nLab
topological T-duality

Idea

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

• a smooth manifold $X$

  with the structure of a torus-principal bundle $T^n\to X\to X/T^n$

• equipped with a

  circle 2-bundle with connection

• and in twisted K-theory

  refined to elements in differential twisted K-theory

• and notably equipped with

  a (pseudo)Riemannian metric

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 $X_1{\times }_B{X}_2$:

$$\begin{array}{ccccccc}& & {\mathrm{pr}}_1^{*}{G}_1& & \stackrel{u}{\leftarrow }& & {\mathrm{pr}}_2^{*}{G}_2\\ & \swarrow & & \searrow & & \swarrow & & \searrow \\ G_1& & & & X_1{\times }_B{X}_2& & & & G_2\\ & \searrow & & \swarrow & & \searrow & & \swarrow \\ & & X_1& & & & X_2\\ & & & \searrow & & & \swarrow \\ & & & & B\end{array}$$\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)

Related concepts

• T-duality

  • topological T-duality

  • differential T-duality

References

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

• Peter Bouwknegt-Jarah Evslin-Varghese Mathai (hep-th/0306062)

and the torus bundle case was introduced in

• Bouwknegt-Hannabus-Mathai hep-th/0312284.

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

• Ulrich Bunke and Thomas Schick, On the topology of T-duality (pdf)

and

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

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

• V. Mathai and Jonathan Rosenberg, T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology (arXiv)

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

• Gor Sarkissian and Christoph Schweigert, Some remarks on defects and duality (arXiv)

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

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory

Blog resources

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

• Mathai on T-duality:

  • I: Overview

  • II: T-dual K-classes by Fourier-Mukai.