## June 2, 2006

### Mathai on T-Duality, II: T-dual K-classes by Fourier-Mukai

#### Posted by urs

The first part of my transcript of V. Mathai’s talk.

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

(1)$X=M×T\phantom{\rule{thinmathspace}{0ex}}.$

Let $\stackrel{̂}{T}$ denote the dual torus, dual to $T$. Technically (since there is no metric in this topological game) this means that we take $\stackrel{̂}{T}$ to be the space of characters ($\to$) of $T$, 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 ($\to$)

(2)$\begin{array}{ccc}M×T×\stackrel{̂}{T}& \stackrel{{p}_{2}}{\to }& M×\stackrel{̂}{T}\\ {p}_{1}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ M×T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

(3)$\begin{array}{ccc}ℝ×\stackrel{̂}{T}×ℂ& \stackrel{n\in ℤ}{\to }& ℝ×\stackrel{̂}{T}×ℂ\\ \left(r,\rho ,z\right)& ↦& \left(r\cdot n,\rho ,\rho \left(n\right)\cdot z\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$

such that

(4)$\begin{array}{c}P=\left(ℝ×\stackrel{̂}{T}×ℂ\right)/ℤ\\ ↓\\ T×\stackrel{̂}{T}\phantom{\rule{thinmathspace}{0ex}}.\end{array}$

The curvature of this bundle is

(5)$F=d\theta \wedge d\stackrel{̂}{\theta }\phantom{\rule{thinmathspace}{0ex}}.$

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

Let

(6)$G\in {\Omega }^{•}\left(M×T\right)$

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

(7)$G\in {\Omega }^{\mathrm{even}}\left(M×T\right)$

for type IIA and

(8)$G\in {\Omega }^{\mathrm{odd}}\left(M×T\right)$

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

(9)${T}_{*}G:={\int }_{T}\phantom{\rule{thickmathspace}{0ex}}{e}^{F\wedge }G\phantom{\rule{thinmathspace}{0ex}}.$

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

(10)${e}^{F}=\mathrm{ch}\left(P\right)$

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 $M×T$ and $M×\stackrel{̂}{T}$

(11)${T}_{*}:{H}^{•}\left(M×T\right)\simeq {H}^{•}\left(M×\stackrel{̂}{T}\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(12)$X\stackrel{f}{\to }Y$

there is an obvious way to pull back a class in $K\left(Y\right)$ to a class in $K\left(X\right)$, 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

(13)$\begin{array}{c}P\\ ↓\\ M×T×\stackrel{̂}{T}& \stackrel{\stackrel{̂}{p}}{\to }& M×\stackrel{̂}{T}\\ p↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ M×T\end{array}$

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 $\stackrel{̂}{p}$:

(14)$\begin{array}{ccc}{T}_{!}:{K}^{•}\left(M×T\right)& \simeq & {K}^{•+1}\left(M×\stackrel{̂}{T}\right)\\ E& ↦& {\stackrel{̂}{p}}_{*}\left(\left({p}^{*}E\right)\otimes P\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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 $M×T$).

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

(15)$\begin{array}{ccc}{K}^{•}\left(M×T\right)& \stackrel{{T}_{!}}{\to }& {K}^{•+1}\left(M×\stackrel{̂}{T}\right)\\ \mathrm{ch}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\mathrm{ch}\\ {H}^{•}\left(M×T\right)& \stackrel{{T}_{*}}{\to }& {H}^{•+1}\left(M×\stackrel{̂}{T}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$
Posted at June 2, 2006 12:58 PM UTC

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