View Full Version : weakened nonabelian bundle gerbes and 2-bundles
David Roberts - 1078662
Apr13-05, 08:52 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>Please note, the kernel of the idea here comes from a comment in an\nemail from B Jurco to my supervisor, Michael Murray.\n\nIn Baez and Schreiber\'s paper `2-connections on 2-bundles\', they talk\nabout the automorphism 2-group AUT(H) corresponding to the crossed\nmodule t:H Aut(H). I\'ve been looking at non-abelian bundle gerbes\n(NABG) and one way to define them is to look at an H-bitorsor (or\nprincipal H-bibundle) P defined on the fibre product Y^[2] of a\nsubmersion Y \\to M. In that case there is a local automorphism (comes\nfrom a section, and here i indexes a cover of Y^[2] = \\coprod U_i)\n\nu_i : U_i \\to Aut(H)\n\n(called in Aschieri-Cantini-Jurco (hep-th/0312154) \\varphi_e - I\'ve\n`locally trivialised\' my bitorsor) which obeys\n\nh.s_i = s_i.u_i(h).\n\nHowever, an automorphism is just a self-diffeomorphism respecting the\ngroup structure of the Lie group H. When we look at the contracted\nproduct of two H-bitorsors\n\nE_12 \\wedge^H E_23 = E_12 \\times_Y^[2] E_23 / ~,\n\nfor the equivalence relation (xh,y)~(x,hy), in the case where u_i \\in\nAut(H) this product is associative (due to the homomormorphism\nproperty of u_i). This example being taken from the NABG product (E_12\n\\wedge E_23 \\to E_13), E_ij = pulled back bitorsor along the\nprojection\n\n\\pi_ij: Y^[3] \\to Y^[2] i,j=1,2,3.\n\nHowever, for a general diffeomorphism f:H \\to H (here\'s the point,\nfinally), this is not so.\n\nAnd this is what we would _like_, considering that for general\nbitorsors (thinking more alg. geom here), the product (where defined,\nif we want H-G-bitorsors) is _not_ associative. Also, dealing with a\n\\in Aut(H) which satisfies\n\na(h_1).a(h_2) = a(h_1.h_2)\n^\n|\n|\nbad!\n\nseems very uncategorylike. Diffeomorphisms are also just arrows in the\ncategory of smooth manifolds and so seem a bit more natural when\nturning to smooth spaces etc than automorphisms, which satisfy an\n`on-the-nose\' equation.[Later: I think something between the two is\nmore appropriate, given we are still dealing with groups]\n\nOne complaint is that the diffeomorphisms may satisfy the homomorphism\nproperty up to equivalence, but only pointwise, as opposed to in each\nelement of the open cover. I think, however, that Jurco has worked\nthis bit out.\n\nOne `snag\' - H \\to Diff(H) won\'t give us a Lie crossed module, but\nsomething weaker. This is probably where the coherent\nLie-2-group/?something like a crossed module? correspondence comes\nin.\n\nCan this concept be made a bit less hand-wavy?\n\nDM Roberts\nPure Maths\nUniv. Adelaide\nSouth Australia\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>Please note, the kernel of the idea here comes from a comment in an
email from B Jurco to my supervisor, Michael Murray.
In Baez and Schreiber's paper `2-connections on 2-bundles', they talk
about the automorphism 2-group AUT(H) corresponding to the crossed
module t:H Aut(H). I've been looking at non-abelian bundle gerbes
(NABG) and one way to define them is to look at an H-bitorsor (or
principal H-bibundle) P defined on the fibre product Y^[2] of a
submersion Y \to M. In that case there is a local automorphism (comes
from a section, and here i indexes a cover of Y^[2] = \coprod U_i)u_i : U_i \to Aut(H)
(called in Aschieri-Cantini-Jurco (http://www.arxiv.org/abs/hep-th/0312154) \varphi_e - I've
`locally trivialised' my bitorsor) which obeys
h.s_i = s_i.u_i(h).
However, an automorphism is just a self-diffeomorphism respecting the
group structure of the Lie group H. When we look at the contracted
product of two H-bitorsors
E_{12} \wedge^H E_{23} = E_{12} \times_Y^[2] E_{23} / ~,
for the equivalence relation (xh,y)~(x,hy), in the case where u_i \in
Aut(H) this product is associative (due to the homomormorphism
property of u_i). This example being taken from the NABG product (E_{12}\wedge E_{23} \to E_{13}), E_{ij} = pulled back bitorsor along the
projection
\pi_ij: Y^[3] \to Y^[2] i,j=1,2,3.
However, for a general diffeomorphism f:H \to H (here's the point,
finally), this is not so.
And this is what we would _like_, considering that for general
bitorsors (thinking more alg. geom here), the product (where defined,
if we want H-G-bitorsors) is _not_ associative. Also, dealing with a
\in Aut(H) which satisfies
a(h_1).a(h_2) = a(h_1.h_2)^
|
|
bad!
seems very uncategorylike. Diffeomorphisms are also just arrows in the
category of smooth manifolds and so seem a bit more natural when
turning to smooth spaces etc than automorphisms, which satisfy an
`on-the-nose' equation.[Later: I think something between the two is
more appropriate, given we are still dealing with groups]
One complaint is that the diffeomorphisms may satisfy the homomorphism
property up to equivalence, but only pointwise, as opposed to in each
element of the open cover. I think, however, that Jurco has worked
this bit out.
One `snag' - H \to Diff(H) won't give us a Lie crossed module, but
something weaker. This is probably where the coherent
Lie-2-group/?something like a crossed module? correspondence comes
in.
Can this concept be made a bit less hand-wavy?
DM Roberts
Pure Maths
Univ. Adelaide
South Australia
Urs Schreiber
Apr14-05, 06:42 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>"David Roberts - 1078662" <droberts@maths.adelaide.edu.au> schrieb im\nNewsbeitrag news:200504140111.j3E1BA9u028138@staff.maths.adela ide.edu.au...\n\n> Please note, the kernel of the idea here comes from a comment in an\n> email from B Jurco to my supervisor, Michael Murray.\n>\n> In Baez and Schreiber\'s paper `2-connections on 2-bundles\', they talk\n> about the automorphism 2-group AUT(H) corresponding to the crossed\n> module t:H Aut(H). I\'ve been looking at non-abelian bundle gerbes\n> (NABG) and one way to define them is to look at an H-bitorsor (or\n> principal H-bibundle) P defined on the fibre product Y^[2] of a\n> submersion Y \\to M. In that case there is a local automorphism (comes\n> from a section, and here i indexes a cover of Y^[2] = \\coprod U_i)\n>\n> u_i : U_i \\to Aut(H)\n>\n> (called in Aschieri-Cantini-Jurco (hep-th/0312154) \\varphi_e - I\'ve\n> `locally trivialised\' my bitorsor) which obeys\n>\n> h.s_i = s_i.u_i(h).\n>\n> However, an automorphism is just a self-diffeomorphism respecting the\n> group structure of the Lie group H. When we look at the contracted\n> product of two H-bitorsors\n\n> However, for a general diffeomorphism f:H \\to H (here\'s the point,\n> finally), this is not so.\n>\n> And this is what we would _like_, considering that for general\n> bitorsors (thinking more alg. geom here),\n\n\nAs far as I am aware this idea arose in a discussion when I was visiting\nBranislav Jurco and Paolo Aschieri in Torino/Italy last year. I was\nmentioning how it seemed to me that 2-bundles with weak coherent structure\n2-groups (as opposed to strict 2-groups), whose product is not associative\non the nose, would capture the idea of a base-space-dependent group product\nin some sense and hence account for the \'algebra bundle\'-freedom that Andrew\nNeitzke identified as a plausible candidate for the n^3-scaling behaviour on\n5-branes:\n\nhttp://golem.ph.utexas.edu/string/archives/000461.html\n\nBranislav Jurco and Paolo Aschieri noted that this idea might correspond to\nthe "weak automorphisms" in a NABG that you are discussing in your post.\n\n\n> the product (where defined,\n> if we want H-G-bitorsors) is _not_ associative. Also, dealing with a\n> \\in Aut(H) which satisfies\n>\n> a(h_1).a(h_2) = a(h_1.h_2)\n> ^\n> |\n> |\n> bad!\n>\n> seems very uncategorylike.\n\n\nRight, and the way to do it is to go to coherent 2-groups instead. But\ncoherent 2-groups are much less well understood than strict ones. Since we\nknow that a strict one is just a crossed module, we would want to know which\nweak form of a crossed modules describes a coherent 2-group. I have once\nstarted working that out\n(http://golem.ph.utexas.edu/string/archives/000471.html), but it\'s not\nreally finished yet.\n\n\n> One `snag\' - H \\to Diff(H) won\'t give us a Lie crossed module, but\n> something weaker. This is probably where the coherent\n> Lie-2-group/?something like a crossed module? correspondence comes\n> in.\n\nYes, that\'s what I am talking about above.\n\n\n> Can this concept be made a bit less hand-wavy?\n\nThere is a precise way to define a coherent 2-group, a 2-bundle with a\ncoherent 2-group as structure 2-group as well as what connection and curving\non such a 2-bundle would be. There is also a known way how to get a\nnonabelian gerbe from a strict 2-bundle.\n\nWhat is hard is to fill the definition of a 2-connection for a coherent\n2-group with life by given concrete realizations in terms of local data. One\npossible approach I am discussing here:\nhttp://golem.ph.utexas.edu/string/archives/000542.html. There is also a\ncomplementary approach using connections on path space which is more\ndirectly related with Hofman\'s ideas\n\nI thought I\'d have lots of time to work this out more completely. But now\nthat Adelaide is also working on this... :-)\n\nIf these 2-bundle ideas help you to work out the construction of a\nweakened nonabelian bundle gerbe I don\'t know. But since all this is\nreally just different ways to look at the same thing it should really be\nrelated.\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>"David Roberts - 1078662" <droberts@maths.adelaide.edu.au> schrieb im
Newsbeitrag news:200504140111.j3E1BA9u028138@staff.maths.adela ide.edu.au...
> Please note, the kernel of the idea here comes from a comment in an
> email from B Jurco to my supervisor, Michael Murray.
>
> In Baez and Schreiber's paper `2-connections on 2-bundles', they talk
> about the automorphism 2-group AUT(H) corresponding to the crossed
> module t:H Aut(H). I've been looking at non-abelian bundle gerbes
> (NABG) and one way to define them is to look at an H-bitorsor (or
> principal H-bibundle) P defined on the fibre product Y^[2] of a
> submersion Y \to M. In that case there is a local automorphism (comes
> from a section, and here i indexes a cover of Y^[2] = \coprod U_i)
>
> u_i : U_i \to Aut(H)
>
> (called in Aschieri-Cantini-Jurco (http://www.arxiv.org/abs/hep-th/0312154) \varphi_e - I've
> `locally trivialised' my bitorsor) which obeys
>
> h.s_i = s_i.u_i(h).
>
> However, an automorphism is just a self-diffeomorphism respecting the
> group structure of the Lie group H. When we look at the contracted
> product of two H-bitorsors
> However, for a general diffeomorphism f:H \to H (here's the point,
> finally), this is not so.
>
> And this is what we would _like_, considering that for general
> bitorsors (thinking more alg. geom here),
As far as I am aware this idea arose in a discussion when I was visiting
Branislav Jurco and Paolo Aschieri in Torino/Italy last year. I was
mentioning how it seemed to me that 2-bundles with weak coherent structure
2-groups (as opposed to strict 2-groups), whose product is not associative
on the nose, would capture the idea of a base-space-dependent group product
in some sense and hence account for the 'algebra bundle'-freedom that Andrew
Neitzke identified as a plausible candidate for the n^3-scaling behaviour on
5-branes:
http://golem.ph.utexas.edu/string/archives/000461.html
Branislav Jurco and Paolo Aschieri noted that this idea might correspond to
the "weak automorphisms" in a NABG that you are discussing in your post.
> the product (where defined,
> if we want H-G-bitorsors) is _not_ associative. Also, dealing with a
> \in Aut(H) which satisfies
>
> a(h_1).a(h_2) = a(h_1.h_2)
> ^
> |
> |
> bad!
>
> seems very uncategorylike.
Right, and the way to do it is to go to coherent 2-groups instead. But
coherent 2-groups are much less well understood than strict ones. Since we
know that a strict one is just a crossed module, we would want to know which
weak form of a crossed modules describes a coherent 2-group. I have once
started working that out
(http://golem.ph.utexas.edu/string/archives/000471.html), but it's not
really finished yet.
> One `snag' - H \to Diff(H) won't give us a Lie crossed module, but
> something weaker. This is probably where the coherent
> Lie-2-group/?something like a crossed module? correspondence comes
> in.
Yes, that's what I am talking about above.
> Can this concept be made a bit less hand-wavy?
There is a precise way to define a coherent 2-group, a 2-bundle with a
coherent 2-group as structure 2-group as well as what connection and curving
on such a 2-bundle would be. There is also a known way how to get a
nonabelian gerbe from a strict 2-bundle.
What is hard is to fill the definition of a 2-connection for a coherent
2-group with life by given concrete realizations in terms of local data. One
possible approach I am discussing here:
http://golem.ph.utexas.edu/string/archives/000542.html. There is also a
complementary approach using connections on path space which is more
directly related with Hofman's ideas
I thought I'd have lots of time to work this out more completely. But now
that Adelaide is also working on this... :-)
If these 2-bundle ideas help you to work out the construction of a
weakened nonabelian bundle gerbe I don't know. But since all this is
really just different ways to look at the same thing it should really be
related.
DM Roberts
Apr20-05, 09:36 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>\nFurther to the above:\n\nAll the work I\'ve seen so far on 2-bundles with connection has been in\nterms of local data - Lie(G) 1-forms etc. Could we instead define a\nconnection as in the 1-bundle case as a sort of "bundle of subspaces"?\n(heuristic definition only) That is, a splitting into horizontal and\nvertical parts T = H \\oplus V. We have a concept of 2-vector space from\nHDA VI (math.QA/0307263) and there is the concept of a sub-2-vector\nspace (I think one could work backward from the direct sum of two\n2-vector spaces to get apropriate definitions, or else in terms of\n"images" and "kernels" of "linear transformations").\n\nThen we have the more geometric image as we do for principal bundles,\nthe only trouble would be to show equivalence of the two definitions.\nAh, I see where the nonabelian surface parallel transport rears its\nugly head - how can we generalise the proof as per bundles without a\ndecent definition of this?\n\nWe could work from a position of physical insight perhaps. But as the\n"physics" of this (H-flux in string theory, say) is not yet complete\n(the fault of the mathematicians, physicists or mathematical\nphysicists? Which came first, the chicken or the egg?) I don\'t know if\nthat will help.\n\nDavid\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>Further to the above:
All the work I've seen so far on 2-bundles with connection has been in
terms of local data - Lie(G) 1-forms etc. Could we instead define a
connection as in the 1-bundle case as a sort of "bundle of subspaces"?
(heuristic definition only) That is, a splitting into horizontal and
vertical parts T = H \oplus V. We have a concept of 2-vector space from
HDA VI (math.QA/0307263) and there is the concept of a sub-2-vector
space (I think one could work backward from the direct sum of two
2-vector spaces to get apropriate definitions, or else in terms of
"images" and "kernels" of "linear transformations").
Then we have the more geometric image as we do for principal bundles,
the only trouble would be to show equivalence of the two definitions.
Ah, I see where the nonabelian surface parallel transport rears its
ugly head - how can we generalise the proof as per bundles without a
decent definition of this?
We could work from a position of physical insight perhaps. But as the
"physics" of this (H-flux in string theory, say) is not yet complete
(the fault of the mathematicians, physicists or mathematical
physicists? Which came first, the chicken or the egg?) I don't know if
that will help.
David
Urs Schreiber
Apr20-05, 11:09 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>"DM Roberts" <droberts@physics.adelaide.edu.au> schrieb im Newsbeitrag\nnews:Pine.LNX.4.62.0504201035210.1380 1@feynman.harvard.edu...\n\n\n> All the work I\'ve seen so far on 2-bundles with connection\n\nIs there any other work on that than hep-th/0412325 ?\n\n\n> has been in terms of local data - Lie(G) 1-forms etc.\n\nMore precisely, in hep-th/0412325 this is given in terms of a local holonomy\n2-functor which is then decoded to yield local p-form data.\n\n\n> Could we instead define a\n> connection as in the 1-bundle case as a sort of "bundle of subspaces"?\n> (heuristic definition only) That is, a splitting into horizontal and\n> vertical parts T = H \\oplus V. We have a concept of 2-vector space from\n> HDA VI (math.QA/0307263) and there is the concept of a sub-2-vector\n> space (I think one could work backward from the direct sum of two\n> 2-vector spaces to get apropriate definitions, or else in terms of\n> "images" and "kernels" of "linear transformations").\n\n\nI expect that this should work and should be equivalent to the existing\ndefinition. But as far as I know so far nobody has tried to spell that out\nin detail.\n\n\n> Ah, I see where the nonabelian surface parallel transport rears its\n> ugly head - how can we generalise the proof as per bundles without a\n> decent definition of this?\n\n\nThere is in fact a decent definition of nonabelian surface parallel\ntransport in strict G-2-bundles. This is unfortunately only hinted at in\nhep-th/0412325, but I have reported on more details here:\n\nhttp://golem.ph.utexas.edu/string/archives/000503.html\n\nand, upon request, have clarified the context here:\n\nhttp://golem.ph.utexas.edu/string/archives/000547.html#c002194 .\n\nA more detailed exposition is underway:\n\nhttp://www-stud.uni-essen.de/~sb0264/2NCG.pdf .\n\nWhat I haven\'t shown yet, though, is indepence of this construction on the\nchoice of cover. I expect the proof to be completely analogous to the well\nknown abelian case.\n\n\n> We could work from a position of physical insight perhaps. But as the\n> "physics" of this (H-flux in string theory, say)\n\n\nH-flux gives rise to _abelian_ gerbes coupled to F-strings. Holonomy for\nabelian 2-gerbes is well understood, parallel transport has recently been\nstudied by Picken. This is a special case of the nonabelian surface\ntransport mentioned above.\n\nThe challenge is to identify the physics that gives rise to _non_abelian\ngerbes/2-bundles. The ordinary F-string in 10D does not couple to any\nnonabelian 2-form, so it must be something else.\n\nSeveral people expect this to be related to theories on stacks of N\nM5-branes, where we have end-strings of open membranes on the 5-branes. For\nN>1 these should sort of carry Chan-Paton-like degrees of freedom and couple\nto nonabelian 2-forms which are known to be part of the spectrom on these\nbranes.\n\nEdward Witten called the effective field theories for these branes once\ntentatively "nonabelian gerbe theories":\n\nhttp://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/71.bmp\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>"DM Roberts" <droberts@physics.adelaide.edu.au> schrieb im Newsbeitrag
news:Pine.LNX.4.62.0504201035210.13801@feynman.har vard.edu...
> All the work I've seen so far on 2-bundles with connection
Is there any other work on that than http://www.arxiv.org/abs/hep-th/0412325 ?
> has been in terms of local data - Lie(G) 1-forms etc.
More precisely, in http://www.arxiv.org/abs/hep-th/0412325 this is given in terms of a local holonomy
2-functor which is then decoded to yield local p-form data.
> Could we instead define a
> connection as in the 1-bundle case as a sort of "bundle of subspaces"?
> (heuristic definition only) That is, a splitting into horizontal and
> vertical parts T = H \oplus V. We have a concept of 2-vector space from
> HDA VI (math.QA/0307263) and there is the concept of a sub-2-vector
> space (I think one could work backward from the direct sum of two
> 2-vector spaces to get apropriate definitions, or else in terms of
> "images" and "kernels" of "linear transformations").
I expect that this should work and should be equivalent to the existing
definition. But as far as I know so far nobody has tried to spell that out
in detail.
> Ah, I see where the nonabelian surface parallel transport rears its
> ugly head - how can we generalise the proof as per bundles without a
> decent definition of this?
There is in fact a decent definition of nonabelian surface parallel
transport in strict G-2-bundles. This is unfortunately only hinted at in
http://www.arxiv.org/abs/hep-th/0412325, but I have reported on more details here:
http://golem.ph.utexas.edu/string/archives/000503.html
and, upon request, have clarified the context here:
http://golem.ph.utexas.edu/string/archives/000547.html#c002194 .
A more detailed exposition is underway:
http://www-stud.uni-essen.de/~sb0264/2NCG.pdf .
What I haven't shown yet, though, is indepence of this construction on the
choice of cover. I expect the proof to be completely analogous to the well
known abelian case.
> We could work from a position of physical insight perhaps. But as the
> "physics" of this (H-flux in string theory, say)
H-flux gives rise to _abelian_ gerbes coupled to F-strings. Holonomy for
abelian 2-gerbes is well understood, parallel transport has recently been
studied by Picken. This is a special case of the nonabelian surface
transport mentioned above.
The challenge is to identify the physics that gives rise to _non_abelian
gerbes/2-bundles. The ordinary F-string in 10D does not couple to any
nonabelian 2-form, so it must be something else.
Several people expect this to be related to theories on stacks of N
M5-branes, where we have end-strings of open membranes on the 5-branes. For
N>1 these should sort of carry Chan-Paton-like degrees of freedom and couple
to nonabelian 2-forms which are known to be part of the spectrom on these
branes.
Edward Witten called the effective field theories for these branes once
tentatively "nonabelian gerbe theories":
http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/71.bmp
Urs Schreiber
Apr20-05, 01:57 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>My last message in this thread was truncated for some reason. The full\nmessage has correctly appeared on sci.physics.strings. It continues as\nfollows:\n\n\n[...]\n\nBut I was being told that he has given up on making this precise. (?)\n\nHisham Sati is still arguing for this, e.g. in\n\nI. Kriz and H. Sati\nM-Theory, Type IIA Superstrings and Elliptic Cohomology\nhep-th/0404013\n\nH. Sati\nM-theory and characteristic classes\nhep-th/0501245\n\nThe most direct argument that this must be true that I know of is that in\nsection 5 of\n\nP. Aschieri & B. Jurco,\nGerbes, M5-Brane Anomalies and E_8 Gauge Theory\nhep-th/0409200 .\n\nRecall that they argue as follows:\n\nThe M2 brane couples to the SUGRA 3-form. There seems to be no choice but\nthat this coupling is globally described by an abelian 2-gerbe/3-bundle,\njust like in 1-dimension lower the coupling of the string to the KR 2-form\nis globally described by an abelian 1-gerbe/2-bundle.\n\nFor the string we can derive from the fact alone that its bulk couples to an\nabelian 1-gerbe the fact that its boundary couples to a nonabelian\n0-gerbe/1-bundle, namely that living on the D-brane that the string ends on.\n\nSchematically this works by noting that every abelian 1-gerbe G can be\nwritten as a trivial gerbe G0 plus a lifting gerbe D(B) of a twisted\nnonabelian 0-gerbe/1-bundle:\n\nG = D(B) + G0.\n\nB is the nonabelian 0-gerbe/bundle on the D-brane.\n\nA similar relation holds for abelian 2-gerbes. They can be realized as a\nlifting 2-gerbe of a twisted nonabelian 1-gerbe plus something else.\n\nBy analogy it is to be expected that this possibly twisted nonabelian\n1-gerbe is that living on the 5-branes that the membrane ends on.\n\nBut what is interesting is that one can say more: The abelian 2-gerbe\ncoupled to the M2 brane is in fact a Chern-Simons 2-gerbe classified by the\nPontryagin class. These 2-gerbes are known to be the lifting 2-gerbes for\nlifting an (Omega G)-gerbe to a \\hat(Omega G)-gerbe, where Omega G is the\nloop group of G and \\hat(Omega G) its Kac-Moody central extension.\n\nIncidentally, the \\PG-2-bundles that we find in\n\nBaez,Crans,Schreiber&Stevenson\n>From Loop Groups to 2-Groups\nmath.QA/0504123\n\nto be related to the group String(n) are known (not rigorously proven yet,\nthough) to be the same as these \\hat(Omega G)-1-gerbes.\n\nCombined with the argument by Aschieri&Jurco this would say that what lives\non a stack of M5-branes are these \\PG-2-bundles. Since they also seem to be\nrelated to elliptic cohomology (due to the appearance of String(n), for\none), this gives precisely the picture that Hisham Sati is arguing for in\nthe above papers.\n\nBut the details here still need to be written down.\n\n\n> (the fault of the mathematicians, physicists or mathematical\n> physicists? Which came first, the chicken or the egg?) I don\'t know if\n> that will help.\n\nUnderstanding the physical setups that give rise to nonabelian\ngerbes/2-bundles would certainly help the general understanding.\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>My last message in this thread was truncated for some reason. The full
message has correctly appeared on sci.physics.strings. It continues as
follows:
[...]
But I was being told that he has given up on making this precise. (?)
Hisham Sati is still arguing for this, e.g. in
I. Kriz and H. Sati
M-Theory, Type IIA Superstrings and Elliptic Cohomology
http://www.arxiv.org/abs/hep-th/0404013
H. Sati
M-theory and characteristic classes
http://www.arxiv.org/abs/hep-th/0501245
The most direct argument that this must be true that I know of is that in
section 5 of
P. Aschieri & B. Jurco,
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
http://www.arxiv.org/abs/hep-th/0409200 .
Recall that they argue as follows:
The M2 brane couples to the SUGRA 3-form. There seems to be no choice but
that this coupling is globally described by an abelian 2-gerbe/3-bundle,
just like in 1-dimension lower the coupling of the string to the KR 2-form
is globally described by an abelian 1-gerbe/2-bundle.
For the string we can derive from the fact alone that its bulk couples to an
abelian 1-gerbe the fact that its boundary couples to a nonabelian
0-gerbe/1-bundle, namely that living on the D-brane that the string ends on.
Schematically this works by noting that every abelian 1-gerbe G can be
written as a trivial gerbe G0 plus a lifting gerbe D(B) of a twisted
nonabelian 0-gerbe/1-bundle:
G = D(B) + G0[/itex].
B is the nonabelian 0-gerbe/bundle on the D-brane.
A similar relation holds for abelian 2-gerbes. They can be realized as a
lifting 2-gerbe of a twisted nonabelian 1-gerbe plus something else.
By analogy it is to be expected that this possibly twisted nonabelian
1-gerbe is that living on the 5-branes that the membrane ends on.
But what is interesting is that one can say more: The abelian 2-gerbe
coupled to the M2 brane is in fact a Chern-Simons 2-gerbe classified by the
Pontryagin class. These 2-gerbes are known to be the lifting 2-gerbes for
lifting an (\Omega G)-gerbe to a \hat(\Omega G)-gerbe, where \Omega G is the
loop group of G and \hat(\Omega G) its Kac-Moody central extension.
Incidentally, the \PG-2-bundles that we find in
Baez,Crans,Schreiber&Stevenson
>From Loop Groups to 2-Groups
math.[itex]QA/0504123
to be related to the group String(n) are known (not rigorously proven yet,
though) to be the same as these \hat(\Omega G)-1-gerbes.
Combined with the argument by Aschieri&Jurco this would say that what lives
on a stack of M5-branes are these \PG-2-bundles. Since they also seem to be
related to elliptic cohomology (due to the appearance of String(n), for
one), this gives precisely the picture that Hisham Sati is arguing for in
the above papers.
But the details here still need to be written down.
> (the fault of the mathematicians, physicists or mathematical
> physicists? Which came first, the chicken or the egg?) I don't know if
> that will help.
Understanding the physical setups that give rise to nonabelian
gerbes/2-bundles would certainly help the general understanding.
DM Roberts
Oct11-06, 02:40 PM
Further to the above:
All the work I've seen so far on 2-bundles with connection has been in
terms of local data - Lie(G) 1-forms etc. Could we instead define a
connection as in the 1-bundle case as a sort of "bundle of subspaces"?
(heuristic definition only) That is, a splitting into horizontal and
vertical parts T = H \oplus V. We have a concept of 2-vector space from
HDA VI (math.QA/0307263) and there is the concept of a sub-2-vector
space (I think one could work backward from the direct sum of two
2-vector spaces to get apropriate definitions, or else in terms of
"images" and "kernels" of "linear transformations").
Then we have the more geometric image as we do for principal bundles,
the only trouble would be to show equivalence of the two definitions.
Ah, I see where the nonabelian surface parallel transport rears its
ugly head - how can we generalise the proof as per bundles without a
decent definition of this?
We could work from a position of physical insight perhaps. But as the
"physics" of this (H-flux in string theory, say) is not yet complete
(the fault of the mathematicians, physicists or mathematical
physicists? Which came first, the chicken or the egg?) I don't know if
that will help.
David
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