group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $T^n \to X \to X/T^n$ – modelling spacetime
equipped with a circle 2-bundle with connection – modelling the Kalb-Ramond field
and in twisted K-theory refined to elements in differential twisted K-theory – modelling the RR-field
and notably equipped with a (pseudo)Riemannian metric – modelling the field of gravity.
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 $(X_i \to B, G_i)_{i = 1,2}$ 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 $X_1 \times_B X_2$:
of a certain prescribed integral transform-form (Bunke-Rumpf-Schick 08, p. 9)).
topological T-duality
The concept was introduced on the level of differential form data in
Peter Bouwknegt, Jarah Evslin, Varghese Mathai, T-Duality: Topology Change from H-flux, Commun. Math. Phys. 249:383-415, 2004 (hep-th/0306062)
Peter Bouwknegt, Keith Hannabus Varghese Mathai, T-duality for principal torus bundles, JHEP 0403 (2004) 018 (hep-th/0312284)
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
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 $T$-duality for $T^n$-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
Ulrich Bunke, Thomas Nikolaus, T-Duality via Gerby Geometry and Reductions (arXiv:1305.6050)
Thomas Nikolaus, T-Duality in K-theory and Elliptic Cohomology, talk at String Geometry Network Meeting, Feb 2014, ESI Vienna (website)
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
Jonathan Rosenberg has also written a little introductory book for mathematicians:
Another discussion that instead of noncommutative geometry uses topological groupoids is in
The bi-brane perspective on T-duality is amplified in
Discussion in rational homotopy theory/dg-geometry is in
Ernesto Lupercio, Camilo Rengifo, Bernardo Uribe, T-duality and exceptional generalized geometry through symmetries of dg-manifolds (arXiv:1208.6048)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes (arXiv:1611.06536)
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