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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: firstname.lastname@example.org (john baez) Date: 21 Jan 1999 00:00:00 GMT Approved: email@example.com (sci.physics.research) Newsgroups: sci.physics.research Organization: University of California, Riverside References: <firstname.lastname@example.org> In article <email@example.com>, <firstname.lastname@example.org> 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 this subject - for more information on this, try: 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!