nLab
topological T-duality

Context

String theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 iB,G i) i=1,2(X_i \to B, G_i)_{i = 1,2} consisting of a T nT^n-bundle X iX_i over a manifold BB and a line bundle gerbe G iX iG_i \to X_i over XX are topological T-duals if there exists an isomorphism uu of the two line bundle gerbes pulled back to the fiber product correspondence space X 1× BX 2X_1 \times_B X_2:

pr 1 *G 1 u pr 2 *G 2 G 1 X 1× BX 2 G 2 X 1 X 2 B \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 XX 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,)H^3(X,\mathbb{Z}) of XX has torsion in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in

and

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 *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:

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