The n-Category Café

To look at the very simplest case first: a 1-dimensional Riemannian cobordism is essentially a number $$t \in \mathbb{R}_+ \,,$$ it’s length. Composition of such cobordisms is addition of these numbers $$t_1 \circ t_2 = t_1 + t_2 \,.$$ For a super-Riemannian structure on our 1-dimensional cobordisms, we would like each cobordism to be characterized by a tuple $$(t, \theta)$$ of sorts, such that composition is described by a fomula of the kind $$(t_1, \theta_1) \circ (t_2, \theta_2) = (t_1 + t_2 + \theta_1 \theta_2, \; \theta_1 + \theta) \,.$$ Whatever that means.

In Elke Markert’s thesis Connective 1-dimensional Euclidean field theories (which I once mentioned here) this is done in great detail by understanding 1-dimensional super-Riemannian cobordisms as $\mathbb{Z}_2$-graded locally ringed spaces equipped with a nondegenerate super 1-form. There it takes 35 pages to understand and derive the above formula.

If we think of a (pseudo-)Riemannian metric as encoded in a $\mathrm{ISO}(m,n)$ connection, we find the following simple situation in one dimension.

Here $$\mathrm{ISO}(1) \simeq \mathbb{R}$$ and a $\mathrm{ISO}(1)$-connection on a 1-dimensional cobordism $C$, given by its parallel transport functor $$\array{ P_1(C) \\ \downarrow^g \\ \Sigma \mathrm{ISO}(1) }$$ is simply a smooth functor $$\array{ P_1(C) \\ \downarrow^g \\ \Sigma \mathbb{R} }$$ which measures the length of $C$. It comes from a $\mathbb{R}$-valued 1-form: the vielbein (here an einbein).

Two such Riemannian cobordisms $(C,g)$, $(C',g')$ are isometric if they are isomorphic in the obvious over category $$\array{ P_1(C) &&\stackrel{\sim}{\to}&& P_1(C') \\ & {}_g\searrow && \swarrow_{g'} \\ && \Sigma \mathrm{ISO}(1) } \,.$$

Now replace $\mathrm{ISO}(1)$ with the corresponding super Lie group. Its super Lie algebra has the two generators $$e, \; \; \psi$$ of even and odd degree, respectively, with the only nonvanishing bracket being $$[\psi , \psi] = 2\; e \,.$$

A connection on $C$ with values in this comes from a $\mathbb{R} \oplus \mathbb{R}$-valued 1-form: the einbein $E$ and the gravitino $\psi$. The parallel transport functor $$P_1(C) \to \Sigma \mathrm{ISO}(1)$$ sends the cobordisms $C$ to the path ordered exponential $$\begin{aligned} &\exp( \mathrm{length}(C) + \mathrm{superlength}(C)) \\&:= P \mathrm{exp}( \int_C ( E + \Psi )) \\&= \text{"} \mathrm{lim}_{\epsilon \to 0} \exp( E(0) + \Psi(0)) \exp( E(\epsilon) + \Psi(\epsilon)) \cdots \exp( E(1) + \Psi(1)) \text{"} \end{aligned}$$ of these connection 1-forms over the cobordism.

Using the Baker-Campbell-Hausdorff formula and remembering the only nontrivial bracket $[\psi,\psi] = 2 e$ on the generators , we find the super Lie group group structure $$\begin{aligned} &\exp( t_1 e + \theta_1 \psi ) \exp( t_2 e + \theta_2 \psi) \\ &= \exp( (t_1 + t_2 + \frac{1}{2}[\theta_1 \psi, \theta_2 \psi] + \; (\theta_1 + \theta_2) \psi ) ) \\ & = \exp( (t_1 + t_2 + \theta_1 \theta_2 + \; (\theta_1 + \theta_2) \psi ) ) \end{aligned}$$ for all real coefficients $t_1,t_2, \theta_1, \theta_2 \in \mathbb{R}$.

So composing our cobordisms equipped with smooth parallel transport functors with values in super $\mathrm{ISO}(1)$ in the obvious way (by gluing, doing this in detail requires introducing collars) it seems to me we do get the desired category structure on super-Riemannian cobordisms.

I tend to like this for various reasons. On the one hand, I think that in fact all fields in $d$-dimensional QFT should be thought of as $d$-functors on $d$-paths inside $d$-dimensional cobordisms. We would get the above super-Riemannian structure on our cobordisms by freezing out the component of these functors with values in $\Sigma \mathrm{ISO}(1,d-1)$.

For this to make fully good sense, we should expect higher versions of the super-Poincare Lie algebra. And indeed, these seem to be precisely what we ought to be looking at.