Classifying Rational Conformal Field Theories
Yesterday afternoon was quite a chilly day in London, the kind of day
when being crammed into a packed and warm lecture room below ground
level in the basement of Queen Mary college from where you can hear
the tube rattle by was quite an attractive prospect. So at three in
the afternoon yesterday that's where I and other London theoretical
physicists gathered to hear Terry Gannon talk about "The
classification of RCFTs".
First off, it gives me great pleasure to report to you that the "damn
book" is finished :) after five years of hard slog Terry's book,
Moonshine Beyond the Monster and available to buy from the 31st
August, 2006. Hurrah. There's an excellent documentary by Ken Burns on
the American Civil War that took longer to make than the war itself, I
have no doubt that it will take me inestimably longer to understand
this 538 page book than it took to write. Fortunately noone died in
the making of the book, to the best of my knowledge. For some history
of the Monster see Terry's Monstrous Moonshine: The First Twenty-Five
Years.
Terry described his approach to trying to classify Rational Conformal
Field Theories (you could look at Wikipedia for a brief definition of
a RCFT, or a much better idea might be to start learning about CFT
from scratch with Paul Ginsparg's Les Houches lectures, Applied
Conformal Field Theories or Krzysztof Gawedzki's Lectures on Conformal
Field Theory) by searching for invariants of the chiral algebra, or
Frobenius algebra, that underlies the RCFT. By way of comparison,
Terry said that the very succesful classification of the Lie algebras
rested upon the invariant of the Dynkin diagram. But what invariants
are worth considering, whose discovery will tell us most of the
information about the algebra? Terry suggested two:
modular invariants (i.e. partition function on the torus)
NIM representations (i.e. partition function on the cylinder) But he
only had enough time to talk a little about the first and describe to
us the modular functions that appear.
To commence one must settle upon a chiral algebra, or a vertex
operator algebra, and Terry told us that some very nice choices are
the affine Kac-Moody algebras (see Fuchs' Lectures on conformal field
theory and Kac-Moody algebras section 16 for the definitions). A
level, k, must also be picked. We were told that one way to imagine a
chiral algebra is as a complexification, or 2-dimensionalisation, of a
Lie algebra. If we denote all the objects appearing in a Lie algebra
by a tree diagram, having all the properties of the Lie bracket at the
branch (i.e. antisymmetric...) then the complexified version of the
algebra turns each of the branches of the tree diagram into a
cylinder: For more about this way of complexifying to get loop
algebras we were referred to the work of Yi-Zhi Huang, in particular
his book Two-Dimensional Conformal Geometry and Vertex Operator
Algebras.
Returning to the CFT, the Hilbert space is described by irreducible
representations of our affine algebra (left moving and right moving
copies) which for a given level k, are paramaterised by highest weight
labels. For the example of affine SU(2), the highest weights are
characterised by two labels ( , ) such that + = k. The Hilbert space
may be written as: Where M is the multiplicity, and the one-loop
partition function for this RCFT may be written in terms of the
characters, : It turns out that the characters are modular functions,
and are subject to the familiar S and T transformations: Furthermore,
the partition function is modular invariant and characterised by its
multiplicities, M.
At this point in the talk, Terry had about six minutes remaining and
had arrived at what he thought of as the start of his talk, and
defined the "modular invariant" he hoped to use to classify RCFTs:
Given some affine algebra at level k, a modular invariant is a
matrix M of multiplicities describing the partition function, Z,
such that,
Terry told us that these conditions gave rise to RCFTs that are "just
barely" classifiable.
Terry finally asked us why bother classifying? Or, in his words, "who
cares?" His answer was that the classification leads to interesting
results. What more could you want? He gave us the example from
Cappelli-Itzykson-Zuber from 1986 of the classification of affine
su(2), which is completely classified for the levels, k, 4/k, k/2 is
odd, k=10,16,28, and he told us a story he heard twice; once from
Zuber about a correspondence he had with Victor Kac, and a second time
the same story from Kac - so, he said, it must be a true story. It
went like this: After having written down some of the classifications
of affine su(2) in 1986, Zuber wrote to Kac about the results, who
replied and pointed out the classification for k=10, which he said
contained some exceptional numbers - literally numbers he thought came
from the exceptional group E_6. Zuber said he didn't understand Kac
nor pay it much heed until someone else repeated it years later and he
dug out the letter, headed to the library and confirmed that all the
numbers appearing in the classification do indeed have an intimate and
mysterious (to this day...) relation with the groups A, D, E, and the
symmetries of their Dynkin diagrams. At this point Terry bemoaned the
fact that God was manifestly not benevolent since he insisted on
making 2 a prime number...Terry's discomfort with 2 didn't seem
justifiable until later on when he mentioned that his wife is
expecting twins (excuse me for this weak pun) so I just put two and
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