View Full Version : orbifolds
Ian Agol
Apr8-04, 05:50 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nIf as hypothesized in some versions of string theory, space time\nis 10 dimensions, with six curled up dimensions being a Calabi-Yau\n3-fold, it seems reasonable that there might be further curled up\ndimensions which account for the orbifold structure. 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. Have string/M-theorists considered such\npossibilities, or is there a good physical reason for having an orbifold\n\nrather than a manifold, and that there might not be further curled\nup dimensions?\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>If as hypothesized in some versions of string theory, space time
is 10 dimensions, with six curled up dimensions being a Calabi-Yau
3-fold, it seems reasonable that there might be further curled up
dimensions which account for the orbifold structure. If one considers
a circle acting on a space, then the quotient is usually an orbifold. So
if one had a 7-dimensional space, and a circle action with very small
diameter orbits, then the whole thing would look very much like a
6 dimensional orbifold. Have string/M-theorists considered such
possibilities, or is there a good physical reason for having an orbifold
rather than a manifold, and that there might not be further curled
up dimensions?
Florian Gmeiner
Apr8-04, 02:27 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Ian Agol wrote:\n> If as hypothesized in some versions of string theory, space time\n> is 10 dimensions, with six curled up dimensions being a Calabi-Yau\n> 3-fold, it seems reasonable that there might be further curled up\n> dimensions which account for the orbifold structure. If one considers\n> a circle acting on a space, then the quotient is usually an orbifold.\n\nSorry, I don\'t get the point. What further dimensions should be curled\nup? I think we agree that our Minkowski space is not curled and if you\nthink of the compactification manifold - there is 11d supergravity/\nM-theory, but more seems to be impossible (at least in the context of\nsuperstrings). An orbifold is a singular space - and we don\'t want\nsomething like this - so what you do in fact is to blow up the orbifold\nsingularities to get a CY.\n\n> So if one had a 7-dimensional space, and a circle action with very small\n> diameter orbits, then the whole thing would look very much like a\n> 6 dimensional orbifold.\nI don\'t understand. Let\'s take M^7 be some compact manifold and act by\na discrete group, e.g. Z_2 to get O^7 = M^7/Z_2 - what we get is a\nseven-dimensional orbifold. why should it look six-dimensional?\nMaybe you want to take a product like M^6 x S^1/Z_2 ? then you can go to\na small radius limit in the seventh dimension - this is what is\nactually done by some people in M-theory.\n\n> Have string/M-theorists considered such\n> possibilities, or is there a good physical reason for having an orbifold\n> rather than a manifold, and that there might not be further curled\n> up dimensions?\n\nThe reason for taking an orbifold rather then a manifold are chiral\nfermions - in heterotic orbifold theories (which were done in the 80s,\nafter two papers by Dixon,Harvey,Witten&Vafa ) it was the possibility to\nget chiral fermions which made orbifolds attractive. more recently they\nare used in intersecting brane world models of type I/type II, where\nthey are also used to get chiral matter. How this works is actually easy\nto see, already in a 5d field theory toy model: take R^(3,1)\\times\nS^1/Z_2, where the Z_2 acts as a reflection along the 5th dimension. Now\ntake fermions and look what happens... You see that only one chirality\n(in the effective 4d theory) survives, because the 5d spinors are\nsplitted according to their \\gamma_5 "parity" (which is chirality in 4d)\n- and only one half has zero-modes.\n\nFlorian\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Ian Agol wrote:
> If as hypothesized in some versions of string theory, space time
> is 10 dimensions, with six curled up dimensions being a Calabi-Yau
> 3-fold, it seems reasonable that there might be further curled up
> dimensions which account for the orbifold structure. If one considers
> a circle acting on a space, then the quotient is usually an orbifold.
Sorry, I don't get the point. What further dimensions should be curled
up? I think we agree that our Minkowski space is not curled and if you
think of the compactification manifold - there is 11d supergravity/
M-theory, but more seems to be impossible (at least in the context of
superstrings). An orbifold is a singular space - and we don't want
something like this - so what you do in fact is to blow up the orbifold
singularities to get a CY.
> So if one had a 7-dimensional space, and a circle action with very small
> diameter orbits, then the whole thing would look very much like a
> 6 dimensional orbifold.
I don't understand. Let's take M^7 be some compact manifold and act by
a discrete group, e.g. Z_2 to get O^7 = M^7/Z_2 - what we get is a
seven-dimensional orbifold. why should it look six-dimensional?
Maybe you want to take a product like M^6 x S^1/Z_2 ? then you can go to
a small radius limit in the seventh dimension - this is what is
actually done by some people in M-theory.
> Have string/M-theorists considered such
> possibilities, or is there a good physical reason for having an orbifold
> rather than a manifold, and that there might not be further curled
> up dimensions?
The reason for taking an orbifold rather then a manifold are chiral
fermions - in heterotic orbifold theories (which were done in the 80s,
after two papers by Dixon,Harvey,Witten&Vafa ) it was the possibility to
get chiral fermions which made orbifolds attractive. more recently they
are used in intersecting brane world models of type I/type II, where
they are also used to get chiral matter. How this works is actually easy
to see, already in a 5d field theory toy model: take R^(3,1)\times
S^1/Z_2, where the Z_2 acts as a reflection along the 5th dimension. Now
take fermions and look what happens... You see that only one chirality
(in the effective 4d theory) survives, because the 5d spinors are
splitted according to their \gamma_5 "parity" (which is chirality in 4d)
- and only one half has zero-modes.
Florian
Florian Gmeiner
Apr8-04, 06:40 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Ian Agol wrote:\n> If as hypothesized in some versions of string theory, space time\n> is 10 dimensions, with six curled up dimensions being a Calabi-Yau\n> 3-fold, it seems reasonable that there might be further curled up\n> dimensions which account for the orbifold structure. If one considers\n> a circle acting on a space, then the quotient is usually an orbifold.\n\nSorry, I don\'t get the point. What further dimensions should be curled\nup? I think we agree that our Minkowski space is not curled and if you\nthink of the compactification manifold - there is 11d supergravity/\nM-theory, but more seems to be impossible (at least in the context of\nsuperstrings). An orbifold is a singular space - and we don\'t want\nsomething like this - so what you do in fact is to blow up the orbifold\nsingularities to get a CY.\n\n> So if one had a 7-dimensional space, and a circle action with very small\n> diameter orbits, then the whole thing would look very much like a\n> 6 dimensional orbifold.\nI don\'t understand. Let\'s take M^7 be some compact manifold and act by\na discrete group, e.g. Z_2 to get O^7 = M^7/Z_2 - what we get is a\nseven-dimensional orbifold. why should it look six-dimensional?\nMaybe you want to take a product like M^6 x S^1/Z_2 ? then you can go to\na small radius limit in the seventh dimension - this is what is\nactually done by some people in M-theory.\n\n> Have string/M-theorists considered such\n> possibilities, or is there a good physical reason for having an orbifold\n> rather than a manifold, and that there might not be further curled\n> up dimensions?\n\nThe reason for taking an orbifold rather then a manifold are chiral\nfermions - in heterotic orbifold theories (which were done in the 80s,\nafter two papers by Dixon,Harvey,Witten&Vafa ) it was the possibility to\nget chiral fermions which made orbifolds attractive. more recently they\nare used in intersecting brane world models of type I/type II, where\nthey are also used to get chiral matter. How this works is actually easy\nto see, already in a 5d field theory toy model: take R^(3,1)\\times\nS^1/Z_2, where the Z_2 acts as a reflection along the 5th dimension. Now\ntake fermions and look what happens... You see that only one chirality\n(in the effective 4d theory) survives, because the 5d spinors are\nsplitted according to their \\gamma_5 "parity" (which is chirality in 4d)\n- and only one half has zero-modes.\n\nFlorian\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Ian Agol wrote:
> If as hypothesized in some versions of string theory, space time
> is 10 dimensions, with six curled up dimensions being a Calabi-Yau
> 3-fold, it seems reasonable that there might be further curled up
> dimensions which account for the orbifold structure. If one considers
> a circle acting on a space, then the quotient is usually an orbifold.
Sorry, I don't get the point. What further dimensions should be curled
up? I think we agree that our Minkowski space is not curled and if you
think of the compactification manifold - there is 11d supergravity/
M-theory, but more seems to be impossible (at least in the context of
superstrings). An orbifold is a singular space - and we don't want
something like this - so what you do in fact is to blow up the orbifold
singularities to get a CY.
> So if one had a 7-dimensional space, and a circle action with very small
> diameter orbits, then the whole thing would look very much like a
> 6 dimensional orbifold.
I don't understand. Let's take M^7 be some compact manifold and act by
a discrete group, e.g. Z_2 to get O^7 = M^7/Z_2 - what we get is a
seven-dimensional orbifold. why should it look six-dimensional?
Maybe you want to take a product like M^6 x S^1/Z_2 ? then you can go to
a small radius limit in the seventh dimension - this is what is
actually done by some people in M-theory.
> Have string/M-theorists considered such
> possibilities, or is there a good physical reason for having an orbifold
> rather than a manifold, and that there might not be further curled
> up dimensions?
The reason for taking an orbifold rather then a manifold are chiral
fermions - in heterotic orbifold theories (which were done in the 80s,
after two papers by Dixon,Harvey,Witten&Vafa ) it was the possibility to
get chiral fermions which made orbifolds attractive. more recently they
are used in intersecting brane world models of type I/type II, where
they are also used to get chiral matter. How this works is actually easy
to see, already in a 5d field theory toy model: take R^(3,1)\times
S^1/Z_2, where the Z_2 acts as a reflection along the 5th dimension. Now
take fermions and look what happens... You see that only one chirality
(in the effective 4d theory) survives, because the 5d spinors are
splitted according to their \gamma_5 "parity" (which is chirality in 4d)
- and only one half has zero-modes.
Florian
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