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Dirac operators on Lorentzian manifolds
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0504020105.1073fab3@posting.google.com...
Quote:
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<3b2kt0F6er029U1@news.dfncis.de>...

The underlying philosophy the way Akhmedov presents it is rather similar
to Thomas Larsson's ideas (http://de.arxiv.org/abs/math-ph/0205017) on
2-form gauge theory, though different in the details.

An obvious difference is that Akhmedov uses triangles and I squares, but
this
is hardly important.


The triangles are crucial in making the setup independent of the
latticization, since using them there is a way to get a TFT using structure
constants C_ijk.


Quote:
More significantly, in order to be able to contract indices Akhmedov
introduces
a metric kappa^ij (bottom of page 5). I avoid that by putting two
four-index
quantities, and their inverses, on each plaquette. In that way I can
arrange
things so that always one up and one down index are contracted, and there
is no
need for a metric.

The metric is no extra structure in Akhmedov's setup, since it follows from
the C_ijk.

To me it seems as if Akhmedov provides a way to make your idea of
associating an "internal index" with each edge consistent, i.e. to get a
well-defined continuum theory independent of the choice of discretization
used to define it.

I also believe there is a nice way to describe that continuum limit:

Let G be the space in which the internal indices take values. For instance
for the TFT defined from discrete groups, G could be the limiting Lie group.
But it might be something else, too (and one should not confuse this group
with any "gauge group" here).

Pick any connected collection of triangles, i.e. a small surface element. It
defines a tensor with n incoming and m outgoing indices. In the limit this
gives a map from pairs of smooth paths in G to the base field

T(gamma1, gamma2)

such that composition is given by "continuous index contraction"

(T o T')(gamma1, gamma2)
=
int D[gamma] T(gamma1,gamma) T(gamma,gamma2) .

(Here the functional integral is over a reparameterization gauge slice, the
choice of which is free due to the properties mentioned above.)

These have to be invertible and we can impose a "star-condition". But since
these T are nothing but integral kernels we see that the group they form is
that of unitary operators on something like L^2(PG), where PG is the path
space over G. But in fact without any loss of generality we can assume that
all paths start and end at a given point in G and have sitting instant at
that point and are parameterized by a parameter in [0,1], so we really get
OmegaG, the based loop space over G and unitary operators on L^2(Omega G).

In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by

(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)

where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.

In fact, I believe it can be checked that this defines on the Ts the
structure of a weak monoidal category with all morphisms invertible. By
throwing in weak formal horizontal inverses (representing the "zig-zag
symmetry" of holonomy which is otherwise not captured) we get a weak
2-group.

I had speculated before that weak 2-groups allow for realizing n^3 scaling
behavior, and here indeed it is explicit, due to the TFT definition of the
whole thing. That's quite interesting.
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Theories are increasingly theoretical Reply with quote

On Wed, 9 Feb 2005, Kea wrote:

Quote:
Urs Schreiber Wrote: [Warning: text about category theory. LM]

Right, this is getting off-topic. We have moved the discussion to

http://golem.ph.utexas.edu/string/archives/000509.html
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Ohne
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Joined: 26 Apr 2005
Posts: 1

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: String theory as a generalization of gauge theory Reply with quote

I found the article very interesting but it gives mathematical
motivation for studying string theory. I was looking for physical
motivation.

My attitude is that if string theories are just generalized gauge
theories of the type described above (pointless gauge theories or maybe
they are called noncommutative field theories?) it would constitute
physical motivation for studying string theory since gauge theories are
so fundamental to the standard model and to general relativity.

By generalizing the fiber from an abelian to a nonabelian group we go
from QED to the standard model, describing three of the known forces of
nature. This in itself constitutes strong physical motivation for
studying generalizations of nonabelian gauge theories.

It seems a very basic question to ask: If the base space of a
nonabelian gauge theory is generalized so that the displacement and
gauge algebras no longer commute is this thing something we can
quantize? Is it physically interesting? Is it a string theory?

Maybe someone who knows noncommutative field theories can chime in
here?
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jcgonsowski@yahoo.com
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Joined: 26 Apr 2005
Posts: 5

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Bubbling AdS space and 1/2 BPS geometries Reply with quote

Urs Schreiber wrote:
Quote:
On Wed, 2 Feb 2005, jcgonsowski@yahoo.com wrote:

As for Smith, he calls his model a "work in progress", but I don't
think he has any reason to go away from the E7/E6xU(1) 27 complex
dimensions that he was using even before he put them into the
context
of M-theory.

Probably once you try to understand quantized gravity in this context
further constraints will appear.

I know how Smith dealt with the problems of bosonic strings but I don't
know what extra problems M-theory brings. Smith uses M-theory (and
F-theory) to describe interactions between brane universes.
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jcgonsowski@yahoo.com
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Joined: 26 Apr 2005
Posts: 5

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Bubbling AdS space and 1/2 BPS geometries Reply with quote

kneemo wrote:
Quote:
I recall Smith using the complexified exceptional Jordan algebra for
his model. This is a (complex) 27-dimensional algebra, over the
bioctonions. Now, is it Smith's intention to use the (complex) 11
dimensions of the complexified Jordan algebra, or use the 11 (real)
dimensions of the self-adjoint part (the exceptional Jordan algebra)
for space-time. This would seem to make a difference, as we would be
working with 3 complex dimensions, along with a bioctonionic space,
in
the 11-dimensional complex case.

Would space-time as 11=8+3 complex dimensions be consistent? Or must
space-time dimensions be strictly real?

Real vs. complex depends on whether you are talking about spacetime
within string theory or within M-theory (there's even an F-theory with
28 quaternionic dimensions). The +3 would imply M-theory, +2 would
imply string theory, +4 would imply F-theory. Spacetime itself is kind
of just the 8. Details from Smith's website about the algebra and what
these stringy spaces are used for are at:

http://www.valdostamuseum.org/hamsmith/StringMFbranegrav.html#Stheory
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Kea
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Joined: 26 Apr 2005
Posts: 3

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Theories are increasingly theoretical Reply with quote

Urs Schreiber Wrote: [Warning: text about category theory. LM]

Quote:
I guess the point is that once you realize that category theory is the
language in which god wrote math it becomes clear that at the heart of
it one is dealing with omega-categories

Great! Do many String theorists think this way? I've quoted what
you said about category theory. It's on my door (a collection
point for interesting snippets).

Quote:
This is one way to look at a point.....

If you'll allow me to refer to Ross Street's lectures here: Let
R be a commutative ring with unit. Spec(R) is
the space of prime ideals in R. It turns out that
Spec(R) is a sober space (every irreducible closed subset
is the closure of a unique singleton). Sober spaces are completely
recoverable from the category of elementary toposes.

For instance, when considering a Boolean algebra as a ring,
Spec(R) is the Stone space of the ring. Stone is a very
underrated historical figure. What are Stone spaces?

Recall that one takes the lattice structure
\mathcal{O}(X) of open sets of a space X as the
category underlying sheaves on X, which are contravariant.
Such lattices have a 0 (the empty set) and 1 (the set X).

In the category of topological spaces a point is specified by a
morphism from the one point space, which is an initial object. A
sufficiently general type of distributive lattice with 0 and 1 is
called a frame (see Mac Lane and Moerdijk). In the category of frames
the initial object is the 2 point lattice (0,1) so one defines a
'point' of a generalised space to be a morphism into
this object (remember the contravariance). But by this definition a
space might not have any points at all! A space is said to be
'geometric' if for any two objects of the lattice there exists a point
(morphism) p such that p^{-1} distinguishes
the objects.

Back to Stone. The category of sober spaces is equivalent to the
category of generalised spaces which are 'geometric'. This may be
viewed as a duality in which the two point space plays a special
'self-dual' role (it's called a schizophrenic object). Another example
of these so-called Stone dualities is Pontrjagin duality, for which
U(1) is the schizophrenic object.

So...what about NCG? Well, this is the question, isn't it? We need
2-toposes. This is my pet fundamental thing! To Ross Street a 2-topos
involves 2-stacks, which are, first of all, pseudofunctors from a site
C into Cat. The stack condition is a descent diagram, and
the inclusion of Stack(C) into
Ps(C^{\textrm{op}},Cat) is a nice biadjunction. Gerbes, as
you say, are related to this.

But the lattice theory is more fundamental. The logic of a topos
depends on it. Topos (1-stack) lattices are always distributive.
Quantum lattices are not. But quantum lattices are well understood, and
a proper understanding of 2-toposes means getting the lattice theory
right. I guess this is what I've been trying to say!

Regards Kea

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Urs Schreiber
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Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Bubbling AdS space and 1/2 BPS geometries Reply with quote

On Wed, 2 Feb 2005, jcgonsowski@yahoo.com wrote:

Quote:
As for Smith, he calls his model a "work in progress", but I don't
think he has any reason to go away from the E7/E6xU(1) 27 complex
dimensions that he was using even before he put them into the context
of M-theory.

Probably once you try to understand quantized gravity in this context
further constraints will appear.
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Why higher dimensions? Looking for an intuitive answer Reply with quote

On Tue, 1 Feb 2005, heinrich_neumaier@yahoo.com wrote:

Quote:
Urs Schreiber wrote:

I don't think there is any good heuristic explanation. In fact, the number
of dimensions is fixed by a rather subtle mechanism once you demand
certain desirable properties of your string theory.

Can one give these requirements and properties in simple language?


See Lubos Motl's moderator's comment to another recent post in this
thread.


Quote:
Not sure what a "knotted structure in space-time" is supposed to be. Is
this referring to attempts at modeling particles as topological defects?

Yes; people are speaking of knotted solitons and knotted topological
defects.


There are solitons in field theory, but that's different from saying that
what we currently believe to be fundamental particles are actually
solitons in some deeper theory.


Quote:
For instance with only closed strings and no branes around there is
fundamentally only a gravitational-like force and all other forces come
from the Kaluza-Klein mechanism by suitably compactifying higher
dimensions.

Ok, so the answer would be that higher dimensions are necessary to
have forces other than gravity?


For closed strings, yes. There the Kaluza-Klein mechanism yields the other
forces.


Quote:
When you want supersymmetry on the worldsheet to be equivalent to
supersymmetry in spacetime the string needs 8 transversal dimensions
(meaning 8 dimensions in addition to the 2 of its worldsheet). This gives
a total of 10.

But supersymmetry is broken or at least not exact; does this still mean
that higher dimensions are necessary?


It may be broken spontaneously. But the theory itself is still
supersymmetric.
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bjflanagan
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Joined: 07 May 2005
Posts: 39

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Why higher dimensions? Looking for an intuitive answer Reply with quote

LM:

Quote:
I think that the question is why we need extra dimensions in string
theory.

Yes, but I thought the writer might need some general background, as
would appear to be borne out by his later question.
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jcgonsowski@yahoo.com
science forum beginner


Joined: 26 Apr 2005
Posts: 5

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Bubbling AdS space and 1/2 BPS geometries Reply with quote

Urs Schreiber wrote:

Quote:
This is taken from a discussion about elements of how gauge theories arise
within string theory. I was just recalling some basic facts about effective
actions of D-branes. So you haven't really been inspired by me, but by
standard string theory results!
I am not sure, but it might be helpful to go the other way round: Instead of
trying to integrate string ideas in Smith's universe of ideas you or he
might be interested in seeing how his love for algebraic structures can be
satisfied in string theory. The advantage would be that M-theory, though
incompletely understood, is still much better understood than the 24+3
dimensional bosonic hypotheses that you talked about.

I certainly appreciate that standard string theory includes areas that
are not too far away from the standard model. For me, besides being
hopelessly biased towards Smith's model Smile I kind of have to start with
Smith's model simply beacuse it is what I know best. I have to start
with Smith's model even for the Standard Model or Einstein's gravity.
As for Smith, he calls his model a "work in progress", but I don't
think he has any reason to go away from the E7/E6xU(1) 27 complex
dimensions that he was using even before he put them into the context
of M-theory.
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Kea
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Joined: 26 Apr 2005
Posts: 3

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Theories are increasingly theoretical Reply with quote

Thank you for the references. I think you are one of few people thinking
seriously about where String theory might be going.

But I am a little confused as to why you think we need to stick with
String theoretic foundations. To my way of thinking N = 2
SUSY QM is not fundamental. Categorification isn't about categorifying
bundle structures piece by piece. This is why (I think) Ross Street
says one should look at stack theory and leave gerbes alone.

In particular, recall that the notion of -point- becomes a geometric
morphism

\mathbf{Set} \rightarrow \mathbf{Sh}(M)

into the category of sheaves on a space M. This puts
geometry on a purely axiomatic footing.

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Thomas Larsson
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Joined: 01 May 2005
Posts: 73

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<3c2ar6F6jtgqnU1@news.dfncis.de>...

Quote:
The triangles are crucial in making the setup independent of the
latticization, since using them there is a way to get a TFT using structure
constants C_ijk.

If you have a quadrangulation, you can always cut each square in two triangles.

Quote:
More significantly, in order to be able to contract indices Akhmedov
introduces
a metric kappa^ij (bottom of page 5). I avoid that by putting two
four-index
quantities, and their inverses, on each plaquette. In that way I can
arrange
things so that always one up and one down index are contracted, and there
is no
need for a metric.

The metric is no extra structure in Akhmedov's setup, since it follows from
the C_ijk.

?????

In order to contract two lower indices, you always need something with two
upper indices, i.e. a (possibly degenerate) metric. You don't need a metric
to define holonomies and inverses in 1-gauge theory. However, you do need it
if you want you gauge group to be unitary or orthogonal.

Quote:
Pick any connected collection of triangles, i.e. a small surface element. It
defines a tensor with n incoming and m outgoing indices.

E.g., pick a single triangle. n+m=3, but what are m and n separately?
In C_ijk, which of i,j,k are incoming and which are outgoing?

Quote:
In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by

(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)

where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.

Hm. This part I don't understand.

Quote:
In fact, I believe it can be checked that this defines on the Ts the
structure of a weak monoidal category with all morphisms invertible. By
throwing in weak formal horizontal inverses (representing the "zig-zag
symmetry" of holonomy which is otherwise not captured) we get a weak
2-group.

I had speculated before that weak 2-groups allow for realizing n^3 scaling
behavior, and here indeed it is explicit, due to the TFT definition of the
whole thing. That's quite interesting.

G and dB scales as n^3, but [B,B] like n^4. So we need to contract [B,B]
with something which scales like 1/n, namely a vector parallel to the path.
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

"Thomas Larsson" <thomas_larsson_01@hotmail.com> schrieb im Newsbeitrag
news:24a23f36.0504130328.5036e55e@posting.google.com...


Quote:
In order to contract two lower indices, you always need something with two
upper indices, i.e. a (possibly degenerate) metric.


These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)


Quote:
Pick any connected collection of triangles, i.e. a small surface element.
It
defines a tensor with n incoming and m outgoing indices.

E.g., pick a single triangle. n+m=3, but what are m and n separately?


The edges of these are colored by an orientation which indicates if the
corresponding index is in- or outgoing.


Quote:
In C_ijk, which of i,j,k are incoming and which are outgoing?


As discussed in the literature, e.g. the hep-th/9212154 and as I mentioned
in my original post (http://golem.ph.utexas.edu/string/archives/000542.html)
an upstairs index corresponds to ingoing and a downstairs index to outgoing
(or vice versa, depending on your conventions).


Quote:
In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by

(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)

where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit
that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.

Hm. This part I don't understand.


This is just the continuum version of the obvious horizontal composition of
finite-resolution surface elements (as I have tried to describe here:
http://golem.ph.utexas.edu/string/archives/000542.html#c002138):

For instance let your /\ ~T and /\' ~ T' be single triangles given by
tensors T^ij_k and T'^lm_n. Then we can horizonatlly compose them to get
/\./\' given by the tensor

(T.T')^ijlm_kn = T^ij_k T'^lm_n .



footnote: (*)
This applies to the ordinary TFTs. Of course Akhmedov comes along and wants
to make the C_i^j_k position dependent by replacing them (that's not how he
puts it, though) with

B_i^j_k (x) = C_i^j_k + epsilon b_i^j_j(x).

One would have to work out the conditions on b_i^j_k(x) for this to give a
well-defined continuum limit. This has not been discussed in the literature
as far as I am aware and I also don't see that Akhmedov addresses this
point.
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: weakened nonabelian bundle gerbes and 2-bundles Reply with quote

"David Roberts - 1078662" <droberts@maths.adelaide.edu.au> schrieb im
Newsbeitrag news:200504140111.j3E1BA9u028138@staff.maths.adelaide.edu.au...

Quote:
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.


Quote:
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.


Quote:
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.


Quote:
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.
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Volker Braun
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Joined: 26 Apr 2005
Posts: 6

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

I just want to point out one detail. Lets take the usual 9+1 (or 3+1 for
historical reasons Smile dimensions, and split the tangent bundle TM=V+W
into time and space directions. If the "time-direction" line bundle W is
not trivial, then we are in such deep troubles that closed timelike curves
seem harmless in comparison. So for any physical spacetime, w_1(W) better
be zero.

On Sat, 23 Apr 2005 06:11:10 -0400, Thomas Mautsch wrote:

Quote:
While in Riemannian geometry the condition on a manifold
to carry a spin structure is that the second Stiefel-Whitney class
w_2(TM) be zero, in non-Riemannian geometry
the condition is connected to the splitting of M in space- and time-like
directions, TM = V + W, and it becomes:

w_2(TM) = w_1(V) u w_1(W).

And for any physically sensible M, that just reduces to w_2(TM)=0.
Volker
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