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# Topic: Orbifold

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 Orbifold - Wikipedia, the free encyclopedia Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one stratum corresponds to a set of singular points of the same type. For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a coset of R Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements of G have been identified with the identity. en.wikipedia.org /wiki/Orbifold   (762 words)

 Crystallographic Topology 101 - Orbifold 1 Orbifolding is simply the operation of wrapping, or folding in the case of mirrors, to superimpose all equivalent points. There are times when the orbifolding process itself is important, particularly when we are discussing covering spaces, since in that case we may need to unfold the orbifold partially to obtain some other orbifold or fully unfold it to obtain the original space (i.e., the universal cover). The bottom symbol under each orbifold drawing is the international short crystallographic notation for the point group from which the orbifold is derived, with overbars and m's denoting inversion centers and mirrors, respectively, and with 2, 3, 4, and 6 describing the order of rotation axes. www.ornl.gov /ortep/topology/orbfld1.html   (2624 words)

 Mathematics of The EPINET Project Technically, the orbifold is the quotient space of the pattern by its symmetry group, and the pattern is the universal cover of the orbifold. In the orbifold a reflection produces a boundary component, a rotation induces a cone point, while translations and glides that do not arise from other symmetries of the pattern are encoded by global topological features such as rings, handles, and crosscaps. A 2D orbifold is a 2-manifold with a finite number of punctures, corner, and cone points, so a compact description of these features can encode any discrete symmetry group of the sphere, euclidean, and hyperbolic planes. epinet.anu.edu.au /mathematics/orbifold_notation   (1310 words)

 Crystallographic Topology - Orbifold 2 The spherical orbifolds in the third row are derived by using straight line cuts through 2-axes and appropriate angular cuts at other axes to leave some flaps which are then glued together to produce the 4- and 3-cornered pillow spherical orbifolds. The remaining diskal orbifolds (rows one and two) are derived by cutting along the double lines and along appropriate angles through the single axis pointed to by vectors perpendicular to the ends on double lines, then closing up the cut edges through the axis to form a complete silvered boundary. Much of the orbifold topology literature (e.g., Bonahon and Siebenmann, 1985) uses a Euclidean 2-orbifold as the base orbifold, which is lifted into a Euclidean 3-orbifold using the Seifert fibered space approach (Orlik, 1972) while keeping track of how the fibers (or stratifications) flow in the lifting process. www.ornl.gov /ortep/topology/orbfld2.html   (1387 words)

 Xah: Wallpaper: The 17 Wallpaper Groups To completely understand orbifold and orbifold notation, a background in topology is required. The revolutionary feature of orbifold notation is that it uses topology to explain symmetry, and results a more geometric understanding than groups. Roughly speaking, an orbifold is the quotient of a manifold by a discrete group acting on it. xahlee.org /Wallpaper_dir/c5_17WallpaperGroups.html   (672 words)

 Moerdijk on Orbifolds, II | The String Coffee Table In section 5 Lie groupoids are introduced and the relation to orbifolds is briefly discussed in 5.6. In the physics literature, this special case tends to be regarded as the definition of an orbifold. But, in order to study orbifolds as global objects together with maps between them (and possibly with extra stuff and structure, like for instance fiber bundles, over them) the most convenient language is that of Lie groupoids. golem.ph.utexas.edu /string/archives/000733.html   (1754 words)

 Orbifold String Topology: Paths in Smooth Categories | The String Coffee Table Motivated by parallel transport along paths in orbifolds as well as by the study of strings propagating on orbifolds, one would like to similarly understand paths and loops in orbifolds in terms of the representing groupoids. representing an orbifold, this concept refines the loop groupoid given by Lupercio and Uribe in that it suspends a 1-category of loops and cobordisms to a 2-category of points, paths and cobordisms. Regarding this groupoid as an orbifold (it is in fact the embedding groupoid of the trivial orbifold, as defined in section 3.5 of [1]), the cocycle conditions for a locally trivialized 1- or 2-bundle over this space are nothing but the equivariance conditions with respect to this “orbifold” [7, 8]. golem.ph.utexas.edu /string/archives/000735.html   (835 words)

 Conway's orbifold notation - Wikipedia, the free encyclopedia The orbifold notation is a mathematical notation invented by the mathematician John Horton Conway. The advantage of the notation is that it describes these groups in way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. The corresponding orbifold is orientable in the chiral case and non-orientable otherwise. en.wikipedia.org /wiki/Conway's_orbifold_notation   (713 words)

 Princeton University Traditionally, orbifolds were studied as an extension of the theory of smooth manifolds. The motivation of the new theory is from orbifold string theory. The core of the new theory is a new cohomology of orbifold (orbifold cohomology) introduced by Chen-Ruan. www.math.princeton.edu /~seminar/99-2000-sem/4-26-2000weekly.htm   (517 words)

 Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups - Bryan, Fulman ...   (Site not responding. Last check: 2007-11-07) Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups (1998) We define a natural sequence of "orbifold Euler characteristics" for a finite group G acting on a manifold X. Our definition generalizes the ordinary Euler characteristic of X=G and the string-theoretic orbifold Euler characteristic. Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups. citeseer.ist.psu.edu /9396.html   (407 words)

 Re: orbifold - what is it and how is it used   (Site not responding. Last check: 2007-11-07) However, it is an orbifold, and the map p: M -> M/G is still interesting. Think about the example I gave last time, where M is R^3 and G is the group of rotational symmetries of a Platonic solid. Orbifolds, on the other hand, are a bit new - among physicists, it seems to be string theorists who spend the most time talking about orbifolds. www.lns.cornell.edu /spr/1999-01/msg0014223.html   (498 words)

 Research We show that our orbifold genus is given by the same sort of formula as the orbifold ``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable genus to define an analytic equivariant genus in Grojnowski's equivariant elliptic cohomology, and we show that this gives precisely the orbifold two-variable genus. In the summer of 2003 I gave three plenary lectures at the ``International conference on algebraic geomety and topology'' at the Australian National University, and I gave a lecture at the ``Kinosaki international conference on algebraic topology'' in honor of the 60th birthday of Professor Goro Nishida. www.math.uiuc.edu /~mando/node1.html   (894 words)

 Orbifold Pinball   (Site not responding. Last check: 2007-11-07) Orbifold Pinball is a game in which you roll a ball on an unusual playing field. In this report, the game board is revealed as an `orbifold', and the bumpers become `cone points of order 2'. For a gentle introduction to orbifolds, symmetry groups, and the ties between them, see the Geometry and the Imagination course notes available on this Web server. www.geom.uiuc.edu /apps/pinball/about.html   (336 words)

 Ruan Abstract   (Site not responding. Last check: 2007-11-07) There is an emerging new subject of mathematics, which we call "Stringy Geometry and Topology of Orbifold." The motivation of stringy geometry and topology of orbifold comes from orbifold string theory discovered by physicists Dixon, Harvey, Vafa and Witten more than fifteen years ago. However, the classical theory of orbifold is basically an extension of theory of smooth manifold. Orbifold string theory model is a popular model in string theory. www.math.uiuc.edu /Colloquia/01SP/ruan_apr19-01.html   (166 words)

 Re: orbifold - what is it and how is it used   (Site not responding. Last check: 2007-11-07) On 21 Jan 1999, john baez wrote: > 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. I am sorry to always bring coverings and stuff into it, but it is exciting to me to see these connections with physics. www.lns.cornell.edu /spr/1999-01/msg0014210.html   (228 words)

 The Euler characteristic of an orbifold It is important to keep in mind the distinction between the topological Euler characteristic and the orbifold Euler characteristic. The quotient orbifold of for any symmetry pattern in the Euclidean plane which has a bounded fundamental region has orbifold Euler number 0. To complete the connection between orbifold Euler characteristic and symmetry patterns, we would have to verify that each of the possible configurations of parts with orbifold Euler characteristic 0 actually does come from a symmetry pattern in the plane. geom.math.uiuc.edu /docs/education/institute91/handouts/node35.html   (627 words)

 [No title] We introduce the notion of generalized orbifold Euler character* *istic as- sociated to an arbitrary group, and study its properties. Generalized orbifold Euler characteristics A generalization of physicists' orbifold Euler characteristic (1-2) was give* *n in the introduction in (1-3). When a loop passes through orbifold points, its inverse image is not unique, but finite, correspon* *ding to finitely many possibilities of different conjugacy classes of lifts. www.math.purdue.edu /research/atopology/Tamanoi/orbifold.txt   (7112 words)

 A field guide to the orbifolds These names are far from standard, and while they are unlikely ever to enter common use, we have found from our own experience that they are not wholly useless as a method for recognizing the patterns. The quotient orbifold of the diglide pattern is a projective plane with two cone points; the quotient of the monoglide patterns is a Klein bottle. Yet another clue is that in a monoglide you can often spot two disjoint Möbius strips within the quotient orbifold, corresponding to the fact that the quotient orbifold for a monoglide pattern is a Klein bottle, which can be pieced together from two Möbius strips. www.geom.umn.edu /docs/education/institute91/handouts/node39.html   (1332 words)

 orbifolds If one considers\na circle acting on a space, then the quotient is usually an orbifold. So\n\nif one had a 7-dimensional space, and a circle action with very small\ndiameter orbits, then the whole thing would look very much like a\n6 dimensional orbifold. a circle acting on a space, then the quotient is usually an orbifold. www.physicsforums.com /showthread.php?t=18768   (1328 words)

 IngentaConnect Orbifold resolutions and fermion localization   (Site not responding. Last check: 2007-11-07) orbifold, where the conical singularities are replaced by suitable spherical caps with constant curvature. This study shows how localized and bulk fermions arise in the orbifold as the resolved space approaches the orbifold limit. It is explicitly shown how a resolution of the orbifold puts severe constraints on the allowed chiralities and U(1) charges of the massless four-dimensional fermions, localized or not, that can be present in the orbifold. www.ingentaconnect.com /content/iop/cqg/2005/00000022/00000021/art00010   (175 words)

 Luboš Motl's reference frame: Orbifold tachyons from SUGRA and other papers   (Site not responding. Last check: 2007-11-07) Consider the nonsupersymmetric orbifold of type II string theory - Adams-Polchinski-Silverstein (APS) type of orbifold - on C/Z_n for large n. Finally, the present authors calculate some interactions of the momentum modes in supergravity - which exist off-shell - and they show that they agree with the couplings of the different tachyons calculated from the CFT - which only exist on-shell. The main thing I worry about is that the result is perhaps not too unexpected because instead of the orbifold, one might work directly with the "limiting CFT" on the thin cone and its T-dual. motls.blogspot.com /2004/11/orbifold-tachyons-from-sugra-and-other.html   (2901 words)

 Orbifold Compactifications   (Site not responding. Last check: 2007-11-07) We can easily construct 4D strings from orbifold compactifications in which the 10D spacetime of the heterotic string is the product of 4D flat spacetime and a six-dimensional orbifold The heterotic string is particularly interesting because we can extend the action of the point group to the 16D lattice of the gauge group by embedding the action of the orbifold twist in the gauge degrees of freedom defined by the We can see now how a vast amount of heterotic string orbifold models can be generated. fisica.usac.edu.gt /public/curccaf_proc/quevedo1/node5.html   (1075 words)

 The orbifold shop   (Site not responding. Last check: 2007-11-07) The Orbifold Shop has gone into the business of installing orbifold parts. There are only a few kinds of features for two-dimensional orbifolds, but they can be used in interesting combinations. If you exactly spend your coupon at the Orbifold Shop, you will have a quotient orbifold coming from a symmetrically repeating pattern in the Euclidean plane with a bounded fundamental domain. www.geom.uiuc.edu /docs/doyle/mpls/handouts/node34.html   (221 words)

 Departmental Colloquium 2006-2007   (Site not responding. Last check: 2007-11-07) Abstract: Einstein metrics are expected to provide the optimal connection between the local geometry and the global properties of a manifold. After an introduction, the talk describes the construction of new Einstein metrics on odd dimensional spheres using complex algebraic orbifolds. Abstract: The asymptotic cone of a metric space X is the Gromov-Hausdorff limit of rescaled copies X/n of X, n=1,2,... www.math.utah.edu /research/colloquia   (931 words)

 Orbifold tachyons from SUGRA and other papers Consider the nonsupersymmetric orbifold of type II string\ntheory - Adams-Polchinski-Silverstein (APS) type of orbifold - on C/Z_n\nfor large n. But Matt and Joris\nfinally consider the obviously interesting limit in which n is sent to\ninfinity, but you keep n times alpha\' fixed. To properly write down the path integral, the first thing\nyou must define what you are integrating over.\n\nLets take some old QFT (as opposed to some fancy theory of gravity),\nand "fields" just some sort of function. www.physicsforums.com /showthread.php?t=54004   (7930 words)

 Softness of supersymmetry breaking on the orbifold T 2/Z 2 Softness of supersymmetry breaking on the orbifold T 2/Z 2 We show that the Scherk-Schwarz breaking in 6d is equivalent to the localized breaking with mass terms along the lines in extra dimensions. In particular, reselling and systematic downloading of files is prohibited. dx.doi.org /10.1088/1126-6708/2005/06/044   (281 words)

 CJM - The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces   (Site not responding. Last check: 2007-11-07) CJM - The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact K\"ahler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces. journals.cms.math.ca /cgi-bin/vault/view/steer1010   (83 words)

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