Discussion:
weakened nonabelian bundle gerbes and 2-bundles
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Urs Schreiber
2005-04-14 10:42:49 UTC
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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 (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.
Urs Schreiber
2005-04-14 23:42:07 UTC
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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 (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
2005-04-20 14:36:57 UTC
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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
2005-04-20 15:45:13 UTC
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Post by DM Roberts
All the work I've seen so far on 2-bundles with connection
Is there any other work on that than hep-th/0412325 ?
Post by DM Roberts
has been in terms of local data - Lie(G) 1-forms etc.
More precisely, in hep-th/0412325 this is given in terms of a local holonomy
2-functor which is then decoded to yield local p-form data.
Post by DM Roberts
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.
Post by DM Roberts
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
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.
Post by DM Roberts
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
.

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
hep-th/0404013

H. Sati
M-theory and characteristic classes
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
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.

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
Post by DM Roberts
From Loop Groups to 2-Groups
math.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.
Post by DM Roberts
(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.

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