<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>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag\nnews:cj8bud\\$sdj\\$1@glue.ucr.edu... \n>\n>\n> I\'m starting to read this paper:\n>\n> P. Aschieri & B. Jurco:\n> Gerbes, M5-Brane Anomalies and E_8 Gauge Theory,\n> hep-th/0409200\n>\n> I\'m pleased at how reminiscent it is of some old remarks\n> on s.p.r about the relation between 2-gerbes and E8 gauge theory:\n\n\nI see. There are lots of things I need to catch up with.\n\n\n> From: baez@galaxy.ucr.edu (John Baez)\n> Newsgroups: sci.physics.research\n> Subject: Re: Cech Cohomology, Gerbes, Differential Geometry, and more!\n> Date: Mon, 23 Sep 2002 19:05:59 +0000 (UTC)\n> Message-ID: <amnj4u\\$gu1\\$1@glue.ucr.edu>\n>\n> In article <mmurray-C93704.21481415092002@localhost>,\n> Michael K Murray <mmurray@maths.adelaide.edu.au> wrote:\n>\n> >Its generally true that every principal G (simple,\n> >simply connected etc) bundle over a manifold\n> >M defines a 2 gerbe whose 4 class is the pontrjagin class of the\n> >original bundle.\n>\n> Good point!\n>\n> >Are you expecting more from an E8 bundle?\n>\n> Perhaps the fact that Aaron Bergman mentioned, namely that\n> after the degree-4 Pontrjagin class an E8 bundle has no\n> more characteristic classes until we get to degree 16,\n> makes it hard to tell if we\'re dealing with an E8 bundle\n> or a 2-gerbe here. A 2-gerbe is just a degree-4 integral\n> cohomology class; an E8 bundle is a degree-4 integral\n> cohomology class plus some extra fuzz that you don\'t notice\n> until you get to very high dimensions.\n\n> Does the stuff I wrote above make sense?\n\n\nI can follow it.\n\n\n> One needs to see\n> why n-gerbes on X are classified by the (n+2)nd integral\n> cohomology of X.\n\n\nI have read that it does and it is plausible by extrapolating from the\n0-gerbe case, but I am not be able to prove this.\n\n\n> E8-bundles are classified by the 4th integral\n> cohomology of X if the dimension of X is small enough - smaller\n> than 16, I guess. So, over low-dimensional manifolds,\n> 2-gerbes are indistinguishable from E8 bundles.\n\n\nI take it that you mean abelian 2-gerbes here. Apparently an abelian p-gerbe\nallows to talk about (twisted) non-abelian (p-1)-gerbes. (At least it seems\nthat Aschieri and Jurco are saying so.) This would mean that E8 bundles are\nalso indistinguishable from (twisted) non-abelian 1-gerbes!?\n\n\n> Let me give it a try. Let me only talk about abelian gerbes\n> for now. Given an abelian group G,\n\n> the category of G-torsors\n\n\nLet\'s see:\n\nA G-torsor (l,X) is a topological space X with free and transitive\n(left-)action l of G on it. In the category of G-torsors objects are\nG-torsors and morphisms should be maps\n\nf : (l,X) -> (l\',X\')\n\nsuch that\n\nl\'_g (f(x)) = f(l_g(x)) .\n\nAt least that\'s what it says in\n\nRomain Attal:\nCombinatorics of Non-Abelian Gerbes with Connection and Curvature,\nmath-ph/0203056 .\n\n\n> has a tensor product\n\n\nI guess in my above notation this must be\n\n(l,X) x (l\',X\') = (l\'\', X x_g X\')\n\nwhere X x_g X\' is the space of equivalence classes with equivalence ~ given\nby\n\n(l_g x, y) ~ (x , l\'_g y)\n\nand l\'\' acts on (say) the first factor as in\n\nl\'\'_g (a , a\') = (l_g a , a\') ~ ( a , l\'_g a\') .\n\nThis dividing out by ~ is necessary in order to keep the action of G free\nand transitive.\n\n\n> which makes it into a 2-group, G-Tor.\n\n\nI assume here you mean that we regard the above tensor product as the\n1-morphisms and the above morphisms between G-torsors as the 2-morphisms of\nG-Tor.\n\n\n> Do you see why we need G to be abelian to tensor two G-torsors\n> and get a G-torsor?\n\n\nIt should be necessary for l\'\' above to really be a group action on the\nquotiont space X x_g X\'. Namely if we apply l\'\' twice\n\nl\'\'_g (l\'\'_g\' (a,a\') ) = ( l_gg\' a , a\' )\n\nand then invoke the similarity relation in the form\n\n( l_gg\' a , a\') ~ ( a, l\'_gg\' a\' )\n\nand again in the form\n\n( l_gg\' a , a\' ) = ( l_g l_g\' a , a\' ) ~ ( a, l\'_g\' l\'_g a\') = (a , l\'_g\'g\na\')\n\nwe should end up with the same thing, which implies\n\nl\'_gg\' = l\'_g\'g\n\nand hence\n\ng g\' = g\' g .\n\nI assume that\'s the reason why in the above mentioned paper the author talks\nabout left and right actions of torsors, because these always commute with\neach other.\n\n\n> If so, perhaps you\'re ready to take stuff\n> about G-bundles and categorify it to get stuff about (G-Tor)-2-bundles,\n> which are usually called "abelian G-gerbes" or sometimes just "G-gerbes".\n>\n> Let\'s sketch how it goes:\n>\n> 1) A trivial G-bundle over X looks like G x X.\n>\n> 2) A general G-bundle is built by gluing together trivial ones using\n> transition functions g_{ab} which must satisfy the cocycle condition\n>\n> g_{ab} g_{bc} = g_{ac}.\n>\n> 1\') A trivial abelian G-gerbe over X looks like G-Tor x X.\n>\n> 2\') A general abelian G-gerbe over X is built by gluing together trivial\nones\n> using transition functors g_{ab} which satisfy the above cocycle condition\n> up to a natural isomorphism\n>\n> h_{abc}: g_{ab} g_{bc} -> g_{ac}.\n>\n> which in turn satisfies a cocycle condition for quadruple overlaps.\n\n\nWhat I do pretty much understand is the definition of abelian p-gerbes (with\nconnection) in terms of elements in the p-th Deligne hypercohomology class.\nThis is just a nifty way to encode these cocycle conditions together with\nthose of the connection 1,2,3...-forms. But how exactly one gets from this\ndescription to that of (G-Tor )-2-bundles and vice versa is not clear to me\nyet.\n\n\n> >Currently I have no good idea of what a connection on a gerbe really is.\n> >There is a definition (4.1) on p. 27 of Breen&Messing (math.AG/0106083)\nbut\n> >I don\'t really understand it yet.\n\n> You actually *do* understand Breen and Messing\'s definition of a\n> connection on a nonabelian gerbe, at least *locally*, where we can\n> assume this nonabelian gerbe is trivial. The reason is that in this\n> case I kindly translated the definition into language mortals can\n> understand, and you read the paper where I did this.\n>\n> Namely:\n>\n> A connection on the trivial nonabelian G-gerbe over a\n> manifold X consists of:\n>\n> 1) a der(\\g)-valued 1-form A on X\n>\n> 2) a \\g-valued 2-form B.\n>\n> Here G is a Lie group, \\g is its Lie algebra, and der(\\g) is the\n> Lie algebra of derivations of \\g, which is also the Lie algebra\n> of Aut(G), the Lie group of all automorphisms of G.\n\n\nHere "trivial" is the crucial qualification, I assume. I am perplexed that\nAschieri et al. in "Nonabelian bundle gerbes" (hep-th/0312154) and "Gerbes,\nM5-brane Anomalies..." (hep-th/0409200) have not one set of 1-forms and\n2-forms, but two of them. I need to understand how that formalism collapses\nfor trivial gerbes to the single 1+2 form of Breen&Messing.\n\n\n> >It seems to involve "arrows" (functors) between fibers (categories) over\nM,\n> >which makes sense.\n>\n> The idea is that any tangent vector v to the point x in the base manifold\nX\n> determines a functor, "infinitesimal parallel transport",\n> from the fiber over x (which is a groupoid)\n> to the fiber over the point infinitesimally far from x in the v direction.\n\n\nYes, that makes good sense to me - in the 0-gerbe=bundle case. There the\n1-form gives rise to the parallel transport in the usual way. Now how does\nthe 2-form B used by Breen and Messing help to make this an infinitesimal\nparallel transport between nearby category fibers?\n\n\n> But fear not, I have translated this into language mortals can\n> understand!\n\n\nGood. Prometheus is always welcome down here! :-)\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>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
news:cj8bud$sdj$1@glue.ucr.edu...
>
>
> I'm starting to read this paper:
>
> P. Aschieri & B. Jurco:
> Gerbes, M5-Brane Anomalies and E_8 Gauge Theory,
> http://www.arxiv.org/abs/hep-th/0409200
>
> I'm pleased at how reminiscent it is of some old remarks
> on s.p.r about the relation between 2-gerbes and E8 gauge theory:
I see. There are lots of things I need to catch up with.
> From: baez@galaxy.ucr.edu (John Baez)
> Newsgroups: sci.physics.research
> Subject: Re: Cech Cohomology, Gerbes, Differential Geometry, and more!
> Date: Mon, 23 Sep 2002 19:05:59 +0000 (UTC)
> Message-ID: <amnj4u$gu1$1@glue.ucr.edu>
>
> In article <mmurray-C93704.21481415092002@localhost>,
> Michael K Murray <mmurray@maths.adelaide.edu.au> wrote:
>
> >Its generally true that every principal G (simple,
> >simply connected etc) bundle over a manifold
> >M defines a 2 gerbe whose 4 class is the pontrjagin class of the
> >original bundle.
>
> Good point!
>
> >Are you expecting more from an E8 bundle?
>
> Perhaps the fact that Aaron Bergman mentioned, namely that
> after the degree-4 Pontrjagin class an E8 bundle has no
> more characteristic classes until we get to degree 16,
> makes it hard to tell if we're dealing with an E8 bundle
> or a 2-gerbe here. A 2-gerbe is just a degree-4 integral
> cohomology class; an E8 bundle is a degree-4 integral
> cohomology class plus some extra fuzz that you don't notice
> until you get to very high dimensions.
> Does the stuff I wrote above make sense?
I can follow it.
> One needs to see
> why n-gerbes on X are classified by the (n+2)nd integral
> cohomology of X.
I have read that it does and it is plausible by extrapolating from the
0-gerbe case, but I am not be able to prove this.
> E8-bundles are classified by the 4th integral
> cohomology of X if the dimension of X is small enough - smaller
> than 16, I guess. So, over low-dimensional manifolds,
> 2-gerbes are indistinguishable from E8 bundles.
I take it that you mean abelian 2-gerbes here. Apparently an abelian p-gerbe
allows to talk about (twisted) non-abelian (p-1)-gerbes. (At least it seems
that Aschieri and Jurco are saying so.) This would mean that E8 bundles are
also indistinguishable from (twisted) non-abelian 1-gerbes!?
> Let me give it a try. Let me only talk about abelian gerbes
> for now. Given an abelian group G,
> the category of G-torsors
Let's see:
A G-torsor (l,X) is a topological space X with free and transitive
(left-)action l of G on it. In the category of G-torsors objects are
G-torsors and morphisms should be maps
f : (l,X) -> (l',X')
such that
l'_g (f(x)) = f(l_g(x)) .
At least that's what it says in
Romain Attal:
Combinatorics of Non-Abelian Gerbes with Connection and Curvature,
http://www.arxiv.org/abs/math-ph/0203056 .
> has a tensor product
I guess in my above notation this must be
(l,X) x (l',X') = (l'', X x_g X')
where X x_g X' is the space of equivalence classes with equivalence ~ given
by
(l_g x, y) ~ (x , l'_g y)
and l'' acts on (say) the first factor as in
l''_g (a , a') = (l_g a , a') ~ ( a , l'_g a') .
This dividing out by ~ is necessary in order to keep the action of G free
and transitive.
> which makes it into a 2-group, G-Tor.
I assume here you mean that we regard the above tensor product as the
1-morphisms and the above morphisms between G-torsors as the 2-morphisms of
G-Tor.
> Do you see why we need G to be abelian to tensor two G-torsors
> and get a G-torsor?
It should be necessary for l'' above to really be a group action on the
quotiont space X x_g X'. Namely if we apply l'' twice
l''_g (l''_g' (a,a') ) = ( l_{gg}' a , a' )
and then invoke the similarity relation in the form
( l_{gg}' a ,[/itex] a') ~ ( a, l'_gg' a' )
and again in the form
( l_{gg}' a , a' ) = ( l_g l_g' a , a' ) ~ ( a, l'_g' l'_g a') = (a , l'_g'g
a')
we should end up with the same thing, which implies
l'_gg' = l'_g'g
and hence
g g' = g' g .
I assume that's the reason why in the above mentioned paper the author talks
about left and right actions of torsors, because these always commute with
each other.
> If so, perhaps you're ready to take stuff
> about G-bundles and categorify it to get stuff about (G-Tor)-2-bundles,
> which are usually called "abelian G-gerbes" or sometimes just "G-gerbes".
>
> Let's sketch how it goes:
>
> 1) A trivial G-bundle over X looks like G x X.
>
> 2) A general G-bundle is built by gluing together trivial ones using
> transition functions g_{ab} which must satisfy the cocycle condition
>
> g_{ab} g_{bc} = g_{ac}.
>
> 1') A trivial abelian G-gerbe over X looks like G-Tor x X.
>
> 2') A general abelian G-gerbe over X is built by gluing together trivial
ones
> using transition functors g_{ab} which satisfy the above cocycle condition
> up to a natural isomorphism
>
> h_{abc}: g_{ab} g_{bc} -> g_{ac}.
>
> which in turn satisfies a cocycle condition for quadruple overlaps.
What I do pretty much understand is the definition of abelian p-gerbes (with
connection) in terms of elements in the p-th Deligne hypercohomology class.
This is just a nifty way to encode these cocycle conditions together with
those of the connection 1,2,3...-forms. But how exactly one gets from this
description to that of (G-Tor )-2-bundles and vice versa is not clear to me
yet.
> >Currently I have no good idea of what a connection on a gerbe really is.
> >There is a definition (4.1) on p. 27 of Breen&Messing (math[itex].AG/0106083)
but
> >I don't really understand it yet.
> You actually *do* understand Breen and Messing's definition of a
> connection on a nonabelian gerbe, at least *locally*, where we can
> assume this nonabelian gerbe is trivial. The reason is that in this
> case I kindly translated the definition into language mortals can
> understand, and you read the paper where I did this.
>
> Namely:
>
> A connection on the trivial nonabelian G-gerbe over a
> manifold X consists of:
>
> 1) a der(\g)-valued 1-form A on X
>
> 2) a \g-valued 2-form B.
>
> Here G is a Lie group, \g is its Lie algebra, and der(\g) is the
> Lie algebra of derivations of \g, which is also the Lie algebra
> of Aut(G), the Lie group of all automorphisms of G.
Here "trivial" is the crucial qualification, I assume. I am perplexed that
Aschieri et al. in "Nonabelian bundle gerbes" (http://www.arxiv.org/abs/hep-th/0312154) and "Gerbes,
M5-brane Anomalies..." (http://www.arxiv.org/abs/hep-th/0409200) have not one set of 1-forms and
2-forms, but two of them. I need to understand how that formalism collapses
for trivial gerbes to the single 1+2 form of Breen&Messing.
> >It seems to involve "arrows" (functors) between fibers (categories) over
M,
> >which makes sense.
>
> The idea is that any tangent vector v to the point x in the base manifold
X
> determines a functor, "infinitesimal parallel transport",
> from the fiber over x (which is a groupoid)
> to the fiber over the point infinitesimally far from x in the v direction.
Yes, that makes good sense to me - in the 0-gerbe=bundle case. There the
1-form gives rise to the parallel transport in the usual way. Now how does
the 2-form B used by Breen and Messing help to make this an infinitesimal
parallel transport between nearby category fibers?
> But fear not, I have translated this into language mortals can
> understand!
Good. Prometheus is always welcome down here! :-)