The String Coffee Table

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

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 $(n.m)$ 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 $T^{i}{}_{jk}$ and $T'^{rs}$ their composition (‘vertical product’) is

and their horizontal product is

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

1) $\delta^i{}_j$

2) $g_{ij}$ and $g^{ij}$

3) $C^{ij}{}_k$ .

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.