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Urs Schreiber
Jul16-04, 01:48 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>Suppose a closed string is propagating in some background which somehow\ninvolves a nonabelian 2-form. We expect that this 2-form associates a\n"surface holonomy" with the string, i.e. a group element associated with the\nshape of the worldsheet, being the generalization of an ordinary line\nholonomy.\n\nBut now furthermore assume that the closed string is wrapping a toroidal\ndimension and assume it is wrapping it "tight", i.e. so that it has no\nfluctuations.\n\nThen would we expect that a string wrapped n times receives the same surface\nholonomy as a string wrapped only once (the point set images of their\nworldsheets in target space being idential) or a different one, in general?\n\nI\'d say a different one - otherwise it would be weird.\n\nThen the question is:\n\nAre there non-flat connections on loop space that associate unique surface\nholonomies with such wrapped strings?\n\nI don\'t think this is a trivial question. Can anyone help?\n\nFor some dicussion about this see\n\nhttp://golem.ph.utexas.edu/string/archives/000399.html#c001326 .\n\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Suppose a closed string is propagating in some background which somehow
involves a nonabelian 2-form. We expect that this 2-form associates a
"surface holonomy" with the string, i.e. a group element associated with the
shape of the worldsheet, being the generalization of an ordinary line
holonomy.

But now furthermore assume that the closed string is wrapping a toroidal
dimension and assume it is wrapping it "tight", i.e. so that it has no
fluctuations.

Then would we expect that a string wrapped n times receives the same surface
holonomy as a string wrapped only once (the point set images of their
worldsheets in target space being idential) or a different one, in general?

I'd say a different one - otherwise it would be weird.

Then the question is:

Are there non-flat connections on loop space that associate unique surface
holonomies with such wrapped strings?

I don't think this is a trivial question. Can anyone help?

For some dicussion about this see

http://golem.ph.utexas.edu/string/archives/000399.html#c001326 .
Lubos Motl
Jul18-04, 12:24 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>On Fri, 16 Jul 2004, Urs Schreiber wrote:\n\n&gt; But now furthermore assume that the closed string is wrapping a toroidal\n&gt; dimension and assume it is wrapping it "tight", i.e. so that it has no\n&gt; fluctuations.\n&gt;\n&gt; Then would we expect that a string wrapped n times receives the same surface\n&gt; holonomy as a string wrapped only once (the point set images of their\n&gt; worldsheets in target space being idential) or a different one, in general?\n\nWell, if you wrap something n times, the holonomy (if I deal with Abelian\nstuff) is the n-th power of the holonomy of a singly wrapped object. You\nhave not exactly defined the physical/mathematical theory (or the\nconfiguration in string theory) in whose context you are asking the\nquestion, but I would say that any reasonable theory would confirm the\nanswer "different" regardless whether you wrap the object on exactly the\nsame points. This is the whole point of "multiple wrapping" and "having a\nstuck of branes" that such a configuration remembers that the number of\nthe objects is bigger than one even if you don\'t see it geometrically\n(because they overlap). It would also be a highly obscure theory if the\nholonomy changed drastically under infinitesimal variations of the shapes\nof the surfaces.\n\n&gt; Are there non-flat connections on loop space that associate unique surface\n&gt; holonomies with such wrapped strings?\n\nI probably don\'t know the right context, but the only realization of these\nholonomies I can imagine is exp(i.integral d^2 \\xi B) where the integral\ngoes over the 2D surface - a generalization of an Abelian U(1) gauge\ntheory to higher dimensions of the loops. I suppose that this definition\nis so easy that any question can be easily answered.\n\nBecause I don\'t know how to generalize path-ordering to surfaces, I don\'t\nknow how can you define non-Abelian holonomies - well, I am not the only\none who has this problem with the gerbes etc.\n\nBest\nLubos\n_____________________________ _________________________________________________\ nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 16 Jul 2004, Urs Schreiber wrote:

> But now furthermore assume that the closed string is wrapping a toroidal
> dimension and assume it is wrapping it "tight", i.e. so that it has no
> fluctuations.
>
> Then would we expect that a string wrapped n times receives the same surface
> holonomy as a string wrapped only once (the point set images of their
> worldsheets in target space being idential) or a different one, in general?

Well, if you wrap something n times, the holonomy (if I deal with Abelian
stuff) is the n-th power of the holonomy of a singly wrapped object. You
have not exactly defined the physical/mathematical theory (or the
configuration in string theory) in whose context you are asking the
question, but I would say that any reasonable theory would confirm the
answer "different" regardless whether you wrap the object on exactly the
same points. This is the whole point of "multiple wrapping" and "having a
stuck of branes" that such a configuration remembers that the number of
the objects is bigger than one even if you don't see it geometrically
(because they overlap). It would also be a highly obscure theory if the
holonomy changed drastically under infinitesimal variations of the shapes
of the surfaces.

> Are there non-flat connections on loop space that associate unique surface
> holonomies with such wrapped strings?

I probably don't know the right context, but the only realization of these
holonomies I can imagine is \exp(i.integral d^2 \xi B) where the integral
goes over the 2D surface - a generalization of an Abelian U(1) gauge
theory to higher dimensions of the loops. I suppose that this definition
is so easy that any question can be easily answered.

Because I don't know how to generalize path-ordering to surfaces, I don't
know how can you define non-Abelian holonomies - well, I am not the only
one who has this problem with the gerbes etc.

Best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
kneemo
Jul20-04, 05:49 AM
Suppose a closed string is propagating in some background which somehow
involves a nonabelian 2-form.

Is this the fundamental closed string and its rank two antisymmetric tensor NS-NS field?


But now furthermore assume that the closed string is wrapping a toroidal
dimension and assume it is wrapping it "tight", i.e. so that it has no
fluctuations.

May we also assume the loops (of the n wrapped string) are suitably close together (say, within epsilon); for then the worldsheets--for small initial time intervals--surely are topologically distinct, i.e., away from the packed n wrappings, the worldsheet can be one-dimensional and therefore affect the 2-form.

http://www.cs.csubak.edu/~mrios/tor_ws_wrap.jpg