The String Coffee Table

Here are some preliminary remarks, as a preparation for the talk this afternoon.

Let $X$ be some space. A vector bundle $E\to X$ is like a vector whose entries are vector spaces, each entry being the fiber over one point of $X$. We hence call such a vector bundle a (generalized Kapranov-Voevodsky ($\to$)) 2-vector. They live in the 2-vector space ${\mathrm{Vect}}^X$, the category of vector bundles over $X$.

Alternatively, we may think of this as

where $A_X=C(X)$ is the algebra of functions on $X$ (or the sheaf of such functions, according to taste).

We want a notion of linear 2-map from ${}_{A_X}\mathrm{Mod}$ to ${}_{A_Y}\mathrm{Mod}$.

Consider the product space

and a vector bundle $L$ over that

This is like a $Y\times X$-matrix whose entries are vector spaces. (For $Y$ and $X$ finite sets, this are precisely the ordinary linear 2-maps of Kapranov-Voevodsky ($\to$)).

$L$ is an $A_{Y\times X}$ module, which we may regard as a $A_Y$-$A_X$ bimodule.

We may act with $L$ on $E$ by pulling back along $p_2$, tensoring fiberwise with $L$ and pushing forward along $p_1$.

The result is a bundle $(p_1{)}_{*}(L{\otimes }_{Y\times X}{p}_1^{*}(E))\to Y$ over $Y$, which we may regard as a $A_Y$-module.

I believe that, possibly up to some subtleties, equivalently, we can regard this as the bimodule tensor product

Of course the subtlety is the pushforward of vector bundles, which goes the “wrong way”. For the first of the examples below, the baby toy example, we can think of the push forward as being given by taking direct sums over fibers living over the same fiber of the map that we push along. For base spaces that are finite sets this makes sense. For continuous spaces we get infinite-dimensional fibers this way, which will need to be reduced to something finite using maybe a Dirac operator and its index, or related tricks.

(Thanks to Varghese Mathai for a lot of very helpful discussion on this point. All oversimplifications which I am making here are completely my fault, of course.)

Examples.

1)

One can check that, when $Y$ and $X$ are finite sets, this reproduces the 2-map operation of Kapranov-Voevodsky, where we multiply matrices whose entries are vector space with vectors whose entries are vector spaces by mimicking ordinary matrix multiplication but with sums of numbers replaced by direct sums of vector spaces, and with products of numbers replaced by tensor products of vector spaces.

2)

The Fourier-Mukai transformation is of this form.

Shigeru Mukai
Duality between $D(X)$ and $D(Y)\mathrm{with}\mathrm{its}\mathrm{application}\mathrm{to}\mathrm{Picard}\mathrm{sheaves}$
Nagoya Mathematical J. 81 (1981) 153-175
(pdf)

In that context $Y$ is the dual of $X$ and $L$ the Poincaré-line bundle on $X^{*}\times X$.

We can regard this as a categorified Fourier transformation from a 2-vector space of 2-functions on $X$ to that of 2-functions on the dual of $X$.

3)

The Hecke operator is of this form ($\to$).

In this case the line bundle over the correspondence space is trivial.

4)

The equivalence constituting the Langlands duality is of this form. See p. 53 of Frenkel’s lecture notes.

In fact, at least in the classical limit, the geometric Langlands duality is essentially the same as a Fourier-Mukai transformation ($\to$).

5)

T-duality can be understood as induced by pulling-tensoring-pushing using a certain line bundle on a certain correspondence.

I’ll talk about that in a seperate entry.

But see for instance def. 1.5 in

U. Bunke, P. Rumpf, Th. Schick
The topology of T-duality for $T^n$-bundles
math.GT/0501487 .