The n-Category Café

So - lets play the Kaluza-Klein game… on loop space! […]

What kind of dynamics do we get out?

Beautiful question.

One way to see the Kaluza-Klein mechanism is to consider the Laplace(-Beltrami) operator

which locally reads

insert the metric on $X \times S^1$ and study the field equation

of a given particle.

Now, it is true that the dynamics of the (Nambu-Goto/Polyakov) string is essentially given by the same kind of wave equation, too - but on loop space. (You can find some references collected here.)

While I haven’t checked it, I would not be surprised if (modulo the usual technical issues with differential geometry on loop space) the KK-mechanism goes through here just as well and produces the Kalb-Ramond field.

If one allows oneself to work naively with locally coordinatized versions of $\Delta$ on loop space, it should be a simple matter to check this.

(But I am afraid even if I had the time to start this computation right now, you would come up with yet another deep question before I finished. :-)

Said that, I should remark that from the perspective of physical applications, KK-theory on loop space seems (unless I am missing something) much less compelling than KK-theory on target space.

But something closely related is of considerable interest:

what is the KK-reduction of gravity coupled to a 2-gerbe on a circle bundle?

(The issue is: what is it in terms of globally defined quantities? We know what it is in terms of local data of a trivial 2-gerbe.)

Bruce Bartlett writes:

In addition, if your manifold has a metric (as any self-respecting manifold has), then supposedly this can be lifted to loop space.

Sure. A tangent vector in the space of loops in $X$ is a loop of tangent vectors in $X$. If we have two of these, say

$v, w : S^1 \to X$

there’s an obvious way to define their inner product using a Riemannian metric on $X$, namely

$$\langle v, w \rangle = \int_{S^1} \langle v(t), w(t) \rangle d t$$

So, if $X$ is a Riemannian manifold, the free loop space $L X$ becomes a Riemannian manifold as well. Of course, it’s infinite-dimensional. So, we should be precise: if we use smooth loops, it’s a Riemannian Fréchet manifold.

None of this used anything special about the circle $S^1$ except that we can do integrals of smooth functions on it and be sure they converge. So, we could replace $S^1$ by any compact manifold $\Sigma$ with a volume form!

Theorem: If $\Sigma$ and $X$ are smooth finite-dimensional manifolds, $\Sigma$ is compact and equipped with a volume form, and $X$ is equipped with a Riemannian metric, then the space of smooth maps $C^\infty(\Sigma, X)$ is a Riemannian Fréchet manifold.

There is, however, something special about using the circle for $\Sigma$. Brylinski explains it in his book on gerbes.

In this case, the space of smooth maps $C^\infty(\Sigma, X)= L X$ is not just a Riemannian manifold. It’s a Kähler manifold!

In other words, it gets a complex structure and a symplectic structure which get along nicely with each other and the Riemannian structure.

As discussed in the week 10 lectures of this course on classical mechanics, Kähler manifolds are just what you want for geometric quantization.

So, there’s something espeically nice going on.

And I hope Urs sees why this is so tantalizing. I’d like to know the analogue of geometric quantization for gerbes. Normally geometric quantization wants an integral Kähler structure, which in turn gives a line bundle; then we take the space of holomorphic sections to be our Hilbert space. I don’t know the analogue of all that for gerbes.

But, a Riemannian metric on $X$ gives a Kähler structure on $L X$! And, a gerbe on $X$ gives a line bundle up on $L X$! So, a gerbe on a Riemannian manifold seems in some ways like the right analogue of a line bundle on a Kähler manifold.

But the analogy is weird: I know how to obtain a line bundle (up to isomorphism) from an integral Kähler manifold, but I don’t know how to obtain a gerbe (up to equivalence) from an integral Riemannian manifold. I don’t even know what an “integral Riemannian manifold” is!

Put another way, I know how the curvature 2-form of a line bundle with connection can equal the 2-form that’s part of a Kähler structure - but I don’t know how the curvature 3-form of a gerbe with 2-connection can equal “the 3-form that’s part of a Riemannian structure”. There is no 3-form as part of a Riemannian structure!

Well, I suppose if my Riemannian manifold happened to be oriented and 3-dimensional, its volume form would be a 3-form. Then, if the deRham cohomology class of this 3-form were integral, I could say I had an integral Riemannian 3-manifold - and on this 3-manifold there’d be a gerbe with 2-connection whose curvature 3-form equalled the volume 3-form!

But it’s annoying to be limited to 3-manifolds here.

I just said it was annoying how we’re limited to 3-manifolds if we’re trying to get gerbes from Riemannian manifolds the way we get line bundles from Kähler manifolds.

But, I should emphasize that Brylinski - a smart fellow - makes precisely this limitation at a certain point in his book on gerbes and geometric quantization!

He gets some interesting stuff, namely a gerbey way of thinking about the dynamics of “vortex filaments” in 3d fluid flow. There might even be a nice way of geometrically quantizing these, if we could handle the infinite-dimensional Kähler manifold $L X$, where $X$ is our integral Riemannian 3-manifold.