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Re: orbifold - what is it and how is it used

```Subject: Re: orbifold - what is it and how is it used
From: baez@galaxy.ucr.edu (john baez)
Date: 21 Jan 1999 00:00:00 GMT
Approved: bunn@leporello.berkeley.edu (sci.physics.research)
Newsgroups: sci.physics.research
Organization: University of California, Riverside
References: <784tjl\$ccb\$1@nnrp1.dejanews.com>

In article <784tjl\$ccb\$1@nnrp1.dejanews.com>, <teds@intex.com> wrote:
>As a matter of interest, there are some introductory lectures on string

>theory at www.fys.ruu.nl. While I was reading one of them, they
mentioned >orbifolds. I was wondering if someone could describe what an
orbifold >is and how it's useful (in string theory and beyond).

An orbifold is a slight generalization of a manifold. I forget the
technical definition - which is often presented in a highly terrifying
manner - but a good example of an orbifold is a manifold modulo a finite
group actions. Does that mean anything to you? If not, an example of the
example might help! Each Platonic solid has a finite group G of
symmetries. Let's not consider reflections, just rotations, so that G is
a subgroup of the rotation group SO(3). For the tetrahedron G is a
12-element group, for the cube or octahedron it's a 24-element group,
and for the dodecahedron or icosahedron it has 60 elements. Now let's
take R^3 and decree two points (x,y,z) and (x',y',z') to be equal if we
can get from one to the other by a rotation in G. In math jargon we get
"R^3 modulo the action of G". The resulting space is not a manifold; for
example, it has a singularity corresponding to the point (0,0,0). But
it's an orbifold!

These particular orbifolds have fascinating relationships to a vast
number of subjects - for more, try:

M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The
ubiquity of Coxeter-Dynkin diagrams (an introduction to the ADE
problem), Niew. Arch. Wisk., 25 (1977), 257-307

and also

V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in
Mathematical Analysis and Catastrophe theory from Evolvents to
Quasicrystals, translated from the Russian by Eric J. F. Primrose,
Boston, Birkhauser Verlag, 1990.

Orbifolds are quite important in string theory. We often think of of
string theories as dealing with maps from a surface (the string
worldsheet) into a manifold representing spacetime. But it's also
important to study maps from a surface into an orbifold. For example, in
26-dimensional bosonic string theory, spacetime can be a 26-dimensional
torus - not just any old 26-dimensional torus, just tori of certain
special shapes - and a torus is a manifold. But you can also let
spacetime be a 26-dimensional torus modulo a finite group action - an
orbifold! If you do this just right, you get a marvelous string theory
whose symmetry group is the Monster group - the largest sporadic finite
simple group, which has

808017424794512875886459904961710757005754368000000000

elements. Richard Borcherds recently won a Fields medal for his work on

W. Wayt Gibbs, Monstrous moonshine is true, Scientific American,
November 1998, 40-41. Also available at:
http://www.sciam.com/1998/1198issue/1198profile.html

A less spectacular but more practical fact is that all the so-called
"minimal models" - roughly, basic string theories from which more fancy
ones can be built - can be obtained from theories in which the string
worldsheet is mapped into an orbifold. A good introduction to this is
the book:

Phillippe Di Francesco, Pierre Mathieu, and David Senechal, Conformal
Field Theory, Springer, 1997.

More generally, orbifolds turn out to be important all over mathematics,
because manifolds are not general enough - it's often nice to let your
spaces have some singularities of a reasonably controlled sort. For
example, recently I posted an article about moduli spaces and said that
moduli spaces often fail to be manifolds due to singularities. Sometimes
they are orbifolds!
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