Towards nonabelian membrane action
Urs Schreiber
I am glad to be able to announce a new preprint:
John Baez & Urs Schreiber,
Higher Gauge Theory: 2-Connections on 2-Bundles,
hep-th/0412325 .
I have spent much time lately thinking about 'higher' gauge
theory aka 'nonabelian gerbe theory'.
Looking at string dynamics as SQM on loop space there is
a natural way to couple a nonabelian 2-form to a string,
as I have tried to point out in hep-th/0407122.
It was a certain constraint found in that paper that made me
become interested in an approach known as categorified gauge
theory, where the gauge group is replaced by a 2-group,
which is a 'stringified' group in a sense. 2-groups come in
a 'strict' flavor and in the more general 'weak' flavor.
As first noted by Girelli and Pfeiffer in hep-th/0309173
strict 2-groups impose a certain constraint on any local
connection that takes values in them, which is precisely
the constraint I encountered in loop space.
For these reasons it was natural to have a closer look at the
relation between loop space geometry and categorified gauge
theory.
The framework for that is the theory of 2-bundles recently
introduced by Toby Bartels in math.CT/0410328.
The idea here is simply to take the definition of an ordinary
bundle, which fundamentally is nothing but a map
p : E -> B
and 'stringify' it by letting the total space E and the base
space B be not ordinary spaces consisting of points, but
be 2-spaces which consist of an ordinary point space together
with a space of string configurations in that point space,
roughly. Then the map E -> B becomes a 2-map, which in
math jargon is known as a functor.
By continuing in this spirit one gets stringified locally
trivializable principal G-bundles, for G some 2-group.
Due to the nature of 2-spaces it is clear that loop space
reasoning plays a role in understanding them. It turned
out that it is possible to construct from a local connection
on loop space a 2-connection in a 2-bundle.
Moreover, by re-expressing the abstract arrow-theoretic
construction of principal G-2-bundles in terms of local
p-form data, it is possible to show that under certain
conditions 2-bundles with 2-connections define nonabelian
gerbes with conection and curving.
These nonabelian gerbes have only rather recently been
understood and in particular were shown in hep-th/0312154
to apply to the dynamics of membranes attached to stacks
of 5-branes. That demonstration relies on global anomaly
cancellation arguments and does not provide an acion
for the membrane whose boundary should couple to the
nonabelian 2-form.
The reason for that is that no notion of surface holonomy
has been known for nonabelian gerbes, as opposed to
the case of abelian gerbes.
It is interesting now that a 2-bundle with 2-connection
not only defines a nonabelian gerbe with connection and
curving, but at the same time automatically comes with a
notion of 2-holonomy. This should hence be related to the
description of M2-M5 configurations (but the details have not
been investigated yet).
The above mentioned paper will not be generally available
before next week. A copy of the paper os linked at my
weblog entry discussing it:
Lubos Motl
> I am glad to be able to announce a new preprint:
> hep-th/0412325 .
Congrats, looking forward to see it.
> The idea here is simply to take the definition of an ordinary
> bundle, which fundamentally is nothing but a map
>
> p : E -> B
>
> and 'stringify' it by letting the total space E and the base...
I thought that a bundle is a generalization of a direct product that
becomes just a direct product locally, but allows for nontrivial
monodromies globally, within the structure group.
If you use the base space of closed string loops, is not it simply
connected? Is not it a union of topologically trivial spaces (union
because there can be winding sectors)? In that case, is not your bundle
just a direct product? Aren't you talking about string fields? And if you
do, is your goal to make the string fields F-valued where F is something
different than the complex numbers? Why do you exactly think that this is
a physically consistent procedure? F should be a field with the usual
operations defined on it, in order to have any *-product like operations,
should not it?
> These nonabelian gerbes have only rather recently been
> understood and in particular were shown in hep-th/0312154
> to apply to the dynamics of membranes attached to stacks
> of 5-branes.
Is not it a little bit too strong statement? This paper looks like totally
abstract math without any demonstrated connections to string theory. The
only places in which they say something about the branes is the
introduction in which they cite some technically unrelated stringy papers,
and they conjecture that their work "should be relevant" for the
description of branes in string theory. But I don't see any evidence. I
don't think it's possible to find evidence - or a new description - by
pure thought, without dealing with the real objects (the stringy objects
in this case) and their properties.
> The reason for that is that no notion of surface holonomy
> has been known for nonabelian gerbes, as opposed to
> the case of abelian gerbes.
Is not it a general argument against this whole gerbe approach? If it's
not, what's exactly the loophole that you want to use to revive this
direction of thinking?
> It is interesting now that a 2-bundle with 2-connection
> not only defines a nonabelian gerbe with connection and
> curving, but at the same time automatically comes with a
> notion of 2-holonomy. This should hence be related to the
> description of M2-M5 configurations (but the details have not
> been investigated yet).
When you say that "it should be related to M2-M5", do you nontrivially
use any other property of the M2-brane than the number "2" in it, which is
the same number as the number in 2-bundles and 2-connections?
Thanks.
Happy New Year!
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Aaron Bergman
Motl wrote:
> If you use the base space of closed string loops, is not it simply
> connected?
No. The free loop space is topolgically the product of the based
loop space of the original space and the original space itself.
> Is not it a union of topologically trivial spaces (union
> because there can be winding sectors)? In that case, is not your bundle
> just a direct product?
You can still have nontrivial bundles over simply connected
spaces. Look at instantons on S^4, for example.
> Aren't you talking about string fields?
I think it's clear that bundles over loop space are *not* the
correct answer for string fields. In particular, I think that
most people believe that an elliptic object, ie, something that
lives in elliptic cohomology just as a vector bundle lives in
K-theory, should correspond to a sort of global twisting of a
string. But, bundles over loop space aren't enough to give you an
elliptic object. You need more information. This was the subject
of the recent stuff by Stoltz and Teichner, for example.
I thought about gerbes a bit sometime ago and there are some
intriguing aspects to them. In particular, there is a curvature
term that looks like it has the known invariance in string theory
where you can exchange background B-field for brane F-field. I'd
need to get all my signs straight to say anything for sure, but
I'm worried the B = F condition that Urs has referred to in the
past might mean that that part of the D-brane action vanishes.
Aaron
--
Aaron Bergman
<http://zippy.ph.utexas.edu/~abergman/>
Lubos Motl
> No. The free loop space is topolgically the product of the based
> loop space of the original space and the original space itself.
Oh yeah... You're right.
> You can still have nontrivial bundles over simply connected
> spaces. Look at instantons on S^4, for example.
In string field theory, the gauge field is just one specific mode or part
of the full open string field. What I don't understand is why would one
talk about bundles over the loops in the first place. Aren't the
function(al)s enough? Moreover, if one talks about something defined over
the loop space, then it is a kind of string field, and the natural gauge
symmetry is not SU(N) but its full open-stringy generalization, is not it?
> I think it's clear that bundles over loop space are *not* the
> correct answer for string fields.
Right, that's my feeling, too.
> I thought about gerbes a bit sometime ago and there are some
> intriguing aspects to them. In particular, there is a curvature
> term that looks like it has the known invariance in string theory
> where you can exchange background B-field for brane F-field. I'd
> need to get all my signs straight to say anything for sure, but
> I'm worried the B = F condition that Urs has referred to in the
> past might mean that that part of the D-brane action vanishes.
Hmm. Incidentally there's now a new paper by Jeff Harvey and Anirban Basu
about the M2-M5 system,
http://www.arxiv.org/abs/hep-th/0412310
Best, Lubos
Urs Schreiber
Happy New Year! And thanks for all the questions.
"Lubos Motl" <mo...@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.041230...@feynman.harvard.edu...
> On Thu, 30 Dec 2004, Urs Schreiber wrote:
> Aren't you talking about string fields?
I am not talking about string fields. (But see at the very end for a related
comment.)
The motivation is rather this:
The coupling of type II strings in the presence of D-branes to the B-field
is known to be described by *abelian gerbes*. This follows from
considerations of "global anomaly cancellation", i.e. by checking that the
action for the string is a well defined functional not only locally but also
globally.
The proof of that is reviewed on pp 8-9 of Adchieri&Jurco's hep-th/0409200.
See also for instance
Carey, Johnson and Murray:
Holonomy on D-Branes,
hep-th/0204199
and references given there.
It is strongly expected that strings on 5-branes (from whatever equivalent
point of view they are regarded) couple to nonabelian 2-forms. The goal is
to find out how exactly. On pp. 14 of the above hep-th/0409200 it is shown
that using abomalies in 11D sugra with 5-branes one similarly finds that
this nonabelian 2-form is described by nonabelian gerbes.
Alternatively, this result should be reproducible by checking that the
action functional for the M2 brane with its boundary coupled to the
nonabelian 2-form is globally well defined. This could so far not be checked
because it was unknown how to cook up an action for a nonabelian 2-form
coupled to the worldsheet of a membrane boundary because no notion of global
nonabelian surface holonomy was available.
One result of our paper is to demonstrate that for a special case of
nonabelian gerbes a notion of nonabelian surface holonomy does exist and how
it works.
>> These nonabelian gerbes have only rather recently been
>> understood and in particular were shown in hep-th/0312154
>> to apply to the dynamics of membranes attached to stacks
>> of 5-branes.
>
> Is not it a little bit too strong statement? This paper looks like totally
> abstract math without any demonstrated connections to string theory.
Please see section 5 on pp. 14 of that paper.
>> The reason for that is that no notion of surface holonomy
>> has been known for nonabelian gerbes, as opposed to
>> the case of abelian gerbes.
>
> Is not it a general argument against this whole gerbe approach? If it's
> not, what's exactly the loophole that you want to use to revive this
> direction of thinking?
As far as I am aware it is just a fact that abelian gerbes describe strings
coupled to abelian 2-forms and that nonabelian gerbes describe nonabelian
2-forms on 5-branes. Maybe there are other languages to describe the same
thing, but I don't think that the applicability of gerbes for strings is
hypothetical.
Regarding the "loophole": Surface holonomy for *abelian* gerbes is well
understood, provides the correct action for the string in the presence of
branes and is reproduced by our results.
On the other hand, nonabelian gerbes have not received much attention yet.
When I say that no notion of nonabelian surface holonomy has been known for
nonabelian gerbes I don't mean that there is a proof that none exists. On
the contrary, notions of *local* nonabelian surface holonomy have been
discussed before, as summarized at the my Coffee Table entry.
We show in our paper how these notions of local nonabelian surface holonomy
by Alvarez, Pfeiffer et al, when embedded into the context of 2-bundles,
generalize to a globally defined nonabelian surface holonomy that comes from
the connection and curving of a nonabelian gerbe.
> When you say that "it should be related to M2-M5", do you nontrivially
> use any other property of the M2-brane than the number "2" in it, which is
> the same number as the number in 2-bundles and 2-connections?
Yes, I think so.
This is also not my idea. As you know, it is generally believed that
4-dimensional super YM has a decompactification limit to a six-dimensional
theory which involves nonabelian 2-forms, and that this six-dimensional
theory is equivalently described as the worldvolume theory of a stack of
5-brans with membranes ending on them.
Didn't you recently mention somewhere that you talked to A.J. Tolland about
precisely this?
Let me end with a brief comment on the relation of surface holonomy with
string fields:
In order to compute ordinary holonomy one glues together pieces of worldline
to get a closed curve or something. Similarly, in order to compute surface
holonomy one glues together pieces of worldsheet to get a closed surface or
something. This involves operations that are quite reminiscent of string
field operations.
In particular, in our paper we use the "2-groupoid of bigons" which is the
structure obtained by taking pieces of surface under the obvious operation
of gluing such pieces together. The boundaries of these pieces are
parametrized pieces of string and gluing them together looks very much like
some string field vertex operation. It is non-associative in that the
parametrization depends on the order in which three pieces of string are
glued this way. So this is not like the cubic OSFT vertex but similar to
operations in closed SFT.
So this might have a relation to string field theory in the long run. But
right now it is just a formal similarity.
Urs Schreiber
news:slrnct9kg4.mnp....@cardinal3.Stanford.EDU...
> In article <Pine.LNX.4.31.041230...@feynman.harvard.edu>,
> Lubos Motl wrote:
>
>> If you use the base space of closed string loops, is not it simply
>> connected?
>
> No. The free loop space is topolgically the product of the based
> loop space of the original space and the original space itself.
One interesting result of looking at 2-bundles is that in order to derive
nonabelian gerbes as defined by Breen, Messing, Aschieri, Cantini & Jurco
from them one can restrict to the special class of 2-bundles where the base
2-space (but not the total space) is that of free "infinitesimal loops",
where an "infinitesimal loop" based at x is something modeled by an element
of the dual space of Omega^2_x.
For this case of course the topology of the "loop space" simplifies
drastically, as there are no windings.
2-bundles are easily formulated for more general base 2-spaces, but then
their expression in terms of local data becomes more subtle. In any case, we
don't have a concise formulation yet.
>> Is not it a union of topologically trivial spaces (union
>> because there can be winding sectors)? In that case, is not your bundle
>> just a direct product?
>
> You can still have nontrivial bundles over simply connected
> spaces. Look at instantons on S^4, for example.
Yes, and in particular one can have *twisted bundles* and *twisted gerbes*.
The most general bundle on a D-brane is a twisted one. The most general
nonabelian gerbe on a 5-brane is a twisted one.
>> Aren't you talking about string fields?
>
> I think it's clear that bundles over loop space are *not* the
> correct answer for string fields.
Just for the record I would like to emphasize that I never claimed that they
are.
Nor did I claim that a 2-bundle is the same thing in general as a bundle
over loop space. A 2-bundle with base 2-space a loop space can be locally
described in terms of trivial bundles over loop space, but I am not sure
that this remains true globally in general.
Ah, but I recall that weeks ago on sci.physics.research I mentioned the idea
of looking at YM theory over loop space. Maybe that is causing the
discussion here. But that was in the context of a general discussion of loop
space stuff. Of course YM on loop space cannot be related to string field
theory. An ordinary local field theory on loop space (if it can be well
defined) would describe string interactions where the elementary vertex
consists of two strings completely overlapping and becoming a single string,
or vice versa. This is of course not the interaction of SFT. The latter
instead is non-local on loop space, because there two different strings
(seperated by a finite distance in loop space) merge to become a single
string, or vice versa.
> In particular, I think that
> most people believe that an elliptic object, ie, something that
> lives in elliptic cohomology just as a vector bundle lives in
> K-theory, should correspond to a sort of global twisting of a
> string. But, bundles over loop space aren't enough to give you an
> elliptic object. You need more information. This was the subject
> of the recent stuff by Stoltz and Teichner, for example.
Do you have a reference for that? Thanks.
> I thought about gerbes a bit sometime ago and there are some
> intriguing aspects to them. In particular, there is a curvature
> term that looks like it has the known invariance in string theory
> where you can exchange background B-field for brane F-field. I'd
> need to get all my signs straight to say anything for sure, but
> I'm worried the B = F condition that Urs has referred to in the
> past might mean that that part of the D-brane action vanishes.
Yes, this condition requires more thought. But before jumping to any
conclusions one should carefully look at the nature of the fields involved
here. For one, I do not think that the 1-form field A that enters into B =
F_A (using your notation) is related to an ordinary gauge field on a
D-brane.
To see what I mean, look at abelian gerbes:
An open string on a D-brane with a B-field turned on is described by the
abelian gerbe
(B_i, a_ij, lambda_ijk)
with
B_i the Kalb-Ramond field
a_ij an auxiliary 1-form field
and
lambda_ijk the gerbe "transition function".
The gauge field on the D-brane is not part of this data but enters
seperately.
For nonabelian gerbes the above data is enlarged and now includes also
fields of deRham degree 1 and Chech degree 1, namely the 1-forms A_i . There
is no reason to expect these to be ordinary D-brane gauge fields either.
So what the constraint
dt(B_i) + F_A_i = 0
(as I write is) really says is this:
The nonabelian h-valued 2-form B_i cannot be any h-valued 2-form but must be
such that the part of it which takes values in the complement of the kernel
of the Lie algebra homomorphisms dt : h -> g can be written as the curvature
of a g-valued 1-form.
So while this does not imply the restriction that it seems you have in mind,
I know very well that this constraint is unexpected and worrisome. I spent a
lot of thought into the question how it could be alleviated. It is easy to
write down zillions of local connections on loop space (path space) which do
not have this constraint. But the problem is that in the end they need to be
glued together to give global connections with a well-defined gloabal
surface holonomy. This is where 2-groups come in, since their exchange law
ensure that composition is well defined. And they also imply this
constraint. In order to alleviate it one would have to find a notion of
surface holonomy that does not use strict 2-groups. Coherent 2-groups might
be a first step, but they still sort of have this constraint.
That's why I began to try to see if there is anything else in favor of this
funny constraint. As I mentioned before, it might be related to the
self-duality condition.
The curvature 3-form H on the nonabelian gerbe describing the theory on a
stack of 5-branes should be Hodge self-dual, as far as I know. Now have a
look at equations (52)-(59) of Aschieri&Jurco's hep-th/0409200 and see if
you can find any globally defined self-dual H when the fake curvature does
not vanish. You'll get a formidable set of consistency conditions. Being
immensely involved, they might have no non-trivial solution. The only
obvious solution is obtained by setting dt(B_i) + F_A_i = 0, I think. I have
no prove that there are no other solutions. But those with dt(B_i) + F_A_i =
0 certainly seem to play a special role.
Aaron Bergman
>> In particular, I think that
>> most people believe that an elliptic object, ie, something that
>> lives in elliptic cohomology just as a vector bundle lives in
>> K-theory, should correspond to a sort of global twisting of a
>> string. But, bundles over loop space aren't enough to give you an
>> elliptic object. You need more information. This was the subject
>> of the recent stuff by Stoltz and Teichner, for example.
>
> Do you have a reference for that? Thanks.
JB has written a couple of twf's about this. The paper is
available at
<http://www.math.ucsd.edu/~teichner/papers.html>
>> I thought about gerbes a bit sometime ago and there are some
>> intriguing aspects to them. In particular, there is a curvature
>> term that looks like it has the known invariance in string theory
>> where you can exchange background B-field for brane F-field. I'd
>> need to get all my signs straight to say anything for sure, but
>> I'm worried the B = F condition that Urs has referred to in the
>> past might mean that that part of the D-brane action vanishes.
>
> Yes, this condition requires more thought. But before jumping to any
> conclusions one should carefully look at the nature of the fields involved
> here. For one, I do not think that the 1-form field A that enters into B =
> F_A (using your notation) is related to an ordinary gauge field on a
> D-brane.
>
> To see what I mean, look at abelian gerbes:
>
> An open string on a D-brane with a B-field turned on is described by the
> abelian gerbe
>
> (B_i, a_ij, lambda_ijk)
>
> with
>
> B_i the Kalb-Ramond field
>
> a_ij an auxiliary 1-form field
>
> and
>
> lambda_ijk the gerbe "transition function".
>
> The gauge field on the D-brane is not part of this data but enters
> seperately.
No. They're linked. This is clearly seen in the example of a
torsion three form. Then the gauge field on the brane is no
longer part of a vector bundle, but instead a section of a
SU(N)/Z_n bundle which is classified by the torsion three
form. This is described in Witten's paper on K-theory for
example. The three form twists the gauge field even in the
abelian case.
ISTR that this twisting is in accord with the two form for a
gerbe, but I could be wrong about that. I think Kapustin goes
through the cancellation of the Freed-Witten anomaly in this
case. I would think that for non torsion H, a similar argument
would show that the field theory living on the brane is described
by a connection on an abelian gerbe.
Some evidence for this is that, in the DBI action, F and B
always appear in the combination B+F which looks a lot like the
'fake curvature' you refer to (again up to signs).
Urs Schreiber
> In article <33soqpF43pv...@individual.net>, Urs Schreiber wrote:
> > The gauge field on the D-brane is not part of this data but enters
> > seperately.
>
> No. They're linked.
Yes, they are linked, but the 1-form on the D-brane is not the 1-form in
the gerbe cocycle.
So in the simplest case of an abelian 2-form and everything globally
defined we integrate B over the worldsheet and A over its boundary, which
is the same as integrating B+F over the bulk, of course. What I wanted to
point out is that the B+F that appears in the discussion of nonabelian
gerbes involves not (at least not in any obvious way) this same F but
another F.
> Some evidence for this is that, in the DBI action, F and B
> always appear in the combination B+F which looks a lot like the
> 'fake curvature' you refer to (again up to signs).
Yes, it looks very similar. Maybe its related.
Aaron Bergman
<Pine.LNX.4.31.050104...@feynman.harvard.edu>,
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> On Mon, 3 Jan 2005, Aaron Bergman wrote:
>
> > In article <33soqpF43pv...@individual.net>, Urs Schreiber wrote:
>
> > > The gauge field on the D-brane is not part of this data but enters
> > > seperately.
> >
> > No. They're linked.
>
> Yes, they are linked, but the 1-form on the D-brane is not the 1-form in
> the gerbe cocycle.
I'll have to read your paper. When I read JB's paper back when, I
thought it was clear that it was.
Aaron
Urs Schreiber
Do you mean "Higher Yang-Mills theory" hep-th/0206130 ?
In that paper the connection considered was what from the present point of
view would be called a 2-connection on a trivial 2-bundle (which we now
showed to be the same as a fake-flat connection in a trivial nonabelian
gerbe). Then an action principle for the 2-form B_i = B and the 1-form A_i =
A defining that 2-connection was postulated and studied, namely
S = int tr ( curvature(A)^2 + "curvature"(B)^2 ) .
This action is natural from a certain point of view, but probably not
directly related to any 'nonabelian gerbe theory' as expected to describe
brane physics, largely because it is not invariant under the higher order
gauge transformation
A -> A + dt(a)
B -> B - (da + a/\a) .
To date, as far as I know, nobody has yet written down a satisfactory action
principle governed by a nonabelian gerbe. Though other proposals have been
made by Lahiri and Hofman, for instance.
But maybe you are right. All I am saying is that to me it looks like an open
question and that I don't see a compelling hint that the A_i of the
nonabelian gerbe cocycle data has to be identified with a D-brane gauge
field.
As a plausibility argument I mentioned that for strings on D-branes where
the Kalb-Ramond field is described by abelian gerbes the gauge field is not
part of the gerbe cocycle data, even though it is not unrelated.
For the nonabelian case I don't see that in general there are any D-branes
in the game at all, because the M5 might become just NS5s depending on the
compactification.
Instead, the identification of the A_i in the nonabelian gerbe "cocycle"
data seems to receive its physical interpretation by way of equations like
(66) in A&J's hep-th/0409200. This equation relates all the nonabelian gerbe
data to the abelian supergravity 3-form.
But if you think I am missing something please be so kind and let me know.
P.S.
I have begun looking into elliptic cohomology. I hope to get to the point to
ask some reasonable questions soon.
Urs Schreiber
> In article <33soqpF43pv...@individual.net>, Urs Schreiber wrote:
Aaron Bergman wrote:
>>> In particular, I think that
>>> most people believe that an elliptic object, ie, something that
>>> lives in elliptic cohomology just as a vector bundle lives in
>>> K-theory, should correspond to a sort of global twisting of a
>>> string. But, bundles over loop space aren't enough to give you an
>>> elliptic object. You need more information. This was the subject
>>> of the recent stuff by Stoltz and Teichner, for example.
>>
>> Do you have a reference for that? Thanks.
>
> JB has written a couple of twf's about this. The paper is
> available at
>
> <http://www.math.ucsd.edu/~teichner/papers.html>
Ok, I have skimmed TWF 197, 149 and 150
http://math.ucr.edu/home/baez/week149.html
http://math.ucr.edu/home/baez/week150.html
http://math.ucr.edu/home/baez/week197.html
as well as briefly looked at the Stolz and Teichner text linked above.
I haven't read enough yet to really know what I am talking about, but let me
venture a vague observation that may help me to get started:
Early on John Baez has speculated that there could be or should be a
relation between categorified gauge theory and elliptic cohomology. Both
deal with associating holonomies in one sense or another to paths in loop
space modulo re-parametrization of the corresponding surfaces. No details
concerning such a relation are known so far. Now we have been asked a
related question by Edward Witten and I would like to at least get a rough
idea of what the question entails.
So in the context of elliptic cohomology one makes use of the fact that path
integrals of covariant theories on d-dimensional manifolds are natural
candidates to be interpreted as d-dimensional parallel transport, roughly.
When conformal field theory (in two dimensions, say) is formulated in the
functorial TFT-like style following Graeme Segal (e.g. as described in
http://www.math.umn.edu/~voronov/18.276/) we associate Hilbert spaces with
loops and Hilbert-Schmidt operators between these Hilbert spaces with 2-d
cobordisms (worlsheets) between these loops - the "propagators".
In one dimension lower one can show that a similar association of Hilbert
spaces to points and operators to lines (known as quantum mechanics) is
related to complex K-theory. The analogue of this in two dimensions is
elliptic cohomology.
So in this context one is dealing with Hilbert space bundles over a free
loop space over spacetime and CFT "propagators" (path integrals) between
different fibers play the role of parallel transport.
The question is: Can this be captured by 2-bundles?
I don't know, but from what I do know the following observation looks rather
suggestive:
Notwithstanding my current non-familarity with the Stolz&Teichner work
(which hopefully improves soon) I dare to say that it seems that in terms of
bundle theory we have it to do with structure groups something like U(H),
the group of unitary operators on a complex seperable Hilbert space H.
Right?
In order to make contact to 2-bundles it is useful to have a central
extension in hand. The natural central extension here is
U(1) -> U(H) -> PU(H) .
We know that 2-bundles are related to nonabelian gerbes and in
hep-th/0409200 Aschieri&Jurco argue (pp.16-17) that this central extension
is the most general one in this context. For base spaces of dimensions
smaller than 14 (as they appear in string theory) this can with respect to
homotopy be reduced to
U(1) -> tilde Omega E_8 -> Omega E_8,
where Omega E_8 is the based loop group of E_8.
2-bundles provide a notion of holonomy for nonabelian gerbes with the strict
structure 2-group coming from an automorphism crossed module. Could it be
that this 2-holonomy for an automorphism crossed module coming from U(H) or
Omega E_8 is related to elliptic cohomology?
Certainly the constraint of vanishing fake curvature again looks bothersome
here. But since this depends on a special choice of standard 2-fiber in our
2-bundle which could be replaced, it would be interesting for me if the
above could roughly be true up to that constraint.
I'll be grateful for any input.
Aaron Bergman
> As a plausibility argument I mentioned that for strings on D-branes where
> the Kalb-Ramond field is described by abelian gerbes the gauge field is not
> part of the gerbe cocycle data, even though it is not unrelated.
Now that I'm back in Austin, I'll see if I can reconstruct why I thought
the opposite was true.
> For the nonabelian case I don't see that in general there are any D-branes
> in the game at all, because the M5 might become just NS5s depending on the
> compactification.
It's not clear to me that gerbes are the proper description of the
theory on a stack of M5 branes. Can you reproduce the N^3 scaling?
Aaron
Urs Schreiber
It is expected to be so, e.g.
http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/71.bmp
and proof (evidence ?) is presented in hep-th/0409200 (see section 5).
> Can you reproduce the N^3 scaling?
There is an argument that it should be possible to reproduce the N^3
scaling and I have a plan for how to make it explicit, but it is not done
yet. I have written about that plan here
http://golem.ph.utexas.edu/string/archives/000461.html .
It was an observation by Andrew Neitzke that the product Hochschild operator
M in Hofman's paper hep-th/0207017, which essentially encodes the product in
the Lie algebra, seems like a natural candidate for a carrier of ~n^3
degrees of freedom. (In fact without any restriction it carries even more
degrees of freedom, I believe).
These same degrees of freedom can be recognized also in the context of a
slight generalization of nonabelian bundle gerbes as well as, I think, in
2-bundles with *coherent* 2-groups, which is he sort of 2-group that should
ultimately be used in 2-bundles.
Understanding 2-bundles with coherent 2-groups is the obvious next step
anyway. In our recent paper we work with the weak groupoid of bigons
(surface elements) but let our 2-holonomy be a functor from that to a strict
2-group. That's the easier case, but really a functor with source a weak
2-group wants to have as target a weak 2-group and hence coherent 2-groups
are what should be the really interesting case.
Urs Schreiber
S. Stolz & P. Teichner,
What is an elliptic object?
http://www.math.ucsd.edu/~teichner/papers.html .
Here is a very rough survey followed by a question:
G. Segal said that an elliptic object should be a functor
C(X) -> V
from the category of 2-bordisms with conformal structure whose objects are
loops in X and whose morphisms are worldsheets between these to the category
V of topological vector spaces (Hilbert spaces).
This is essentially a bosonic 2d conformal field theory, i.e a closed
bosonic string theory.
However, this is not quite what one wants because it fails to capture a
property of elliptic cohomology known as "excision".
In order to fix that Stolz and Teichner "enriched" this definition by
a) including fermions
b) going from closed to open strings
roughly.
After defining this and that they say that an enriched elliptic object is a
*2-functor*
D(X) -> vN .
Here D(X) is roughly something like a "2 category of open strings" (I won't
mention the fermions here). Its objects are points in X (string endpoints),
its morphism are "curves" in X and its 2-morphisms are surfaces between
these.
That's not surprising. What is a little unexpected first is the target vN.
vN is the bicategory of von Neumann algebras with objects von Neuman
algebras, morphisms bimodules of them and 2-morphisms intertwiners between
these.
That this really is a generalization of Segal's definition in a sense is due
to the fact that we can have an isomorphism from Segal's Hilbert spaces
associated with loops to the "fusion product" of two von Neuman bimodules
associated with two arcs that this loop can made up from.
I am glossing over ~70 pages of details here... (and may have stated things
in a way that is almost but not quite right).
The point I would like to get at is the question mentioned before, if this
"enriched elliptic object" could be related to 2-connections in 2-bundles.
Now, 2-bundles are an awfully general concept. More precisely, it would be
nice to know if a 2-connection in a *principal* 2-bundle, i.e. one whose
typical 2-fiber is some sort of "2-group", is related to enriched elliptic
objects.
For the only case really understood where the typical fiber is a strict
2-group (i.e. a category internalized in the category of groups) a
2-connection in the respective 2-bundle is a 2-functor
P_2(X) -> G
from the 2-groupoid of bigons P_2(X) to the 2-group G, as described in
hep-th/0412325.
This is superficially somewhat similar to an enriched elliptic object, but
really different. I would like to know if it could be made more similar by
doing something to it.
The main similarity, apart from the fact that both concepts are 2-functors,
is the source 2-category, which in both cases involves points and strings
and worldsheets. The main difference is, I believe, that for the
2-connection there are lots of notions of *inverses*, which don't seem to be
there for elliptic objects.
So first of all bigons in P_2(X) are surface elements up to thin homotopy.
This implies that there is an "inverse bigon" to a given bigon, both with
respect to horizontal as well as vertical composition. If I follow correctly
Stolz&Teichner's definitions this is not the case for the 2-morphisms in
D(X).
More importantly, G is a 2-group and hence has horizontal and vertical
inverses. The crucial question in deciding whether 2-holonomy can be related
to elliptic objects seems to me to be to what degree and in which sense
(strict, weak) the 2-category vN has inverses.
I am not sure. Can anyone help me? Do there exist any inverses to the
horizontal composition defined by Stolz&Teichner on p. 60?
Aaron Bergman
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> That's not surprising. What is a little unexpected first is the target vN.
>
> vN is the bicategory of von Neumann algebras with objects von Neuman
> algebras, morphisms bimodules of them and 2-morphisms intertwiners between
> these.
I've been told that the von Neumann algebra part may not be necessary,
but I don't remember the details.
[...]
>
> Now, 2-bundles are an awfully general concept. More precisely, it would be
> nice to know if a 2-connection in a *principal* 2-bundle, i.e. one whose
> typical 2-fiber is some sort of "2-group", is related to enriched elliptic
> objects.
I would hope that a principal 2-bundle would entail some sort of action
(2-action, god help us?) by a 2-group just as a circle bundle is not the
same as a U(1) principal bundle.
> bigons
I can't see that term without thinking about
<http://www.findarticles.com/p/articles/mi_m1511/is_n4_v17/ai_18107914>
Aaron
Urs Schreiber
Let me try to make this more precise, using the stuff from hep-th/0409200:
An F-string on a stack of D-branes in the presence of a Kalb-Ramond field is
related to an abelian 1-gerbe G such that G restricted to the D-branes,
denoted G_Q with Deligne class
[G_Q] = [lambda_ijk, alpha_ij, beta_i]_Q ,
fulfills the relation
[G_Q] = [1,0,B_Q] + [D(G_ij, A_i)] + [omega_ijk,0,0] .
On the left is the Deligne class of the abelian 1-gerbe, on the right is
[1,0,B_Q] - the abelian gerbe coming from the Kalb-Ramond field B
restricted to Q where it can be taken to be globally defined
[D(G_ij,A_i)] - the abelian lifting gerbe of the (possibly twisted)
nonabelian bundle (G_ij, A_i) on the branes (D denotes a nonabelian
generalization of the Deligne coboundary operator)
[omega_ijk,0,0] - something related to spinors that I am going to ignore in
the following
This is how the gerbe, the Kalb-Ramond field and the D-brane gauge field A
are related to each other. In particular, the 1-forms alpha_ij in the gerbe
G cocycle (lambda_ijk, alpha_ij, beta_i) are not equal to the gauge field
1-forms but related to them as
alpha_ij = G_ij(d+A_j)G_ij^-1 - A_i
i.e., they measure the twist in the connection on the branes.
As Aschieri and Jurco emphasize in their paper, the form of the right hand
side essentially follows from a theorem that the gerbe on the left can be
expressed in terms of one of the form [1,0,B] with global B plus a lifting
gerbe of a possibly twisted bundle. Hence this can be viewed as one way to
*derive* the coupling of the boundary of the string to a nonabelian gauge
field from its coupling to the abelian KR 2-form.
This is of importance for the step to one dimension higher.
A (twisted) nonabelian bundle can be called a (twisted) nonabelian 0-gerbe.
Hence we have here that a p-"brane" ending on a stack of branes is described
by
- an abelian p-gerbe coupled to the bulk
and
- a (twisted) nonabelian (p-1)-gerbe coupled to the boundary
of the p-"brane", for p=1.
Going up one dimension the above scenario becomes concerned with a membrane
ending on a stack of 5-branes with a coupling to the abelian supergravity
3-form C.
C now plays a role analogous to B before.
The holonomy of the abelian 3-form C on a 3-cycle is computed by abelian
2-gerbe holonomy. From results by Diaconescu, Moore and Freed
(hep-th/0312069) it follows that this 2-gerbe is a "Chern-Simons 2-gerbe"
with respect to an E_8 Chern-Simons form. So this 2-gerbe class is denoted
[CS.....] =: [CS].
By a reasoning completly analogous to that above, [CS] can be written as
[CS] = [D G] + [1,0,0,C] + [theta_ijkl,0,0,0].
Here D is again the nonabelian Deligne operator and G now denotes a possibly
twisted *nonabelian* 1-gerbe . D G is the abelian lifting 2-gerbe of that
possibly twisted nonabelian 1-gerbe.
By comparison with the above, the nonabelian gerbe G with cocylce data G =
(f_ijk, phi_ij, a_ij, d_ij, A_i, B_i) should describe the coupling of the
boundary of the membrane to the 2-form B_i.
This 2-form is not the Kalb-Ramond 2-form but the nonabelian generalization
of the abelian 2-forms found in the six-dimensional SCFT on the 5-brane
worldvolume.
Similarly, the A_i enetering the nonabelian gerbe cocycle here are not in
any obvious way related to D-brane gauge fields. They instead appear just as
auxiliary as the transition functions phi_ij, for instance.
So once one understands holonomy of nonabelian 2-gerbes one should get that
a 2-brane ending on a stack of branes is described by
- an abelian 2-gerbe coupled to the bulk
and
- a (twisted) nonabelian (2-1 = 1)-gerbe coupled to the boundary
of the 2-brane,
just as above for 1-dimension lower.
The closest relation of the nonabelian 1-gerbe cocylce data to a D-brane
gauge field that I can see is that after compactifying the 5-branes on a
circle or torus the nonabelian B-field with one index in the compact
direction should give rise to the nonabelian gauge field on the resulting D4
or D3 brane. This is at least what happen in the abelian case.
Kea
Nice to see the paper, Urs. Let me try again, Lubos. By the way,
it takes days for my posts to appear here, which is a little
disconcerting.
When Grothendieck was thinking about Weil cohomology, he realised
that he need to think of topological spaces as categories of open
sets (this is basic algebraic geometry) and functors from (the
dual of) this category to the category of Sets (defined by a
Grothendieck universe)....this is just what a sheaf is really
about.....again: basic algebraic geometry.
The notion of 'topos' was invented by Grothendieck......so let's
look at some basic logic.....
In classical (Boolean) logic the law of the excluded middle
U \vee \neg U = \textrm{true}
holds. Consider the following examples of its failure.
1. The collection of subspaces of a Hilbert space
\mathcal{H} forms an algebra under intersection
\wedge and U \vee V the union of all
subspaces whose intersection with U is contained in V.
\neg is the orthogonal complement (hence the need for
inner product). False is the zero subspace and true is the full
space \mathcal{H}.
2. The open sets of the two dimensional sphere under the
operations of intersection and union, with \neg U the
interior of the complement (it can't be the complement because
that's not an open set). False is the empty set, and true the
whole sphere.
This second example is interesting because it describes the two
sphere as a category (with an arrow between open sets when there
is an inclusion) which underlies the sheaf of germs of analytic
functions...anyway, think of it as the celestial sphere of an
observer in GR and it turns up in twistor theory.
One might also consider the axiom of choice. The 'local' version
of this also forces Booleanness on one's logic.
So we're going to have to be rather careful with which mathematics
we use if we want a description of the quantum world that is
sophisticated enough to describe decoherence, for instance.
As far as I understand it (correct me if I'm wrong, Lubos) String
theorists are quite keen on twistors at the moment, but they do
not address these questions.
Now - what has all this to do with String theory and 2-bundles?
Lubos - the '2' is a by no means trivial idea, as it refers to the
2-categorical structure of the representation theory in question.
The connection between this and superalgebras (and more
generalised algebras for higher n) is know quite well understood
in the context of omega-categories (see Verity's latest paper if
you're really interested).
Lubos: are you going to answer my previous questions?
Happy New Year
Kea
:devil:
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Urs Schreiber
> By the way, it takes days for my posts to appear here, which is a little
> disconcerting.
[...]
> ------------------------------------------------------------------------
> This post submitted through the LaTeX-enabled physicsforums.com
> To view this post with LaTeX images:
> http://www.physicsforums.com/showthread.php?t=58174#post415448
Since the moderators of sci.physics.strings operate in very different
time zones any submission to SPS is likely to be moderated well within
24 hours (usually much less, except maybe on weekends) *after* it
appears in our inbox.
The delays that people submitting via PhysicsForums are experiencing
are caused by PhysicsForums, not by the SPS moderators.
It seems that instead of directly forwarding every single message the
PhysicsForums sofware accumulates SPS submissions for a couple of days and
then submits a bunch of them to the SPS moderators at a time. Maybe there
is even a threshold number of accumulated messages which has to be reached
before any one of them arrives here.
Those who use PhysicsForums to submit messages to sci.physics.strings and
who are bothered by these delays could do one of the following things:
- Send a request to the people in charge of PhysicsForums and kindly ask
them if it were possible to modify their forum software such that SPS
submissions through PF are handled more quickly.
- Use the Google Groups web interface to SPS instead.
- Subscribe to a newsserver and access SPS more directly using a
newsreader software.
More details on how to access SPS are given at the group's homepage that
Lubos Motl maintains:
Urs Schreiber
<Some random examples involving categories.>
> Now - what has all this to do with String theory and 2-bundles?
Nothing! :-) Let's try not to get carried away with the c-word.
But in this context, it is maybe amusing to note that today appeared a
discussion of the 4 color theorem using 2-connections:
Romain Attal:
Combinatorial Stacks and the Four-Colour Theorem,
math.CO/0501231 .
The author computes the number of colorings that you can draw on a
polyhedron by regarding that polyhedron as a triangualized worldsheet and
computing a certain combinatorial 2-holonomy of a string with that worldsheet,
roughly.
Romain Attal introduced his idea of 2-connections in
Two-Dimensional Parallel Transport:
Combinatorics and functoriality,
math-ph/0105050
and specified it to Lie-group valued 2-connections in
Combinatorics of Non-Abelian Gerbes with Connection and Curvature,
math-ph/0203056.
Locally and once you specify a gauge and up to issues of differentiability
the 2-holonomy defined in that paper is essentially the same as we used in
our paper.
> Lubos - the '2' is a by no means trivial idea, as it refers to the
> 2-categorical structure of the representation theory in question.
> The connection between this and superalgebras (and more
> generalised algebras for higher n) is know quite well understood
> in the context of omega-categories (see Verity's latest paper if
> you're really interested).
So what's the relation between superalgebras and 2-categories, roughly?
Kea
I must apologise for not explaining myself. I am posting from
PF rather than here.
I should begin by making it clear that while 'standard M-theory'
appears to be a reasonable approximation to the 'true' M-theory,
there are those of us who are strongly of the opinion that the
'true' M-theory is purely category theoretic and hence one cannot
say 'let us not get carried away with the C word'.
First of all, a quick answer to the question of superalgebras.
Following the papers
Quark State Confinement as a Consequence of the Extension of the
Bose-Fermi Recoupling to SU(3) Colour, W. P. Joyce,
http://arxiv.org/abs/hep-th/0306256
and
Recoupling Lie algebra and universal omega-algebra, W. P. Joyce,
J. Math. Phys. 45, 10 (2004) 3859-3877
consider the constraints on phase factors for a (unital) symmetric
premonoidal structure. For instance, for reps of SU(2) (as a
monoidal category) one has the freedom to choose $\gamma_{1,1} =
-1$. In general, this phase defines a bracket [a,b] = - gamma
[b,a]. In the context of omega-algebras one can take this
generalisation and prove a PBW theorem.
Duality for n-stacks:
I mentioned QGC (quantum general covariance) casually but
nonetheless expecting that this would prompt a question. The way
that I understand the category theoretic picture (and this may
well be naive) the physics is quite DIFFERENT to the physics of
string theory. And this is why one cannot say, at the end of the
day, that the standard M-theory is in any way 'correct'.
To begin with, in String theory (at least in my very brief
Heterotic String notes) T-duality is derived, but physically it
should be a fundamental principle. That is to say, to disallow an
objective 'over the horizon' observer is to make a statement about
physics beyond the SM and GR. To relate UV and IR scales is a
physical picture which one can visualise from considering
renormalisation in the SM: or rather, ask oneself the question
'what is renormalisation approximating?'. As one approaches the
Planck scale one imagines a 'universe of possibilities'. Perhaps I
might elaborate on this a little bit....
One is quite used to the notion of indistinguishibility for
'fundamental' particles in the standard model. Some time ago
Bekenstein suggested that quantum gravity should similarly be
constructed out of a notion of fundamental state for spacetime
degrees of freedom. But it is difficult to remove oneself from the
prejudices of the classical picture for spacetime, be it
continuous or discrete.
When one observes a hydrogen line through an optical telescope on
Earth one presumes that this photon was emitted by a hydrogen atom
in the star at which the telescope is pointed. This star has been
observed a LARGE number of 'times' to occupy a known classical
(decohered) position in the sky.
Quantum numbers live in abstract spaces. And yet many discussions
of quantum gravity speak of the Early Universe, for instance, as
if it lived in a physical domain independent of observations of
it. The classical theory (GR) must be recoverable from quantum
gravity, but the starting point cannot be an ontological view of
spacetime. To remove oneself from this prejudice it appears one
must employ purely category theoretic ideas (see Markopoulou,
Isham and others) because even the best String theory stuff (that
I'm aware of, anyway) strongly prejudices the geometry of
spacetimes.
The first interesting step towards a modern category theoretic
understanding of mass is perhaps the study of the Klein-Gordon
equation in Twistor theory in
L.P. Hughston T.R. Hurd, A cohomological description of massive
fields, Proc. Roy. Soc. Lond. A378 (1981) 141-154
In this paper, Hughston and Hurd combine two solutions to the
massless equations for spin $s$ particles thought of as elements
of a sheaf cohomology group $H^{1}(\mathbb{T}^{+} , \mathcal{S}(-2
s - 2))$ on a twistor space. The Klein-Gordon equation solutions
then belong to a second cohomology group $H^{2}(\mathbb{T}^{+}
\times \mathbb{T}^{+} , \mathcal{S}_{m,s}(- \mu - 2 , - \eta -
2))$ for $s - \frac{1}{2} | \mu - \eta | \in \{ 0,1,2,3 \cdots
\}$.
Naively at least, therefore, a quantisation of this origin of mass
involves a non-Abelian sheaf theoretic second cohomology group. On
this we agree! However, by beginning with physical considerations
there is no reason whatsoever to talk about bundle gerbes rather
than stacks in the sense of Grothendieck: 1-stacks are sheaves
(functors if you like), 2-stacks are pseudofunctors (see Street on
Descent theory) etc. And if one thinks this way, one doesn't
destroy the logic of causality underlying twistors.
The first cocycle condition
\[ f_{12} - f_{31} + f_{23} = 0 \] may be thought of as the diagram
$K_{2}$
\[ \xymatrix { & 2 \ar[dr]^{f_{23}} & \\ 1 \ar[rr]_{f_{13}}
\ar[ur]^{f_{12}} && 3 } \] Such diagrams make sense in any
category (see Street's classic orientals paper), so the
coefficients for $H^{1}$ may be generalised, in particular to
non-Abelian groups. The difficulty arises in understanding
categories deeply enough to develop a sufficiently subtle higher
dimensional analogue. This is what descent theory is about.
The concrete removal of base spaces is beautifully captured by the
idea of the Big Zariski topos. This category $\mathcal{Z}$
classifies spaces (affine schemes) by replacing them by arrows
from the (opposite of the) category of finitely generated
commutative algebras, as in the commutative diagram
\begin{equation} \xymatrix { & \mathbf{Alg}^{\textrm{op}} \ar[dl]_{s}
\ar[d]^{y} \\
\mathcal{Z} \ar[r]_{i} & \mathbf{Set}^{\mathbf{Alg}} }
\end{equation}
where $y$ denotes the Yoneda embedding and $i$ an inclusion. The
functor $s$ is the algebro-geometric operation of taking the
spectrum. $\mathbf{Set}^{\mathbf{Alg}}$ is the classifying topos
for commutative algebras.
As pointed out in Mac Lane, determinants are really a natural
transformation between the two functors $GL_{n}: \mathbf{Rng}
\rightarrow \mathbf{Grp}$ and $\ast: \mathbf{Rng} \rightarrow
\mathbf{Grp}$ from the category of (commutative) rings to the
category of groups. The first functor assigns the obvious matrix
group to a ring, and the second the group of units $K^{\ast}$ to a
ring $K$. This fact hints at the need to consider even elementary
constructs from linear algebra in a setting where the base number
field is not fixed a priori.
The Mac Lane pentagon describes a 1-dimensional relation on a weak
associativity arrow $\psi : (A \otimes B) \otimes C \Rightarrow (A
\otimes B) \otimes C$. The arrow is 2-dimensional, as you know,
because the tensor product composition of objects means that
objects should be thought of as 1-dimensional entities.
The association of the broken pentagon with both the confinement
mechanism (mentioned above) and a deformation parameter (as for
quantum groups) is a nice way of saying (albeit not rigorously)
that the mass gap arises from quantisation.
That is: objects in a category such as $\mathbf{Rep_{SU(2)}}$ are
representation spaces rather than particle states, so to capture
the notion of a state in category theoretic terms it is necessary
to internalise this picture and replace $\psi$ by its
tricategorical analogue, that is to say raise the dimension by
one. The truly fascinating thing is that tensor products in higher
dimensional categories are no longer stable dimensionally. In
particular, the horizontal composition of 2-arrows in a
tricategory gives a 3-arrow. For the pentagon this leads to a
symmetry breaking that can only be understood in terms of the weak
nerve of a tricategory. This 3-arrow satisfies a 4-cocycle
condition, and may therefore be interpreted as an appropriate
analogue to the integrand of path integrals in four dimensional
QFT.
Somewhere here (sorry, forget where) I read the statement: the
universe is not a collection of S-matrices. The author was
complaining about a picture that he/she clearly considered simple
minded and inconsistent. But the point is that QGC does in fact
say something like this, only the word 'collection' is replaced by
something more subtle. A 'particle' cannot be represented by a set
(= 0-category). I believe you are quite happy with the importance
of 2-categories, at least.
Now, what is Hodge duality? Positive definite metrics can be taken
care of by Lawvere's ideas in
F. W. Lawvere, Metric Spaces, Generalised Logic and Closed
Categories, Repr. Th. Applic. Cat. 1 (2002) 1-37
The 2 point object of truth values for Set is replaced by the
positive reals (with infinity). But to get Minkowski space working
properly we need axiomatics for stacks with which one can do
'linear logic' (actually I'm working on this at the moment).
Then 'non-abelian Hodge duality' is a quantum lattice descent theoretic
cohomology pairing.
Observe that in this picture the gauge symmetries are essentially
derived from more fundamental physical considerations. U(1) for
instance just happens to be a special object in the lattice theoretic
description of Pontrjagin duality.....
How does that sound?
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Urs Schreiber
[...]
> How does that sound?
Wild.
To be honest, I am getting the impression that what you wrote consists
mostly of speculation and free association. Prove me wrong! :-)
> I must apologise for not explaining myself. I am posting from
> PF rather than here.
I know. And in case that produces any problems I listed some alternatives,
like using Google Groups. But I am under the impression that the people at
PF have already removed the posting delay in their software.
> I should begin by making it clear that while 'standard M-theory'
> appears to be a reasonable approximation to the 'true' M-theory,
Does it indeed?
> First of all, a quick answer to the question of superalgebras.
> Following the papers
>
> Quark State Confinement as a Consequence of the Extension of the
> Bose-Fermi Recoupling to SU(3) Colour, W. P. Joyce,
> http://arxiv.org/abs/hep-th/0306256
>
> and
>
> Recoupling Lie algebra and universal omega-algebra, W. P. Joyce,
> J. Math. Phys. 45, 10 (2004) 3859-3877
>
> consider the constraints on phase factors for a (unital) symmetric
> premonoidal structure.
I haven't read these papers and don't know what you mean here. I know what
a (symmetric) monoidal structure is, but I don't know in which way you
want to associate phase factors with that.