### Differential n-Geometry

#### Posted by Urs Schreiber

I need it all the time - and yet, I still don’t have it:

a nice arrow-theoretic way to talk about differential $n$-geometry.

I know that greater minds than me have thought about this thoroughly before #. That I still don’t feel like having the satisfactory tools at my disposal probably has two reasons:

a) I am ignorant of what has been done already;

b) I feel the need for something somewhat different than what has been done already.

In order to find out which of the two it is, I want to start going through some gymnastics here in the $n$-Café.

I am thoroughly (maybe hopelessly) motivated by

i) quantum physics

and

ii) the belief that $n$-dimensional QFT lives in $n\mathrm{Cat}$.

Maybe this explains point b) above. As far as I am aware, previous work in arrow-theoretic differential geometry was motivated by *classical* physics and the belief that $\mathrm{Cat}$ suffices.

For instance, I believe that we want a notion of differential $n$-forms that take values in $n$-categories, like $n$-functors do.

This belief is a consequence of particular physics applications that I have in mind, which I roughly know how to do already, but which need a more systematic underpinning. In particular, one of my goals is to give a good arrow-theoretic description of an $n$-Dirac operator twisted by an $n$-vector bundle with $n$-connection. Unless I am confused, such a concept is at the heart of $n$-dimensional supersymmetric quantum field theory (at least for $n=1$ and $n=2$).

Okay. The gymnastics starts below.

I think I want to work inside the $(n+1)$-category whose

- - objects are $n$-categories equipped with a sub-$n$-graph of their underlying $n$-graph of $n$-morphisms
- - 1-morphisms are $n$-functors that respect these sub-$n$-graphs

Of course, the choice of sub-$n$-graph is supposed to describe a sub-collection of those $n$-morphisms that have “a small extension”.

Being the physicist that I am, I have a couple of immensely naive examples for this in mind for illustrating (and motivating) all of my constructions below. I am hoping, though, that what I do here is formulated in a manner general enough to accomodate sophisticated topos-theoretic technology as used in synthetic differential geometry - even though this tends to scare me, admittedly.

Mostly, I allow myself to be immensely naive and usually think of categories that are freely *generated* from their graph of small morphisms. That’s like what you’d be content with for applications in lattice field theory.

But I am hoping that there are good implementations in between this very crude model and the very high-brow technology. For instance: let $X$ be a $d$-dimensional Riemannian manifold. Let $\epsilon \in \mathbb{R}$ be a small real number such that all points $x \in X$ have an $\epsilon$-ball around them in which geodesics through $x$ provide good coordinates. Choose an orthonormal system of $d$ vector fields in $TX$. Now identify in the category of thin-homotopy classes of paths in $X$ the subgraph consisting of all those geodesic paths that have length $\epsilon$ and are tangent in their source point to one of the chosen orthonormal vectors.

Presumably, I should now make a definition like this:

**Definition.** *Denote by $d^r$ for $r \in \mathbb{N}$ the $r$-category obtained from the smallest $r$-fold category with a single nontrivial $r$-morphism.*

The first three of these look like

Let $P X$ be my default symbol for some $n$-category with specified sub-$n$-graph.

I should then say that a **tangent $p$-vector** to some point in $P X$ is an $n$-functor

It should be noteworthy that a morphism

in $[d^p,P X]$ looks like a $p$-dimensional cobordism in $PX$ which starts tangential to $v$ and ends tangential to $v'$. For QFT applications it seems # we want to turn $p$-transport functors on globular $p$-paths in $P X$ into “generalized phases” on $p$-dimensional cobordisms, and it looks to me that it’s the structure of morphisms in $[d^p,P X]$ which makes this happen naturally.

And in fact, there is now nothing more immediate than

**Definition.** *For $T$ a $p$-category, a $T$-valued $p$-form $\omega$ on $PX$ is a $p$-functor*

In particular, every $n$-transport $n$-functor

gives rise to a $T$-valued $p$-form on $P X$

We will want to have an idea of wedge products and exterior differentials of $T$-valued $p$-forms. I am not yet completely sure about the general definition. So I start by giving a crucial motivating example:

Let $\tilde \mathrm{Vect}$ be the 2-category which has finite-dimensional hermitian vector spaces as objects, has linear maps between these as morphisms, and has a unique 2-morphisms for every ordered pair of 1-morphisms (for reasons described here). We should think of these 2-morphisms as labeled by the source morphisms and the difference of the target and the source morphism

Let the 1-functor

be a hermitian vector bundle $E$ with connection. We might address $d\mathrm{tra}$ as the corresponding $\tilde \mathrm{Vect}$-valued connection 1-form on $X$.

Now let

be an endomorphism-valued 0-form on $X$. We may “wedge” $d\mathrm{tra}$ with $e$ in order to get the **covariant derivative 1-form** of $e$ with respect to $d\mathrm{tra}$:

Here the right hand side is supposed to denote the functor which sends $d^1 = (a \to b)$ to the morphism $E_x \to E_y$ given by

Notice that this does indeed yield the ordinary covariant exterior derivative to linear order, when $x\to y$ is a short path in an ordinary manifold.

In a similar fashion, we can “wedge” $d\mathrm{tra}$ with itself, to obtain a $\tilde \mathrm{Vect}$-valued 2-form, the **curvature 2-form** of $\mathrm{tra}$:

This, too, does indeed yield the familiar curvature 2-form in cases where it makes sense.

What is important here is that the (covariant) exterior derivation involves something like a graded commutator of a $1$-form with a $p$-form. This is just the first example of a general pattern #, which needs to be systematized.

Also, there is a nice relation to Quillen’s notion of *superconnections* here #.

For nice $T$, our $T$-valued differential forms will naturally be elements of a Hilbert space.

With target being hermitian vector spaces as in the above example, we should equip our space of $T$-valued 0-forms with a scalar product of the form

Similarly for our $\tilde \mathrm{Vect}$-valued 1-forms

The integral sign here denotes some operation of “summing” over all points of $X$, which I haven’t defined yet in general.

But, recall, in case of trouble I’ll always resort to the most naive situation: that of lattice gauge theory. Then $P X$ is just generated from a (finite) square lattice and the sum is literally a sum. And in this case for instance the scalar product

for the above example is the Lagrangian for lattice Yang-Mills theory coupled to a boson in the adjoint rep.

So much for today.

The basic concept I am describing here goes back to work I did a while ago with Eric Forgy. But at that time I was a category illiterate. Part of what I am doing here is trying to find the truly natural setting for these ideas.

## Parity complexes

Just for the record:

I realize that my $n$-category called $d^n$ above is what Ross Street calls the

parity $n$-cube. E.g. section 2 of this.