Contents

cohomology

# Contents

## Idea

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

The idea of topological T-duality (due to Bouwknegt-Evslin-Mathai 04, Bouwknegt-Hannabus-Mathai 04) 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 integral transform (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 (Sarkissian-Schweigert 08).

## 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 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 (Bunke-Rumpf-Schick 08, p. 9)).

The concept was introduced on the level of differential form data in

In these papers the $U(1)$-gerbe (circle 2-bundle with connection) does not appear, but an integral differential 3-form, representing its Dixmier-Douady class 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

A refined version of this using smooth stacks is due to

There is also C*-algebraic version of toplogical T-duality, .e. in noncommutative topology, which sees also topological T-duals in non-commutative geometry:

The equivalence of the C*-algebraic to the Bunke-Schick version, when the latter exists, is discussed in

• Ansgar Schneider, Die lokale Struktur von T-Dualitnätstripeln (arXiv:0712.0260)

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

Another discussion that instead of noncommutative geometry 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

Discussion for non-free torus actions (physically: KK-monopoles) is in

• Ashwin S. Pande, Topological T-duality and Kaluza-Klein Monopoles, Adv. Theor. Math. Phys., 12, (2007), pp 185-215 (arXiv:math-ph/0612034)

Discussion in rational homotopy theory/dg-geometry is in

and a derivation of the rules of topological T-duality from analysis of the super p-brane super-cocycles in super rational homotopy theory (with a doubled supergeometry) is given in

reviewed in

See also

Comprehensive discussion in higher differential geometry:

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