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15-j sybmols

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Igor Khavkine
Oct12-06, 05:01 AM
P: n/a wrote:
> Hi Everyone,
> I was hoping that some expert ;) may enlighten me on this issue.

I don't know if I qualify as an expert, but I did spend a bit of time
looking at the paper by Crane, Kauffman and Yetter (hep-th/9409167).

> I've been reading a lot about the Crane-Yetter TQFT lately, and it
> seems all constructions of it (or of isomorphic TQFT's) use some
> ordering of vertices at an intermediate step. The question is Why?
> Is this because the graph corresponding to a 4-simplex is not
> embeddable in 2d without self intersections? So that if we just
> embedd it randomly in some way, the diagramme (number) corresponding
> to it will be different from the one obtained by embedding it some
> other way.

Basically yes. an ordering on the vertices allows one to canonically
construct the 15j symbol with the appropriate intersections. The
intersections are important in the q-deformed case. Specifically, if
you look at the diagram on page 21 of the paper I referenced above,
it's construction (unfortunately rather opaquely) is described starting
with the last three paragraphs on page 22.

> If the answer is yes to above, why that particular convention is
> chosen? Are there any other consistent conventions for embedding the
> diagramme? (Just to show that i'm totally spoilt, why is the 4
> simplex, for example, put on the "right" of the 0 and the 2 simplex
> on the left? I'm talking about the diagramme in the paper by Crane
> and Yetter "A categorical construction of 4d topological quantum
> field theories")

Unfortunately, the version of the paper you mention that is on the
arXive (hep-th/9301062) doesn't have the figures. So I can't comment on
them. But I would guess that if you look at the figure on page 21 of
the paper I already mentioned, it's probably the same or a similar one.
Besides the sort-of natural construction of this diagram as a
projection of a 4-symplex onto a 2D plane, its main merit is the fact
that it allows the authors to prove the invariance of their state sum
under change of triangulation. This is proved in a sequence of
diagramatic lemmas in the subsequent pages.

> Finally, I've seen MANY books, papers, articles ,etc., discussing the
> 6-j symbols, their relationship to the tetrahedron, identities among
> them etc., but have never seen ANY book which discusses the 15-j
> ones. Can anybody point to a reference which does, and ARE there
> similar identities in the 15-j case (Biedenharn-Elliot,
> orthogonality, etc..)

The reason the 6j symbol is often discussed is because of its relation
to recoupling theory. Diagramatically, the 6j symbol gives the
coefficients that allow us to rewrite

\ / \ /
\____/ as linear combinations of \ / .
/ \ |
/ \ |
/ \
/ \

In other words, you can view it as a sort of change of basis for the
space of intertwiners between pairs of representations of SU(2). Most
of the identities and the 6j's evaluation in terms of the tetrahedral
network comes from this recoupling formula. As far as I know, the 15j
symbol is not associated with any such recoupling formulas. Thus, if
there are identities or recurrence relations for the 15j, they are not
as easy to discover. Mostly, it's just a network with 15 edges that
gives the right sort of amplitude in the Crane-Yetter topological state
sum. That's its most important property.

Hope this helps.


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