nLab topological T-duality

Idea

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).

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)).

Related concepts

• T-duality

  • topological T-duality

  • differential T-duality

  • T-duality 2-group

• spherical T-duality

References

General

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, Reviews in Mathematical Physics 27 05 1550013 (2015) [arXiv:1305.6050, doi:10.1142/S0129055X15500130]

• 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:

• Varghese Mathai, Jonathan Rosenberg, T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology (arXiv:hep-th/0401168)

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)

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

• Calder Daenzer, A groupoid approach to noncommutative T-duality (arXiv:0704.2592)

Comments in relation to T-folds:

• Peter Bouwknegt, Ashwin S. Pande, Topological T-duality and T-folds, Advances in Theoretical and Mathematical Physics 13 5 (2009) [arXiv:0810.4374, doi:10.4310/ATMP.2009.v13.n5.a6]

The bi-brane perspective on T-duality is amplified in

• Gor Sarkissian, Christoph Schweigert, Some remarks on defects and duality (arXiv:0810.3159)

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) 185-215 [arXiv:math-ph/0612034, doi:10.4310/ATMP.2008.v12.n1.a3]

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)

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: T-Duality from super Lie n-algebra cocycles for super p-branes, Adv. Theor. Math. Phys 22 5 (2018) [arXiv:1611.06536, doi:10.4310/ATMP.2018.v22.n5.a3]

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

• Hisham Sati, Urs Schreiber, p. 7 of: Cyclification of Orbifolds [arXiv:2212.13836]

See also:

• Domenico Fiorenza, Mattia Coloma, Eugenio Landi: A very short note on the (rational) graded Hori map, Communications in Algebra 50 5 (2022) [arXiv:2003.13066, doi:10.1080/00927872.2021.2005076]

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)

Equivariant refinement

Discussion in the equivariant generality, ie. as an equivalence of twisted equivariant K-theory groups:

• Tom Dove, Thomas Schick, Equivariant Topological T-Duality [arXiv:2310.06064]

Geometric refinement

On possible geometric refinement of topological T-duality via some form of differential cohomology:

via differential K-theory classes:

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory, Communications in Contemporary Mathematics, 16 02 (2014) [arXiv:0912.2516, doi:10.1142/S0219199713500144]

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, Ann. Henri Poincaré 25 (2024) 1285–1358 [arXiv:2207.11799, doi:10.1007/s00023-023-01295-0]

• Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections [arXiv:2303.16162]