Witten on 3D Quantum Gravity and the Monster Group
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.