### Some Recreational Thoughts on Super-Riemannian Cobordisms

#### Posted by Urs Schreiber

Recently, in the discussions about QFTs and representations of cobordisms categories, once again the following question came up:

What is the best way (the right way?) to conceive categories of Riemannian and super-Riemannian cobordisms? How is that extra metric structure best encoded?

Maybe we need to know: what is a Riemannian metric, *really*?

A priori, there seem to be a couple of alternative choices.

Is it an isomorphism of the tangent space with its dual? Or, more generally (also applicable to superspaces), an isomorphism of the algebra of derivations with its dual?

Or is it most naturally a spectral triple? We sure do expect to get spectral triples (for “target space”) from representations of our cobordisms (the Hilbert space coming from those over the objects, the Laplace or Dirac operato coming from the cylinder) – but do we also want to equip the cobordisms themselves with spectral triple data? How do we compose cobordisms equipped with spectral triples?

Or is it maybe less systematic? A special presciption best suitable for each dimension? A mere (super) 1-form on 1-dimensional cobordisms, for instance? A square root of the canonical line bundle in two dimensions? Something yet to be thought of in three dimensions?

Or is it maybe best thought of as a connection on the cobordisms, with values in the Poincaré Lie algebra $\mathrm{iso}(d)$ (or one of its super versions)?

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.

## 3 recent arXiv papers; Re: Some Recreational thoughts on Super-Riemannian Cobordisms

How best (I earnestly wonder) to connect this with the recent:

arXiv:0706.3055

Title: A primer on the (2+1) Einstein universe

Authors: Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, Karin Melnick

Comments: 56 pages, 6 figures

Subjects: Differential Geometry (math.DG)

and

arXiv:0706.3166

Title: The Equations of Motion of a Charged Particle in the Five-Dimensional Model of the General Relativity Theory with the Four-Dimensional Nonholonomic Velocity Space

Authors: V.R. Krym, N.N. Petrov

Comments: 14 pages, 4 figures

Journal-ref: Vestnik Sankt-Peterburgskogo Universiteta, Ser. 1. Matematika, Mekhanika, Astronomiya, 2007, N1, pp. 62–70

Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc); Metric Geometry (math.MG); Optimization and Control (math.OC); Classical Physics (physics.class-ph)

We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin’s maximum principle we proved that the equations of the horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general relativity theory. This is a Kaluza – Klein problem of classical and quantum physics solved by methods of sub-Lorentzian geometry. We study the geodesics sphere which appears in a constant magnetic field and its singular points. Sufficiently long geodesics are not optimal solutions of the variational problem and define the nonholonomic wavefront. This wavefront is limited by a convex elliptic cone. We also study variational principle approach to the problem. The Euler – Lagrange equations are the same as those obtained by the Pontryagin’s maximum principle if the restriction of the metric tensor on the distribution is the same.

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry.

and

arXiv:0706.3107

Title: Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

Authors: Julien Roth (IECN)

Comments: 35 pages

Subjects: Differential Geometry (math.DG)