## September 20, 2006

### 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
Probably I am thinking of strict $n$-categories here and take an $n$-graph to be the same as an $n$-category but without any rules for composition.

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

(1)$d^0 = \left\{ a \right\} \,,$
(2)$d^1 = \left\{ a \to b \right\} \,,$
(3)$d^2 = \left\{ \array{ a &\to& b \\ \downarrow &\Downarrow& \downarrow \\ c &\to& d } \right\} \,.$

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

(4)$v : d^p \to P X \,.$

It should be noteworthy that a morphism

(5)$v \stackrel{c}{\to} v'$

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

(6)$\omega : [d^p,P X] \to [d^p,T] \,.$

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

(7)$\mathrm{tra} : P X \to T$

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

(8)$\array{ d\mathrm{tra} &:& [d^p,P X] &\to& [d^p,T] \\ && (v : d^p \to P X ) &\mapsto& (v^*\mathrm{tra} : d^p \stackrel{v}{\to} P X \stackrel{\mathrm{tra}}{\to} T) } \,.$

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

(9)$\array{ V &\to& {} \\ g \downarrow\;\;\; &\Downarrow g-f& \;\;\;\downarrow f \\ {} &\to& W } \,.$

Let the 1-functor

(10)$\mathrm{tra} : P X \to \tilde \mathrm{Vect}$

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

(11)$e : [d^0, P X ] \to [d^0 , \mathrm{End}(E)]$

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 $d\mathrm{tra}$:

(12)$\array{ [d\mathrm{tra},e] &:& [d^1, P X] &\to& [d^1 , \tilde \mathrm{Vect}] \\ && \left(x\to y\right) &\mapsto& \left( \array{ E_x &\stackrel{\mathrm{tra}(x\to y)}{\to}& E_y \\ e_x \downarrow\;\;\; &\Downarrow& \;\;\;\downarrow e_y \\ E_x &\stackrel{\mathrm{tra}(x \to y)}{\to}& E_y } \right) } \,.$

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

(13)$\mathrm{tra}(x\to y) \circ e_y - e_x \circ \mathrm{tra}(x\to y) \,.$

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 $\mathrm{tra}$:

(14)$d\mathrm{curv}_\mathrm{tra} = [d\mathrm{tra},d\mathrm{tra}] : \left( \array{ x &\to& y_1 \\ \downarrow &\Downarrow& \downarrow \\ y_2 &\to& z } \right) \mapsto \left( \array{ E_x &\stackrel{\mathrm{tra}_{x,y_1}}{\to}& E_{y_1} \\ \mathrm{tra}_{x,y_2} \downarrow\;\;\;\;\; &\Downarrow& \;\;\;\;\;\downarrow \mathrm{tra}_{y_1,z} \\ E_{y_2} &\stackrel{\mathrm{tra}_{y_2,z}}{\to}& E_z } \right) \,.$

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

(15)$(e_1,e_2) \mapsto \int_{x\in X} \mathrm{Tr}(e_1^\dagger e_2)(x) \,.$

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

(16)$(\omega_1,\omega_2) \mapsto \int_{(x\to y)} \mathrm{Tr}(\omega_1(x\to y)^\dagger \omega_2(x\to y)) \,.$

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

(17)$( [d\mathrm{tra},d\mathrm{tra}], [d\mathrm{tra},d\mathrm{tra}] ) + ( d_\mathrm{tra} e, d_\mathrm{tra} e, )$

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.

Posted at September 20, 2006 6:41 PM UTC

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## 1 Comment & 0 Trackbacks

### 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.

Posted by: urs on September 28, 2006 5:30 PM | Permalink | Reply to this

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