### Re: This Week’s Finds in Mathematical Physics (Week 254)

Here are some questions Christine Dantas raised on Physicsforums, and some attempted answers. Since they’re a bit technical, I prefer to discuss them here, where I think experts are more likely to help us out.

Christine wrote:

Some basic questions.
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).

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 $exp(i k S)$ where $k$ is some coupling constant. The consequence is clear: $exp( i k S)$ remains unchanged under large gauge transformations if and only if $exp(i k C) = 1$, 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!!!

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 Schelleken’s paper — 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 roots of unity. In other words, it’s “modular invariant”.

These two are very important in Witten’s paper.

Third, as another consequence, the Schwinger functions, otherwise known as “$n$-point functions”, are all well-defined meromorphic functions — 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 big 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
(which is the $j$ function minus 744).

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. I mention some problems in week254, and Jacques Distler has mentioned some more.

Witten 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.

### Re: This Week’s Finds in Mathematical Physics (Week 254)

I know there are known relations between exceptional objects, but is there any other theory outside of model theory which goes some way to explain systematically the relationship between classifications in terms of families and sporadic entities? I was speaking recently with Alexander Borovik of Mathematics Under the Microscope, and he mentioned some work along these lines of the model theorists, Lachlan and Cherlin.

Here’s the opening paragraph of a paper by Cherlin – Sporadic homogeneous structures:

When classification results are enlivened by the appearance of uninvited guests in the form of
“sporadic” objects, those who take an interest in these interlopers may be tempted to account for
them in various ways, possibly by viewing them as coming from infinite (perhaps even continuous)
families of more general objects which may be natural from some broader point of view. In pure model theory, Lachlan’s classification theory for finite homogeneous relational structures provides a relatively well understood illustration (or “toy model”, if you will) of this sort of thing. This theory, which will be reviewed below, provides an infinite number of classification theorems of a
general character for combinatorial structures with rich automorphism groups, parametrized by certain bounds on the complexity of the structures. Any finite structure will actually appear at
some stage in one of these classifications, and may well occur as a sporadic structure initially; in the long run, every sporadic structure winds up belonging to a family parametrized by numerical invariants; at any given stage, only finitely many structures occur as sporadics; and finally, one will never “move beyond” the sporadics: we will always encounter new structures making their appearance as (temporarily) sporadic structures.

### E(3|6)

Alas, E(3|8) gets the hypercharges of some fermions wrong. Larsson seems to say this problem also occurs for E(3|6), which would appear to contradict what Kac claims, but I could be misunderstanding.

On page 17-18 of math.QA/9912235, we have

Theorem 3. [KR] The complete list of irreducible degenerate E(3|6)-modules is as follows (m, b in Z+):

(0m, b,-b-2/3m-2), (0m, b, b-2/3m), (m0, b,-b + 2/3m), (m0, b, b+2/3m+2).

In particular, the list contains (01, 0, Y), where Y = -8/3, Y = -2/3.

If we now look in Table 1 on page 19, we see that the second value fits d_R, but Kac uses the value Y = 4/3 for u_R, which is physically correct but does not fit in the list in theorem 3. Similarly, (10, 0, Y) is in the list if Y = 2/3, Y = 8/3, whereas \tilde u_L has Y = -4/3.

## Re: This Week’s Finds in Mathematical Physics (Week 254)

In case people missed it, let me repeat my recommendation of Helgason’s (free!) book on the Radon Transform.