Critical string models
Special and general types
Recall that geometric T-duality is an operation acting on tuples roughly consisting of
The idea of topological T-duality (due toBouwknegt-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).
Two tuples consisting of a -bundle over a manifold and a line bundle gerbe over are topological T-duals if there exists an isomorphism of the two bundle gerbes pulled back to the fiber product correspondence space :
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 -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 of has torsion in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in
Ulrich Bunke, Thomas Schick, On the topology of T-duality, Rev.Math.Phys.17:77-112,2005, (arXiv:math/0405132)
U. Bunke, P. Rumpf, Thomas Schick, The topology of -duality for -bundles, Rev. Math. Phys. 18, 1103 (2006). (arXiv:math.GT/0501487)
Ulrich Bunke, Markus Spitzweck (Regensburg), Thomas Schick, Periodic twisted cohomology and T-duality, Astérisque No. 337 (2011), vi+134 pp. ISBN: 978-2-85629-307-2
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
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
A transcript a a talk by Varghese Mathai on topological T-duality is here: