The String Coffee Table

To start with, consider first spacetimes $X$ which are trivial circle bundles

Let $\hat \mathbf{T}$ denote the dual torus, dual to $\mathbf{T}$. Technically (since there is no metric in this topological game) this means that we take $\hat \mathbf{T}$ to be the space of characters ($\to$) of $\mathbf{T}$, when the latter is regarded as an abelian group.

What governs T-duality is something known as the Poincaré line bundle ($\to$) on the correspondence space ($\to$)

For the present purpose, the Poincaré line bundle on $\mathbf{T}\times \hat \mathbf{T}$ is defined as the quotient of $\mathbb{R}\times \hat \mathbf{T} \times \mathbb{C}$ by $\mathbb{Z}$, where $\mathbb{Z}$ acts as

such that

The curvature of this bundle is

So $P$ is flat when restricted to either $\mathbf{T}$ or $\hat \mathbf{T}$, but is nontrivial on all of $\mathbf{T} \times \hat \mathbf{T}$.

Let

be the total RR-field strength, i.e.

for type IIA and

for type IIB. The the Buscher rules describing the transformation of the RR fields under T-duality read

Notice that $e^F$ is the Chern character of the Poincaré line bundle

and note that $G$ is a closed form if and only if its T-dual $T_* G$ is a closed form. This means that the above Buscher rule(s) may be interpreted as an isomorphism of the deRham cohomology of the T-dual spacetimes $M \times \mathbf{T}$ and $M \times \hat \mathbf{T}$

What we are after is actually a T-duality induced isomorphism of the K-theory of one background with that (possibly up to a shift in degree) of the corresponding T-dual background.

Given any map

there is an obvious way to pull back a class in $K(Y)$ to a class in $K(X)$, simply by pulling back any representative bundle. There is not really a general way to push forward a bundle along $f$, of course. But there is such an operation on K-theory classes. This push forward at the level of K-theory, which can be denoted by $f_*$, is often written $f_!$.

Whenever we have a notion of pulling back and of pushing forward for some kind of object, then we can use correspondences like

to induce maps on the space of these objects ($\to$).

This is how T-duality acts on K-theory classes: we pull them back along $p$, tensor them fiberwise with $P$ and then push the result forward along $\hat p$:

Up to the extra factor of $M$, this is nothing but the Fourier-Mukai transformation, something which may be regarded as a categorified version of ordinary Fourier transformation ($\to$).

Notice that so far no background $H$-flux was considered. So the above transformation exchanges IIA with IIB data without changing the topology (which is that of $M\times \mathbf{T}$).

By way of the Chern character the isomorphism of K-theories and that of deRham cohomology is naturally related, in that this diagram commutes