Archive for June, 2007
I’ll take some time to figure this out , so for the moment I’ll be collecting here some links of interest.
Witten’s paper and talk at Strings 07 about 3D Quantum Gravity. [I've posted about this here, where I also indicate links to blogs that are discussing the talks over at Strings 07, so you can send comments to that post as well].
Wikipedia article on the Monster Group.
MathWorld article on the Monster Group.
Baez’s This Week’s Finds # 66, where he gives an introduction to the Monster Group.
Solomon’s paper On Finite Simple Groups and Their Classification.
A paper by T. Gannon: Monstrous Moonshine: The first twenty-five years.
- Sporadic Groups by Aschbacher.
- Atlas of Finite Groups by Conway.
- Geometry of Sporadic Groups by Ivanov.
- Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics by Gannon (added later).
- Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics by Ronan; non-technical account (added later).
Update: Here are some very basic questions that I have posted over at PF. Answers are very welcomed.
1. Why is k “an integer for topological reasons”? (k is a parameter that appears in a second term — a multiple of the Chern- Simons invariant of the spin connection — added to the action).
2. Further, what is “holomorphic” factorization? (A pointer to the basic literature on this will suffice). Is it the only possible constraint?
3. 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?
4. 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?
Update on question #1: Here is the answer by John Baez (as posted over at PF) –
The Chern-Simons action S is invariant under small gauge transformations (those connected to the identity by a continuous path), but changes by multiples of a certain constant c under large gauge transformations. What shows up in path integrals is the exponentiated action exp(ikS) where k is some coupling constant. The consequence is clear: exp(ikS) remains unchanged under large gauge transformations if and only if exp(ikc) = 1, meaning that k has to be an integer multiple of 2 pi / c.
If you set up all your normalization conventions nicely, c = 2 pi, so k has to be an integer.
This stuff is explained a bit more in my book Gauge Fields, Knots and Gravity, in section II.4, Chern-Simons Theory. Also see the end of section II.5.
In 3d quantum gravity, the consequence is that the cosmological constant can only take certain discrete values!!!
It’s quite clear now! It was the phrase “topological reasons” in his talk that seemed mysterious to me.
Update: Question # 2 can be elucidated in this Wikipedia article on the Weierstrass factorization theorem.
Update: There is of course, Lieven le Bruyn’s excellent blog, formely known as “Neverendingbooks”, but recently reformulated into MoonshineMath, focused on the Monstrous Moonshine.
Update: John Baez also indicates this paper by Gannon.
These two important conferences on high energy theoretical physics and quantum gravity are happening this week. The websites are:
Some bloggers/forums commenting on these conferences:
If you would like to comment specifically on the recent Witten’s paper on 3D Gravity (and his related talk at Strings 07), you are welcomed to do so, but only technical comments will be allowed. All comments are subject to moderation. Thanks!
Update: A fascinating discussion on foundational physics, theoretical versus mathematical physics, and the role of mathematics in physics, spinning off from a post about Witten’s paper over at n-Category Café, is worth reading.
Update: Sabine (Backreaction) has a nice (but non-technical) report from the Loops 07 Conference.
Update: Alejandro Satz over at Reality Conditions has two nice reports on Loops 07.
I have just learned (via Peter Woit’s blog) about a recent article in Wikipedia, written by Christina Sormani, concerning the Poincaré Conjecture and its solution, given by Griori Perelman and Richard Hamilton (Columbia University).
The article is quite pedagogical and contains excellent graphics, with pointers to other articles, on-line lectures and websites.
Science is the human intellectual reaction to Nature’s actions (physical phenomena), so that Nature’s reaction to human actions (experiments or observations) are known or can be predicted.
You can propose your own definition in the previous post.
Given the current “string theory debate” going around the blogosphere (I will not link to any specific sites here), one important issue that often comes about is whether string theory is science or not. There are well known and polemic claims that string theory makes no falsifiable predictions and hence it is not science.
You can pick up a dictionary and look at the canonical definition of the word “science”. For instance, my Apple dictionary says:
Science — the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment .
And here is a Webster’s entry:
Science — knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method.
I’m certain you will find other entries with somewhat different nuances, but the underlying core certainly involves the “scientific method“.
Do you think that the frontiers of physical knowledge got to the point that there is no way to proceed under the “good old” scientific method? Do you think that the notion of science should be revised? What is your notion of science?
I’m curious about the opinion of readers on this matter. You are invited to write here — up to 50 words — what is your definition of science. And you can submit your comment even if you agree with the current definition, but think that you have a better “dictionary entry” than those above.
Important note: I will not be accepting comments on string theory itself.