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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


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

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

(2)M×T×T̂ p 2 M×T̂ p 1 M×T.

For the present purpose, the Poincaré line bundle on T×T̂ is defined as the quotient of ×T̂× by , where acts as

(3)×T̂× n ×T̂× (r,ρ,z) (rn,ρ,ρ(n)z),

such that

(4)P=(×T̂×)/ T×T̂.

The curvature of this bundle is


So P is flat when restricted to either T or T̂, but is nontrivial on all of T×T̂.


(6)GΩ (M×T)

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

(7)GΩ even(M×T)

for type IIA and

(8)GΩ odd(M×T)

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

(9)T *G:= Te FG.

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

(10)e F=ch(P)

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×T̂

(11)T *:H (M×T)H (M×T̂).

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

(13)P M×T×T̂ p̂ M×T̂ p M×T

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

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 p̂:

(14)T !:K (M×T) K +1 (M×T̂) E p̂ *((p *E)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 ().

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)K (M×T) T ! K +1 (M×T̂) ch ch H (M×T) T * H +1 (M×T̂).
Posted at June 2, 2006 12:58 PM UTC

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