Urs Schreiber
Nov18-04, 12:39 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>I\'d like to share some thoughts on \'higher gauge theory\' (just posted to\nhttp://golem.ph.utexas.edu/string/archives/000471.html ):\n\n2-bundles are great\n(http://golem.ph.utexas.edu/string/archives/000457.html). They connect path\nspace/loop space differential geometry and path space bundles with other\nstuff, like nonabelian gerbes. That\'s nice for physics, because it allows to\n\'see\' the string in the nonabelian background described by the gerbe: It\'s\nconfiguration space is the arrow space of the base 2-space of the 2-bundle.\nIts constraints are gauge-covariant deRham operators on that space.\n\nI have recently sketched\n(http://golem.ph.utexas.edu/string/archives/000465.html) a proof for how a\n2-bundle with strict structure 2-group yields a (possibly twisted)\nnonabelian gerbe with curving and connection of a certain kind. In fact, it\nseems that except for one constraint the 2-bundle is more general. (For\ninstance it turns out that the gerbe data encoded in the \\$d_{ij} \\in\n{Lie}(H)\\otimes \\Omega^2(U_{ij})\\$ forms comes from infinitesimal loops in\nthe arrow space of the 2-bundle\'s base 2-space and are enriched for larger\nloops.)\n\nThat one constraint is the infamous \\$dt(B_i) + F_{A_i} = 0\\$, which comes\nfrom the nature of the strict structure 2-group.\n\nBut the most general 2-bundle has a coherent structure 2-group instead, and\nI have now worked out some facts related to surface holonomy using\n<em>coherent</em> 2-connections. There the above constraint is indeed\nalleviated! This might be interesting, since at the same time the data which\nmakes a strict 2-group coherent is encoded in an object with three group\nindices, which might be a candidate carrier of the respective \\$\\sim n^3\\$\ndegrees of freedom seen on 5-branes\n(http://golem.ph.utexas.edu/string/archives/000461.html).\n\nI don\'t know if it is, but I know how to construct a generalization of a\nnonabelian gerbed with consistent surface holonomy which depends on a couple\nof algebra-valued \\$p\\$-forms plus an element of \\$H^3(G,K)\\$, where \\$K \\subset\nH\\$ is an abelian group inside an non-associative algebra \\$H\\$. The key point\nis that in going from strict to coherent structure 2-groups one finds that\nup to ``weakening\'\' the essential equations remain intact:\n\nWhere the strict structure 2-group is described by a crossed module which\ninvolves the semidirect product of two groups, the coherent structure\n2-group is described by what I tend to call a \'weak crossed module\' where\nthe well-known relations hold only up to generalized similarity\ntransformations which are determined by that element of \\$H^3(G,K)\\$. This\nleads to a weakened form of path space connection and hence to a new notion\nof surface holonomy.\n\nArriving at this point involved a lot of work though. The results that I\nhave managed to extract so far are summarized in this set of\nnotes:http://www-stud.uni-essen.de/~sb0264/p14.pdf .\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I'd like to share some thoughts on 'higher gauge theory' (just posted to
http://golem.ph.utexas.edu/string/archives/000471.html ):
2-bundles are great
(http://golem.ph.utexas.edu/string/archives/000457.html). They connect path
space/loop space differential geometry and path space bundles with other
stuff, like nonabelian gerbes. That's nice for physics, because it allows to
'see' the string in the nonabelian background described by the gerbe: It's
configuration space is the arrow space of the base 2-space of the 2-bundle.
Its constraints are gauge-covariant deRham operators on that space.
I have recently sketched
(http://golem.ph.utexas.edu/string/archives/000465.html) a proof for how a
2-bundle with strict structure 2-group yields a (possibly twisted)
nonabelian gerbe with curving and connection of a certain kind. In fact, it
seems that except for one constraint the 2-bundle is more general. (For
instance it turns out that the gerbe data encoded in the $d_{ij} \in{Lie}(H)\otimes \Omega^2(U_{ij})$ forms comes from infinitesimal loops in
the arrow space of the 2-bundle's base 2-space and are enriched for larger
loops.)
That one constraint is the infamous $dt(B_i) + F_{A_i} = 0$, which comes
from the nature of the strict structure 2-group.
But the most general 2-bundle has a coherent structure 2-group instead, and
I have now worked out some facts related to surface holonomy using
<em>coherent</em> 2-connections. There the above constraint is indeed
alleviated! This might be interesting, since at the same time the data which
makes a strict 2-group coherent is encoded in an object with three group
indices, which might be a candidate carrier of the respective $\sim n^3$
degrees of freedom seen on 5-branes
(http://golem.ph.utexas.edu/string/archives/000461.html).
I don't know if it is, but I know how to construct a generalization of a
nonabelian gerbed with consistent surface holonomy which depends on a couple
of algebra-valued $p$-forms plus an element of $H^3(G,K)$, where $K \subset
H$ is an abelian group inside an non-associative algebra $H$. The key point
is that in going from strict to coherent structure 2-groups one finds that
up to ``weakening'' the essential equations remain intact:
Where the strict structure 2-group is described by a crossed module which
involves the semidirect product of two groups, the coherent structure
2-group is described by what I tend to call a 'weak crossed module' where
the well-known relations hold only up to generalized similarity
transformations which are determined by that element of $H^3(G,K)$. This
leads to a weakened form of path space connection and hence to a new notion
of surface holonomy.
Arriving at this point involved a lot of work though. The results that I
have managed to extract so far are summarized in this set of
notes:http://www-stud.uni-essen.de/~sb0264/p14.pdf .
http://golem.ph.utexas.edu/string/archives/000471.html ):
2-bundles are great
(http://golem.ph.utexas.edu/string/archives/000457.html). They connect path
space/loop space differential geometry and path space bundles with other
stuff, like nonabelian gerbes. That's nice for physics, because it allows to
'see' the string in the nonabelian background described by the gerbe: It's
configuration space is the arrow space of the base 2-space of the 2-bundle.
Its constraints are gauge-covariant deRham operators on that space.
I have recently sketched
(http://golem.ph.utexas.edu/string/archives/000465.html) a proof for how a
2-bundle with strict structure 2-group yields a (possibly twisted)
nonabelian gerbe with curving and connection of a certain kind. In fact, it
seems that except for one constraint the 2-bundle is more general. (For
instance it turns out that the gerbe data encoded in the $d_{ij} \in{Lie}(H)\otimes \Omega^2(U_{ij})$ forms comes from infinitesimal loops in
the arrow space of the 2-bundle's base 2-space and are enriched for larger
loops.)
That one constraint is the infamous $dt(B_i) + F_{A_i} = 0$, which comes
from the nature of the strict structure 2-group.
But the most general 2-bundle has a coherent structure 2-group instead, and
I have now worked out some facts related to surface holonomy using
<em>coherent</em> 2-connections. There the above constraint is indeed
alleviated! This might be interesting, since at the same time the data which
makes a strict 2-group coherent is encoded in an object with three group
indices, which might be a candidate carrier of the respective $\sim n^3$
degrees of freedom seen on 5-branes
(http://golem.ph.utexas.edu/string/archives/000461.html).
I don't know if it is, but I know how to construct a generalization of a
nonabelian gerbed with consistent surface holonomy which depends on a couple
of algebra-valued $p$-forms plus an element of $H^3(G,K)$, where $K \subset
H$ is an abelian group inside an non-associative algebra $H$. The key point
is that in going from strict to coherent structure 2-groups one finds that
up to ``weakening'' the essential equations remain intact:
Where the strict structure 2-group is described by a crossed module which
involves the semidirect product of two groups, the coherent structure
2-group is described by what I tend to call a 'weak crossed module' where
the well-known relations hold only up to generalized similarity
transformations which are determined by that element of $H^3(G,K)$. This
leads to a weakened form of path space connection and hence to a new notion
of surface holonomy.
Arriving at this point involved a lot of work though. The results that I
have managed to extract so far are summarized in this set of
notes:http://www-stud.uni-essen.de/~sb0264/p14.pdf .