cohomology

# Contents

## Idea

Recall that ( geometric ) T-duality is an operation acting on tuples roughly consisting of

The idea of topological T-duality is to disregard the Riemannian metric and the connection and study the remaining “topological” structure.

While the idea of T-duality originates in string theory, topological T-duality has become a field of study in pure mathematics in its own right.

In the language of bi-branes a topological T-duality transformation is a bi-brane of a special kind between the two gerbes involved. The induced pull-push operation (in groupoidification and geometric function theory) on (sheaves of sections of, or K-classes of) (twisted) vector bundles is essentially the Fourier-Mukai transformation. More on the bi-brane interpretation of (topological and non-topological) T-duality is in SarkissianSchweigert08.

## Definition

Two tuples $(X_i \to B, G_i)_{i = 1,2}$ consisting of a $T^n$-bundle $X_i$ over a manifold $B$ and a line bundle gerbe $G_i \to X_i$ over $X$ are topological T-duals if there exists an isomorphism $u$ of the two line bundle gerbes pulled back to the fiber product correspondence space $X_1 \times_B X_2$:

$\array{ && pr_1^* G_1 && \stackrel{u}{\leftarrow} && pr_2^* G_2 \\ & \swarrow && \searrow && \swarrow && \searrow \\ G_1 &&&& X_1 \times_B X_2 &&&& G_2 \\ & \searrow && \swarrow && \searrow && \swarrow \\ && X_1 &&&& X_2 \\ &&& \searrow &&& \swarrow \\ &&&& B }$

of a certain prescribed integral transform-form (see p. 9 of (BunkeRumpfSchick08)

## References

The simplest version of topological T-duality, when $X$ is a principal circle bundle, was originally developed in

and the torus bundle case was introduced in

In these papers the gerbe does not appear, but an integral 3-form, representing the Dixmier-Douady class of a gerbe does. Note that if the integral cohomology group $H^3(X,\mathbb{Z})$ of $X$ has torsion in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in

and

• U. Bunke, P. Rumpf and Thomas Schick, The topology of $T$-duality for $T^n$-bundles (arXiv)

A refined description is in

There is a C*-algebraic version of toplogical T-duality, discussed for instance at

Jonathan Rosenberg has also written a little introductory book for mathematicians:

• Jonathan Rosenberg: Topology, $C^*$-algebras, and string duality (ZMATH)

A discussion that instead of noncommutative spaces uses topological groupoids is in

• Calder Daenzer, A groupoid approach to noncommutative T-duality (arXiv:0704.2592)

The bi-brane perspective on T-duality is amplified in

The refinement of topology T-duality to differential cohomology, hence to an operation on the differential K-theory classes that model the RR-field is in

## Blog resources

A transcript a a talk by Varghese Mathai on topological T-duality is here:

• Mathai on T-duality:

Revised on September 8, 2014 05:34:50 by David Corfield (129.12.18.225)