### 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 ${C}^{i}{}_{\mathrm{jk}}$ there is a 2-dimensional topological field theory whose partition sum is computed by triangulating the surface, assigning one ${C}^{i}{}_{\mathrm{jk}}$ 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 ${C}^{i}{}_{\mathrm{jk}}$ with ${C}^{i}{}_{\mathrm{jk}}+{B}^{i}{}_{\mathrm{jk}}(x)$, where the additional tensor $B$ depends on the triangle $x$ 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 $x$-independent quantities ${C}^{i}{}_{\mathrm{jk}}$ it is a theorem that for the number we compute by assigning them to a triangulation of a surface to be well defined, the $C$ have to be the structure constants of a semisimple algebra.

Now if we let these quantities become position dependent by setting $C\to C+B(x)$ there must be *some* condition on the $B$ 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 $C+B$ to be topological anymore. But there must still be *some* requirement.

*What is this sufficient condition on $B$ and how can it be solved?*

## 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.