factored" of that article. Please let me know whether you think that is a right pointer or not. I'll add a link to the new TWF/n-Category Café entry over at my blog opportunely.

Thanks,
Christine