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June 23, 2007

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 iso(d)\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 +, t \in \mathbb{R}_+ \,, it’s length. Composition of such cobordisms is addition of these numbers t 1t 2=t 1+t 2. 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,θ) (t, \theta) of sorts, such that composition is described by a fomula of the kind (t 1,θ 1)(t 2,θ 2)=(t 1+t 2+θ 1θ 2,θ 1+θ). (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 2\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 ISO(m,n)\mathrm{ISO}(m,n) connection, we find the following simple situation in one dimension.

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

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

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

A connection on CC with values in this comes from a \mathbb{R} \oplus \mathbb{R}-valued 1-form: the einbein EE and the gravitino ψ\psi. The parallel transport functor P 1(C)ΣISO(1) P_1(C) \to \Sigma \mathrm{ISO}(1) sends the cobordisms CC to the path ordered exponential exp(length(C)+superlength(C)) :=Pexp( C(E+Ψ)) ="lim ϵ0exp(E(0)+Ψ(0))exp(E(ϵ)+Ψ(ϵ))exp(E(1)+Ψ(1))" \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 [ψ,ψ]=2e[\psi,\psi] = 2 e on the generators , we find the super Lie group group structure exp(t 1e+θ 1ψ)exp(t 2e+θ 2ψ) =exp((t 1+t 2+12[θ 1ψ,θ 2ψ]+(θ 1+θ 2)ψ)) =exp((t 1+t 2+θ 1θ 2+(θ 1+θ 2)ψ)) \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,θ 1,θ 2t_1,t_2, \theta_1, \theta_2 \in \mathbb{R}.

So composing our cobordisms equipped with smooth parallel transport functors with values in super ISO(1)\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 dd-dimensional QFT should be thought of as dd-functors on dd-paths inside dd-dimensional cobordisms. We would get the above super-Riemannian structure on our cobordisms by freezing out the component of these functors with values in ΣISO(1,d1)\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.

Posted at June 23, 2007 10:37 AM UTC

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

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

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)


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.


Title: Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds
Authors: Julien Roth (IECN)
Comments: 35 pages
Subjects: Differential Geometry (math.DG)

Posted by: Jonathan Vos Post on June 24, 2007 6:09 PM | Permalink | Reply to this

Re: Some Recreational thoughts on Super-Riemannian Cobordisms

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

I thought we had already concluded that it was simply something deduced from the Leinster measure :)

Are we back to constructing arbitrary measures again? :)

Posted by: Eric on June 25, 2007 4:59 AM | Permalink | Reply to this

Re: Some Recreational thoughts on Super-Riemannian Cobordisms

I thought we had already concluded that it was simply something deduced from the Leinster measure :)

Good point. But that was for configuration space! Here I am talking about metrics on our parameter space.

Remember # the example of Dijkgraaf-Witten theory: a topological theory with no metric on its parameter spaces. Still, there is a measure controlling the path integral, of course, and it does indeed happen to be the Leinster mesasure.

In addition to that, there may be a metric on our parameter spaces. That’s what I am talking about above.

Notice how the kind of metric which I am talking about here is implicit in the discussion of the canonical 1-particle #: there parameter space is the category of a directed graph. Propagation along tt edges of the graph lead to the standard propagator e tΔ. e^{- t \Delta} \,.

Surely, all this ought to be related, in the end. But for the moment I have to do it piecemeal fashion.

As I had indicated elsewhere already a couple of times, I am also hoping to be able to understand the very appearance of supersymmetry on more basic abstract nonsense grounds. But I am not there yet. For instance: it is obvious how a discrete graph may encode an ordinary metric. I still don’t quite understand what kind of discrete structure would encode a supermetric.

Posted by: urs on June 25, 2007 12:36 PM | Permalink | Reply to this

Re: Some Recreational thoughts on Super-Riemannian Cobordisms

There are some interesting comments made in this talk by Andrew Stacey about the way one sometimes needs to modify one’s understanding of what a “Riemannian metric” is in order to jump the hurdles posed by some infinite dimensional situations.

Posted by: Bruce Bartlett on June 25, 2007 4:13 PM | Permalink | Reply to this

Re: Some Recreational thoughts on Super-Riemannian Cobordisms

Thanks for the link! Interesting. Mostly based on the idea that a metric is an iso TMT *MT M \simeq T^* M.

By the way, I could imagine that an ISO(n,m)\mathrm{ISO}(n,m)-connection would be a more direct route to Dirac operators. After all, in a way the whole point about Dirac operators is to take the square root of the metric: the vielbein.

But then, I still don’t know what a Dirac operator really is. All those metric and Dirac operators on loop and path spaces really ought to be 2-metric and 2-Dirac coperators on 2-spaces, ultimately.

It’s somewhat remarkable that the Poincaré-group is a semidirect product group SO(n) n\mathrm{SO}(n) \ltimes \mathbb{R}^n, hence is asking to be regarded as a 2-group, really.

But not just that, we also have semidirect products SO(n) p n\mathrm{SO}(n) \ltimes \wedge^p \mathbb{R}^n.

In particular SO(n) 2 n\mathrm{SO}(n) \ltimes \wedge^2 \mathbb{R}^n might be interesting. But even though we talked about that a couple of times already, I still don’t really see the puzzle pieces falling into place here.

Posted by: urs on June 25, 2007 6:21 PM | Permalink | Reply to this

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