The n-Category Café

1) First a word on T-duality:

In the context of quantum theory, we conceive the geometry of effective target space, i.e. of the space that our quantum $n$-particle propagates in, entirely in terms of its observable effects on that $n$-particle.

We probe space by letting our quantum $n$-particle roam in it.

At least for $n=1$ (but conceivably also for higher $n$) this is exactly the idea of Alain Connes’ notion of (“spectral”) geometry:

Geometry is whatever is encoded in the spectrum of a suitale Laplace-like operator (the Hamiltonian of our particle) or Dirac-like operator (the supercharge of our superparticle).

Taking this seriously, we are lead to find kinds of geometries that have, in particular, notions of automorphisms that go beyond the diffeomorphisms of ordinary manifolds.

One of the most famous of these “generalized diffeomorphisms” is T-duality. This arises, quite generally, for “2-particles” – also known as strings.

The idea is, roughly, that when probing a space of the form $$X \times \mathbf{T}\,,$$ where $\mathbf{T}$ is the 1-torus, by letting a 2-particle roam in it, we usually cannot distinguish it from $$X \times \hat \mathbf{T} \,,$$ where $\hat \mathbf{T}$ is the dual torus.

The reason is, roughly, that the spectrum of the Hamiltonian of the 2-particle contains contributions from

a) the ordinary momentum of the 2-particle

but also

b) from the linear extension of the 2-particle.

As we pass from a 2-particle on a torus to that on the dual torus, these two contributions interchange their role, but such that the total spectrum (of the loop space Laplace operator) remains unaffected!

So the notion of geometry as induced by the quantum theory of the 2-particle has a notion of automorphism not known to 1-particles. These isomorphisms are called T-dualities. For the obvious reason (“T”=torus).

This is, at least, the standard way to descrtibe T-duality in string theory. However, the basic mechanism is much more general than any mentioning of “momentum” and “linear extension” might suggest: in the context of what is known as Topological T-Duality people study the effect of T-duality in particular on twisted K-theory, without ever considering the dynamics of any 2-particle.

We have seen that K-theory, which classifies Chan-Paton bundles on D-branes (and twisted K-theory, which classifies twisted Chan-Paton bundles known as gerbe modules) is a purely kinematical aspect of the 2-particle.

So: topological T-duality is the kinematical aspect of T-duality.

Now a word on T-Folds:

In as far as T-duality is a “generalized diffeomorphism” of a quantum geometry, it makes sense to regard local such diffeomorphisms.

Like an ordinary manifold is something that locally looks like $\mathbb{R}^n$ such that the transitions are diffeomorphisms of $\mathbb{R}^n$, we can imagine defining a T-fold to be something that locally looks like a manifold, with the transitions being given by T-dualities.

I am not aware that this idea has been adopted seriously by mathematicians and developed rigorously (but maybe it has: I’d be grateful for references!), but string theorists do have thought about this. See for instance

C. M. Hull
A Geometry for Non-Geometric String Backgrounds
hep-th/0406102

Finally: the example

In order to obtain an easily tractable but interesting example of pull-push evolution of 2-particles, I would like to consider a 2-particle on a target space 2-category $\mathrm{tar}$ which models a very simple T-fold structure.

The idea is to model the T-fold transitions in the T-fold in the same spirit that group action transitions are modeled in an orbifold: we simply include additional morphisms between points that are related by the respective transformation. (For how orbifolds are described by groupoids this way see Moerdijk on Orbifolds, II.)

So given a torus $\mathbf{T}$, and its dual torus $\hat \mathbf{T}$ (both really just being $S^1$), consider the 2-category $$P_2^T(S^1 \times S^1)$$ which is defined to be that generated by 2-paths in $\mathbf{T}$ $$P_2(\mathbf{T})$$ and 2-paths in $\hat \mathbf{T}$ $$P_2(\mathbf{\hat T})$$ as well as 1-morphisms that are pairs of points $$\mathbf{T} \times \hat \mathbf{T}$$ and $$\hat \mathbf{T} \times \mathbf{T} \,.$$

More generally, we might consider T-duality on a target space of the form $X \times \mathbf{T}$. Then we would suitably define $$\mathrm{tar} = P_2^T(X \times S^1 \times S^1)$$ in the more or less obvious way.

The 2-particle on that target space will couple to a Kalb-Ramond background field, which can be regarded as nothing but a 2-functor $$\mathrm{tra} : P_2^T(X \times S^1 \times S^1) \to \Sigma(1d\mathrm{Vect}_\mathbb{C}) \,.$$

On the generators $P_2(X\times \mathbf{T})$ and $P_2(X\times \hat \mathbf{T})$ we take this to be any bundle gerbe with connection that we like.

But, since we are now on a T-fold, we will also have to define how $\mathrm{tra}$ acts on the new T-duality morphisms in $\mathbf{T} \times \hat \mathbf{T}$.

Since $\mathrm{tra}$ takes values in $\Sigma(1d\mathrm{Vect}_\mathbb{C})$, this will be nothing but a line bundle $$\array{ P \\ \downarrow \\ \mathbf{T} \times \hat \mathbf{T} } \,.$$

Take this to be the Poincaré line bundle.

Finally, consider the open 2-particle of the form $$\mathrm{par} = (a\to b)$$ and consider a simple cylindrical worldvolume of that $$\mathrm{wolldvol} = \left( \array{ a &\to& a' \\ \downarrow &\Rightarrow& \downarrow \\ b &\to& b' } \right) \,.$$ Take the space of histories $$\mathrm{hist}$$ to be maps from this to all T-duality morphisms.

Then the pull-push evolution on our 2-particle along this worldvolume should give a T-duality transformation on the Chan-Paton bundles over the endpoints $a$ and $b$.

And I claim it does. Feeding this data into the pull-push defined by (2) reproduces, over the 2-particle’s endpoints, precisely the topological T-duality transformation.

I wanted to describe this in more detail. But now I did run out of time. Have to run now. To be continued (maybe in the comment section.)