The n-Category Café

The basic idea is simple and rather obvious:

Our 2-dimensional conformal field theory sends conformal cobordisms to linear maps between vector spaces of states assigned to their boundaries.

Differentiating this assignment on the conformal cylinder – regarded as a parallelogram with diagonal $\tau \in \mathbb{C}$ in the complex plane – with respect to the real part of $\tau$, yields the “generator of infinitesimal time translation”, which is usually denoted $$L_0 + \bar L_0$$ and called “the sum of the left and right part of the zero mode of the Virasoro algebra”.

I once reproduced a rather detailed technical account of how exactly this works in nEFT at Schloss Mickeln, Part II.

This operator $L_0 + \bar L_0$ plays essentially the role that the Hamiltonian does in the quantum mechanics of the point particle. Only that here we have the quantum mechanics of the string. In fact, to a large degree $L_0 + \bar L_0$ may be regarded as something like a Laplace operator on loop space.

At least if we think of the CFT here as describing a string which literally propagates in an ordinary space. In this situation we would say that the CFT arises from a $\sigma$-model (of maps from Riemann surfaces into target space) and that the string background is in a geomtric phase.

The analogous situation of this “geometric phase” would be a functor $1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}$, which sends 1-dimensional Riemannian cobordisms of length $t$ to the operator $$e^{-t L} \,,$$ where $L$ is the Laplace operator of some Riemannian space. It need not. We get a 1-dimensional QFT, a functor $1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}$, for any positive hermitean operator $L$. But if $L$ is a Laplace-like operator on some truly geometric space, then we’d say the background of the particle described by our functor is in a geometric phase.

Now, one important phenomenon is that 2-dimensional CFTs may not exactly come from such “geometric phases”, but that they approximate these closely.

A good 1-dimensional analogy here is actually Connes’ latest spectral triple model for the world we inhabit: this is a spectral space which is close to an ordinary geometric space. In fact, it is such that a single superparticle propagating through this spetral geometry looks like a bunch of different species of particles (namely all elementary particles which we see in particle accelerators, plus the Higgs particle which we expect to see with the new LHC), all of them propagating on an ordinary Minkowski space.

Anyway, the question Soibelman (based on Kontsevich & Soibelman) asks is: how do we understand the limit in which our 2-dimensional QFT $$2\mathrm{Cob}_{\mathrm{conf}} \to 2\mathrm{Hilb}$$ degenerates to a 1-dimensional QFT $$1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb}$$ such that the latter comes from a spectral triple with a commutative algebra – hence being pretty close to an ordinary geometry.

The limit to be taken is quite clear heuristically: we want to shrink the string to a point. This means, given any Riemann surface swept out by the string, we want to decompose it into lots of trinions (three-holed spheres) and lots of cylinders stretching between these, and then let the length of these cylinders tend to infinity.

For this to make sense, we need at the same time let the “target space valume tend to infinity”. Technically, this means that we let the “mass gap”, namely the first nonvanishing eigenvalue of $L_0 + \bar L_0$ tend to zero.

What one arrives at this way is in fact a little more than just a functor $$1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb} \,.$$ Such a functor is defined only on worldlines which are disjoint unions of intervals or circles – since these are the only 1-dimensional manifolds there are!

But the string actually had interactions, where it split into two strings or two merged into one: these were the trinions. In the limit, these trinions do flow to something finite which remains: a certain interaction on a trivialent vertex of a graph.

This way, Kontsevich and Soibelman actually obtain what they call a “graph field theory”, a functor on something like $$1\mathrm{Cob}^{\mathrm{branch}}_{\mathrm{Riem}}$$ whose morphisms are graphs which become 1-dimensional Riemannian cobordisms once all vertices are removed.

(The fact that such funny business is not required for $2\mathrm{Cob}$ (unless we consider limits like here) has traditionally been regarded as an indication for the naturalness of the idea of passing from 1-particles to 2-particles.)

What is remarkable about this is the following:

as long as we are just looking at functors $$1\mathrm{Cob}_{\mathrm{Riem}} \to \mathrm{Hilb} \,,$$ we find that these are given by just two thirds of a spectral triple: a Hilbert space of states and a Hamiltonian (the Laplace operator).

This is for instance what Stolz and Teichner discuss.

Where is the last third: the algebra $A$ represented on that Hilbert space?

Of course Kontsevich and Soibelman now find this realized in the additional ingredient which they have: the (3-valent) vertices!

Apparently for this to make sense we have to assume that there is not just a representation of $A$ by bounded operators on our Hilbert space, but actually a dense embedding of our algebra (regarded as a vector space) into the Hilbert space.

(Of course this is the case in the standard motivating example where $A = C(X)$ and $H = L^2(X)$).

Then we can take the map $$H \otimes H \to H$$ which our functor associated to the 3-valent vertex and read it as a map $$A \otimes A \to A \,,$$ and that’s then the product on our algebra.

This is what I mentioned already in Soibelman on NCG of CFT and Mirror Symmetry.