Akhmedov comments
Posted by urs
Recently I mentioned here an idea by E. Akhmedov to construct a nonabelian surface holonomy using 2D TFTs. Today he has a new preprint on that issue.
When I first read about it I found the
idea interesting:
It is well known that for every associative semi-simple algebra with structure constants there is a 2-dimensional topological field theory whose partition sum is computed by triangulating the surface, assigning one to each surface element and contracting all indices using an orientation on the edges of the triangulation.
It seemed to me that E. Akhmedov was proposing to generalize this by allowing to replace the with , where the additional tensor depends on the triangle that it is assigned to. This will of course no longer give a topological field theory but if this could be given a well defined continuum limit it might have the interesting interpretation as a way to compute some notion of ‘nonabelian’ surface holonomy.
I don’t see yet, though, that this idea has been shown to have a well-defined implementation, some aspects of which I discussed on sci.physics.strings.
After seeing his hep-th/0503234 I had emailed E. Akhmedov asking some technical questions and mentioning that there has been previous work on this question.
Unfortunately I never received a reply. But today a new paper appears on the arXiv:
E. Akhmedov, V. Dolotin & A. Morozov
Comment on the Surface Exponential for Tensor Fields
hep-th/0504160
In this paper some aspect of the previous proposal is being examined more closely.
This paper now does cite previous work on related issues. In the introduction it mentions that
the problem is known under many names, from topological models [2,3] to Connes-Kreimer theory [4]-[6] and that of the 2-categories [7]
This is not quite correct. 2-categories would be needed for volume-holonomy, since using them you can construct 3-groups. What is true is that general 2-groups, which are 1-categories have been considered as a tool for investigating surface holonomy.
The reference number [7] in the above preprint is our From Loop Groups to 2-Groups. This, incidentally, is not concerned with the question that E. Akhmedov is addressing in his papers.
(It rather studies possibly interesting structure 2-groups for 2-bundles. In these 2-bundles we can study nonabelian surface holonomy, but this is discussed in hep-th/0412325).
After having had a closer look at E. Akhmedov’s proposal I have become a little sceptical that the technical details are being appropriately addressed. If anyone thinks I am wrong about this I would kindly ask him or her to help me for instance clarify the following question:
As long as we work with -independent quantities it is a theorem that for the number we compute by assigning them to a triangulation of a surface to be well defined, the have to be the structure constants of a semisimple algebra.
Now if we let these quantities become position dependent by setting there must be some condition on the to ensure that the ‘partition sum’ we compute now is well defined and in particular has a well defined continuum limit.
The requirements may now be a little different, since we don’t want the surface field theory given by to be topological anymore. But there must still be some requirement.
What is this sufficient condition on and how can it be solved?
Posted at April 20, 2005 5:43 PM UTC
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Re: Akhmedov comments
Dear Urs,
I have sent to you already
4 messages including two in reply to your
first message! I have checked with our system administrator the situation with
my messages to you and he says that
for some reason your computer rejects
my e-mails!?
Let me answer on you questions and comments.
First of all, i do not see any problems
for the situation when B-field depends on
x: this is explained in my first paper.
But i will be happy to hear more details
from you.
Second, we have cited that paper which you
mention rather than another because
it is this paper which we have read.
I will read soon another one as well.
Third, every thing we do can be
generalized to four-index tensors and
orderings over 3-volumes and so on:
this is mentioned in the concluding
section of my first paper. I do not
see any conceptual difficulties, but
ofcause one have to find an algorithm
for looking for the analogs of the matrix
I in those situations.
Fourth, concerning the situation
with ambigous two-forms. I can say
the following (at least as far as i understand your question): Consider a section of a
standard vector bundle. It is a “function” defined at
each point of the base of the bundle. I put function
into brakets because it is not globally defined! To deal
with such objects one uses the standard gauge connection,
which tells us how the “function” changes as we take one or
another path on the base. Similarly if a one-form gauge connection is
not globally defined (the only explicite case i know is
the Wu-Yang monopole) i suggest to deal with the connection
on the bundle whose base is the space of loops, i.e.
with the non-Abelian two-tensor field: standard one-form
gauge conncetion is “attached” to paths rather than points
(like the section of a vector bundle). Then the non-Abelian
two-tensor field defines how to “parallel” transport obgects
attached to paths! Then, extrapolating this idea, i suggest
to consider a three-tensor gauge field (with four “color” indices)
to define a parallel transport on the space of closed two-dimensional surfaces, i.e. describes the situation with non-globally defined
B-field attached to two-dimensional surfaces. And so on.
I hope my picture is clear and obvious.
Hopefully it will be useful for you.
Regards,
Emil.
Global Issues, Continuum Limit and Weak 2-Group Formulation
Concerning the issue of how to figure out the correct global definition as well as the continuum limit of this idea, if possible, I would like to repeat how I speculate this construction might be expressible in terms of a weak 2-group, if it works. Once one has a local 2-holonomy in terms of a weak 2-group it is straightforward to get the global construction by means of weak 2-bundles.
I have mentioned the following before on sci.physics.strings:
Start with the reformulation of the TFT-like 2-holonomy at finite resolution in terms of that ‘category of triangles’ that I mentioned recently. There seems to be an obvious structure that this construction asymptotes to in the continuum limit:
Let be the space in which the internal indices take values. For instance
for the TFT defined from discrete group algebras, 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 incoming and outgoing indices. In the limit this
gives a map from pairs of smooth paths in to the base field
(1)
such that composition is given by ‘continuous index contraction’
(2)
(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’ if we like. But since
these are nothing but integral kernels we see that the group they form is
that of unitary operators on something like , where is the path
space over . But in fact without any loss of generality we can assume that
all paths start and end at a given point in and have sitting instant at
that point and are parameterized by a parameter in , so we really get
, the based loop space over and unitary operators on .
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
(3)
where is the path that traces out the first half of at twice
the speed, and similarly gives the right half and it is implicit that
and vanish when their arguments are not based loops with sitting
instant at the base.
In fact, I believe that this defines on the s 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 should get a weak
2-group.
At this vague level this is kind of obvious. The hard part is to take care of subtleties with these functional integrals and things like that.
Re: Akhmedov comments
Dear Urs,
I have sent to you already
4 messages including two in reply to your
first message! I have checked with our system administrator the situation with
my messages to you and he says that
for some reason your computer rejects
my e-mails!?
Let me answer on you questions and comments.
First of all, i do not see any problems
for the situation when B-field depends on
x: this is explained in my first paper.
But i will be happy to hear more details
from you.
Second, we have cited that paper which you
mention rather than another because
it is this paper which we have read.
I will read soon another one as well.
Third, every thing we do can be
generalized to four-index tensors and
orderings over 3-volumes and so on:
this is mentioned in the concluding
section of my first paper. I do not
see any conceptual difficulties, but
ofcause one have to find an algorithm
for looking for the analogs of the matrix
I in those situations.
Fourth, concerning the situation
with ambigous two-forms. I can say
the following (at least as far as i understand your question): Consider a section of a
standard vector bundle. It is a “function” defined at
each point of the base of the bundle. I put function
into brakets because it is not globally defined! To deal
with such objects one uses the standard gauge connection,
which tells us how the “function” changes as we take one or
another path on the base. Similarly if a one-form gauge connection is
not globally defined (the only explicite case i know is
the Wu-Yang monopole) i suggest to deal with the connection
on the bundle whose base is the space of loops, i.e.
with the non-Abelian two-tensor field: standard one-form
gauge conncetion is “attached” to paths rather than points
(like the section of a vector bundle). Then the non-Abelian
two-tensor field defines how to “parallel” transport obgects
attached to paths! Then, extrapolating this idea, i suggest
to consider a three-tensor gauge field (with four “color” indices)
to define a parallel transport on the space of closed two-dimensional surfaces, i.e. describes the situation with non-globally defined
B-field attached to two-dimensional surfaces. And so on.
I hope my picture is clear and obvious.
Hopefully it will be useful for you.
Regards,
Emil.