The String Coffee Table

I am hoping both that there will eventually be more people posting here about a larger variety of topics than I am able to come up with, and also that it is OK with everybody that meanwhile I post here the stuff I do. Hopefully these two hopes are not mutually exclusive. Social dynamics of public forums is a delicate issue.

In any case, I am very grateful to Jacques for providing this weblog. If I am getting on anyone’s nerves with what I am writing about please let me know.

I can’t resist one more comment…

$\Gamma^n$ is polymer space. If we closed the polymer and let $n\to\infty$, we should get loop space (I think).

Yes! Note that this is pretty much what happens in that discrete toy example of that 2-NCG program automotically!. I’ll spell that out in more detail as soon as possible.

I’m looking forward to it :)

There seems to be at least one significant difference though.

so $\delta_{\gamma'} (d\delta_\gamma)$ is a discrete differential 1-form on discrete path space in precisely the same sense as $\delta_{x'} (d\delta_x)$ turned out to be a discrete differential 1-form on ordinary discrete space [2].

In the quoted line above, we are limiting ourselves to 2-nodes that are actually 1-edges in the orginal graph $\Gamma$.

If you look at $\Gamma\times\Gamma$, then we can have 2-edges

where $e_{ij}$ and $e_{kl}$ need not be edges in $\Gamma$. This latter scenario makes more sense to me because the edges of $\Gamma$ are probably going to turn out to be lightlike. If that is the case, then dealing with light-like paths seems a little restrictive.

Just a thought…

Eric

Note on notation: This is a little confusing so I’ll say a quick word on notation. The symbol “$e_{ij}$” denotes an edge in $\Gamma$ from $e_i$ to $e_j$. The symbol “$e_{i,j}$” denotes $e_i \otimes e_j$, and does not require an edge connecting $e_i$ and $e_j$ in $\Gamma$. Maybe I should choose different letters :)

Just a sanity check…

Yes, this sounds good.

I hope to find the time to continue working out my notes on this, but for the moment let me just indicate some more details here in the comment section:

The plan of action is this: For starters, pick a hypercubical graph. Call that an example of a discrete 2-space.

Let a 2-function from this 2-space be a functor from the associated graph category to the category of finite dimensional Hilbert spaces.

This just means that such a 2-function is something

- which assigns natural numbers $n(v) \in \mathbb{N}$ to vertices $v$ in the graph

- which assigns linear maps

to any edge

- and which assigns the obvious composition of such maps to compositions of elementary edges.

Of course these maps are just $n(v_1) \times n(v_2)$ matrices with complex entries.

There is a full-blown theory of 2-Hilbert spaces behind this but actually what matters right now is that the set of all such 2-functions forms an algebra if we define the addition and multiplication of two such 2-maps edgewise as the direct sum and direct product, respectivbely, of these matrices $A$.

So for example consider the case where our graph consist just of two vertices and the single edge

2-functions $f_i$ maps this to

The sum $f_1 + f_2$ is then the 2-function which assigns

to $\gamma$, while the product $f_1 \cdot f_2$ assigns

So you see how all these 2-functions form an algebra. Forget about all the categorification business for a moment. Given an algebra, we can form abstract differential calculi over it by throwing in an abstract $\mathbf{d}$ which we demand to be nilpotent and graded Leibnitz.

So there is a differential calculus with elements of the form

and the like. There is the universal such calculus which is just the free algebra over all these generators. That’s not too interesting. It becomes interesting once we specify certain rules for when such an element is nonvanishing.

In the very simple example where the graph has only a single edge this is not very illuminating. But suppose the graph is actually consisting of four vertices forming a square. Let $f_1$ be a 2-function supported on the top edge of that square and $f_2$ one supported on the bottom edge. Then $f_1 \mathbf{d}f_2$ describes a discrete differential 1-form on path space which says that there is a ‘2-edge’ between the upper and the lower edge, i.e. a plaquette (surface element).

Since every 2-function involves an ordinary function on vertices (assigning the dimension of a Hilbert space to each vertex) the above induces a differential calculus on these ordinary functions. This is just the one that we have studied in our notes! Hence such a plaquette interpolates between two ordinary discrete differential 1-forms as in our notes.

At the level of discrete hypercubical graphs pretty much the only technical difference to what we did before is that now the underlying algebra of the $f_i$ is highly noncommutative by itself. This makes it non-obvious to figure out if there is any set of $f_i$ that canonically and naturally should be regarded as characterizing the elementary paths.