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 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).
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 of topological T-duality 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, doi:10.1007/s00220-004-1115-6]
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 topological/homotopy theoretic data originates in
Ulrich Bunke, Thomas Schick, On the topology of T-duality, Rev. Math. Phys. 17 (2005) 77-112 [arXiv:math/0405132, doi:10.1142/S0129055X05002315]
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, doi:10.1142/S0129055X06002875]
Ulrich Bunke, Markus Spitzweck, 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
Introduction and review:
Jonathan Rosenberg, Topology, $C^*$-algebras, and string duality, Regional Conference Series in Mathematics 111, Amer. Math. Soc. (2009) [doi:10.1090/cbms/111, ZMATH]
§1 in Waldorf 2022
Another discussion that instead of noncommutative geometry uses topological groupoids is in
Comments in relation to T-folds:
The bi-brane perspective on T-duality is amplified in
Discussion for non-free torus actions (physically: KK-monopoles) is in
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
Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-duality in rational homotopy theory via strong homomotopy Lie algebras, Geometry, Topology and Mathematical Physics Journal, Volume 1 (2018) (arXiv:1712.00758)
Domenico Fiorenza, T-duality in rational homotopy theory, talk at 38th Srni Winter School on Geometry and Physics, 2018 (pdf slides)
and further expanded on on
See also
Comprehensive discussion in higher differential geometry:
Luigi Alfonsi, Global Double Field Theory is Higher Kaluza-Klein Theory, Fortsch. d. Phys. 2020 (arXiv:1912.07089, doi:10.1002/prop.202000010)
(relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, non-abelian T-duality, type II geometry, exceptional geometry, …)
Luigi Alfonsi, The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective (arXiv:2007.04969)
Discussion in the equivariant generality, ie. as an equivalence of twisted equivariant K-theory groups:
On possible geometric refinement of topological T-duality via some form of differential cohomology:
via differential K-theory classes:
using adjusted principal 2-connections:
Hyungrok Kim, Christian Saemann, Non-Geometric T-Duality as Higher Groupoid Bundles with Connections [arXiv:2204.01783]
Konrad Waldorf, Geometric T-duality: Buscher rules in general topology [arXiv:2207.11799]
Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections [arXiv:2303.16162]
Last revised on October 11, 2023 at 06:19:39. See the history of this page for a list of all contributions to it.