NetNews Usenet Archive 1992 #30/NN_1992_30.iso/spool/sci/math/17231

Internet Message Format | 1992-12-21 | 2.8 KB

Path: sparky!uunet!olivea!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
From: [e-mail] (Allan Adler)
Newsgroups: sci.math
Subject: Vertex Operator Algebras
Message-ID: <[e-mail]>
Date: 20 Dec 92 13:18:38 GMT
Sender: [e-mail]
Distribution: sci
Organization: M.I.T. Artificial Intelligence Lab.
Lines: 47


From time to time, I pick up the book on Vertex Operator Algebras and
the Monster, by Frenkel, Lepowski and Meurman, and while the results
are admirable, I am nevertheless repelled each time by the notation
and the exposition. Apart from such discourtesies as changing their
notation and conventions in the middle of a book whose  technical details
would be hard enough to follow with consistent notation (something like
Grothendieck changing the meaning of the word scheme in the middle of
writing EGA), the authors often remind us that they are not giving
us the general picture (suggesting that the carefully learned technical details
will have to be unlearned later, either in the same book or in other sources).
The overview is essentially lacking while instead
the authors torture one kind of quadratic form after another, assuming that
the reader will get the general idea of how to proceed without a general
definition of the torturous procedure. Maybe it is the fate of books
on such technical material to be like that but I am not convinced.

There are two hints of a more general setting at the end of the book and
I would like to know where this generalization is carried out, in the
hope that it is done more conceptually, more simply and in a way that
is easier for me to read.

p.469, Remark A.2.3 "In this case, analytic continuation yields a family
of operators parametrized by a configuration of points on a Riemann surface."
(No reference is given)

p.481 Remark A.3.4 "Using the rule (A.3.10) it is not difficult to show
generally that for any homologically trivial sum of configurations of
contours one obtains a formal variable identity."


The first hint suggests that we need a notion of vertex operator living on a
compact Riemann surface or a complex manifold. The second one suggests that we
get relations among vertex operators from 1-cycles on the manifold.


The first hint reminds me of a lecture that Beilinson gave at Harvard,
in which he took compact Riemann surfaces and had a Virasoro algebra
at each point of the Riemann surface and associated an algebra to finite
subsets of the Riemann surface by taking products of the algebras at the
points of the finite set. He then considered functors between the categories
of modules of the algebras associated to different finite sets. But beyond
the vague similarity, I don't know if what he was talking about is really
pertinent to the hints given above.

Allan Adler
[e-mail]