## March 31, 2005

### Akhmedov: Nonabelian 2-Holonomy using TFT

#### Posted by Urs Schreiber

You all know that I have been thinking about 2-holonomy a lot, lately. Hence of course a paper by E. Akhmedov which appeared today

attracted my attention with its abstract, which reads

We present a triangulation-independent area-ordering prescription which naturally generalizes the well known path ordering one. For such a prescription it is natural that the two–form ‘connection’ should carry three ‘color’ indices rather than two as it is in the case of the ordinary one-form gauge connection. To define the prescription in question we have to define how to exponentiate a matrix with three indices. The definition uses the fusion rule structure constants.

I have just read through this paper and I think the idea is what I am going to summarize in the following. My presentation is a little different from E. Akhmedov’s in that I take his last remark right before the conclusions as the starting point and motivate the construction from there.

[Update: Some discussion of these issues is taking place here.]

There is a well-known way to construct 2-dimensional topological field theories on a triangulated surface. It is a 2d version of the Dijkgraaf-Witten model, which I learned about this winter in John Baez’s quantum gravity seminar (week 6 this year) and which is discussed in detail in

M. Fukuma, S. Hosono & H. Kawai, Lattice Topological Field Theory in Two Dimensions (1992) .

The idea is simply to triangulate your manifold and associate to each triangle a given 3-index quantity ${C}_{\mathrm{ijk}}$, with each index associated to one of the edges of the triangle. All edges are labelled either in-going or out-going and if an edge is outgoing we raise the corresponding index of ${C}_{\mathrm{ijk}}$ using a symmetric 2-index quantity ${g}^{\mathrm{ij}}$. Then define the partition function of this setup simply to be the contraction of all the ${C}_{\dots }$ by means of ${g}^{\dots }$ in the obvious way.

This partition function becomes that of a topological theory when the $C$ and $g$ are such that their contraction in the above way is independent of the triangulation of the surface. One can show that this is the case precisely if the ${C}_{\mathrm{ijk}}$ are the structure constants of a semi-simple associative algebra and $g={C}^{2}$ is its ‘Killing form’.

To my mind E. Akhmedov’s central observation is that the formula for computing the holonomy of a non-abelian connection 1-form along a line is like a sum over $n$-point functions of a 1-dimensional topological field theory with the $n$-th powers of the 1-form in the formula for the path-ordered exponential playing the role of the $n$ insertions.

Motivated by this observation, he proposes to compute nonabelian 2-holonomy by taking the analogous sum of $n$-point functions in a 2d TFT of the above type.

An insertion in the above 2d TFT corresponds to removing one of the ${C}_{\mathrm{ijk}}$ labels from one of the triangles and replacing it with a ‘vertex’, which must be a 3-index quantity, too. Guess how we call, it: ${B}_{\mathrm{ijk}}$. Or better yet, when this is inserted in triangle number $a$ call it ${B}_{\mathrm{ijk}}\left(a\right)$.

Denoting by $〈B\left({a}_{1}\right)B\left({a}_{2}\right)\cdots B\left({a}_{n}\right)〉$ the $n$-point function of our theory, I believe that E. Akhmedov proposes (he uses different notation) to define the 2-holonomy ${\mathrm{hol}}_{B}\left(\Sigma \right)$ of the 3-indexed discrete 2-form $B$ over a given closed surface $\Sigma$ to be

(1)${\mathrm{hol}}_{B}\left(\Sigma \right)=\underset{ϵ\to 0}{\mathrm{lim}}\sum _{n=0}^{\infty }\frac{1}{n!}\sum _{\left\{{a}_{i}\right\}}{〈B\left({a}_{1}\right)B\left({a}_{2}\right)\cdots B\left({a}_{n}\right)〉}_{\Sigma }\phantom{\rule{thinmathspace}{0ex}},$

where $ϵ$ is some measure for the coarseness of our triangulation. That’s it.

Plausibly, when things are set up suitably this continuum limit exists and is well defined, i.e. independent of the triangulation.

That sounds good. In particular since the three indices carried by $B$ suggest themselves naturally as a source for the ${n}^{3}$-scaling on 5-branes.

The underlying philosophy the way Akhmedov presents it is rather similar to Thomas Larsson’s ideas on 2-form gauge theory, though different in the details.

Of course for me one big question is: Can this construction be captured using 2-bundles with 2-connection?

In any case, one would have to think about how the above definition of 2-holonomy could be generalized to a situation where there is no gloablly defined 2-form $B$. This can already be seen in the abelian case, where the above 2-holonomy reduces to the 2-holonomy known for abelian gerbes only when everything is defined globally.

Hmm….

Posted at March 31, 2005 2:24 PM UTC

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### Re: Akhmedov: Nonabelian 2-Holonomy using TFT

Let’s see. It should be possible to reformulate this construction in terms of a generalized 2-connection 2-functor and make contact with 2-bundle theory.

So I will define a source 2-category $S$ and a target 2-category $T$ together with a 2-functor

(1)$\mathrm{hol}:S\to T$

such that this data reproduces the 2-holonomy as defined by Akhmedov.

Fix a surface $\Sigma$.

The objects of $S$ are the points in $\Sigma$. The morphisms in $S$ are piecewise smooth parameterized paths in $\Sigma$. The 2-morphism in $T$ between source path ${\gamma }_{1}$ and target path ${\gamma }_{2}$ (where ${\gamma }_{1}$ and ${\gamma }_{2}$ have coinciding endpoints) are a subset of all reparameterization equivalence classes of smooth surfaces $\sigma$ such that $\partial \sigma ={\gamma }_{1}-{\gamma }_{2}$ (where the minus sign means orientation reversal followed by disjoint union).

This subset is the one generated by the following three elementary surfaces under horizontal and vertical composition:

1) When ${\gamma }_{1}={\gamma }_{2}$ we allow the degenerate surface between these paths.

2) When ${\gamma }_{1}$ is a constant path consisting of precisely one smooth segment and ${\gamma }_{2}$ consists of precisely two smooth segments and is furthermore thin homotopy equivalent to the constant path then we allow the degenerate surface with boundary ${\gamma }_{2}$. Similarly for ${\gamma }_{1}$ and ${\gamma }_{2}$ exchanged.

3) When ${\gamma }_{1}$ is any path consisting of precisely two smooth pieces and ${\gamma }_{2}$ is any path consisting of precisely 1 smooth segment, then we allow any surface interpolating between these.

Horizontal and vertical composition of 1- and 2-morphisms is the obvious concatenation.

Item 3) is an elementary triangle of our triangulation, 1) is the identity 2-morphism on any path and 2) allows us to invert the direction of any path. The 2-morphisms generated by these three types of 2-morphisms under composition capture all triangulations of $\Sigma$.

Next I define the target 2-category $T$:

$T$ has only a single object $\star$.

The morphisms in $T$ are the natural numbers. Composition of morphisms is addition of natural numbers.

The 2-morphisms between $n$ and $m$ are rank $\left(n.m\right)$ tensor powers of some fixed vector space $V$. Composition of these 2-morphisms is contraction of tensor indices. The horizontal product is just the ordinary product.

For instance given 2-morphisms ${T}^{i}{}_{\mathrm{jk}}$ and $T{\prime }^{\mathrm{rs}}$ their composition (‘vertical product’) is

(2)$\left(T\circ T\prime {\right)}^{i}=\sum _{j,k}{T}^{i}{}_{\mathrm{jk}}T{\prime }^{\mathrm{jk}}$

and their horizontal product is

(3)$\left(\mathrm{TT}\prime {\right)}^{\mathrm{irs}}{}_{\mathrm{jk}}={T}^{i}{}_{\mathrm{jk}}T{\prime }^{\mathrm{rs}}\phantom{\rule{thinmathspace}{0ex}}.$

We get the bare 2D TFT by allowing only 2-morphism which are generated by these three generators:

1) ${\delta }^{i}{}_{j}$

2) ${g}_{\mathrm{ij}}$ and ${g}^{\mathrm{ij}}$

3) ${C}^{\mathrm{ij}}{}_{k}$ .

Note that these correspond precisely to the three generators for the 2-morphisms in $S$ defined above. One can easily see that using these we can capture every partition function as defined above by a functor $\mathrm{hol}:S\to T$.

It is obvious how to incorporate the $B$-insertions now.

To me the big question is: What type of category is $T$? It is not quite a weak 2-group, even though it comes pretty close.

So far only 2-bundles whose 2-transitions are given by weak 2-groups have been considered, since this gives what one would want to call a principal 2-bundle. Of course the bare concept of a 2-bundle allows for all sorts of other possibilities.

Posted by: Urs Schreiber on March 31, 2005 7:10 PM | Permalink | Reply to this

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