nLab T-duality

Idea

A $d$-dimensional sigma-model is a quantum field theory that is induced from certain differential geometric and differential cohomological data, to be thought of as encoding the background geometry on which quantum objects of dimension $d$ propagate.

The operation of T-duality is a map that interchanges pairs of such geometric data for 2-dimensional conformal field theory sigma-models, such that the induced QFTs are equivalent.

More specifically the space of differential geometric data consisting of

• a smooth manifold $X$

• equipped with the structure of a $k$-torus bundle $X \simeq Y \times T^k$; – the total space of this bundle modelling spacetime;

• and equipped with a Riemannian metric $g$ – modelling the field of gravity;

• and a $U(1)$-gerbe with connection $G$; – modelling the Kalb-Ramond field;

• possibly a cocycle in differential K-theory modelling the RR-field;

admits a certain operation that, roughly, inverts the Riemannian circumference of the torus fibers and mixes the metric with the gerbe data, such that the induced 2-dimensional sigma-model QFTs for these backgrounds are equivalent. This is the operation called T-duality.

This was noticed originally in the study of conformal field theories in the context of string theory: the conformal field theory sigma-models with target space $X$ turn out to be equivalent as quantum field theories for T-dual backgrounds $(X,g,G)$ and $(X',g',G')$ (at least to the approximate degree to which these are realized as full CFTs in the first place).

Further generalisations let $X$ be a nontrivial torus bundle, but the T-dual is then generically a bundle of non-commutative tori. (cite Mathai, Rosenberg and Hannabus)

Path integral heuristics deriving T-duality

We indicate how one can see T-duality from formal manipulations of the path integral for the string sigma-model. We look at the simplest situation, where the torus bundle in question is a trivial circle bundle over a Cartesian space carrying the metric induced from the standard flat metric on $\mathbb{R}^n$ and where there are no other nontrivial background fields. In fact, for the purpose of the following computation we can entirely ignore the base of this bundle and consider target space to be nothing but a circle. Since the sigma-model for this is on the worldsheet just the theory of a single free field with values in $S^1$, this is often also called the “free boson on the circle”.

This means that the only geometric datum determining the background geometry is the circumference $2 \pi R$ of the fiber of the circle bundle. The statement of T-duality in this situation is that the 2-dimensional sigma-model on this background yields the same 2-dimensional CFT as that for this kind of background with circumference of the circle being $2 \pi 1/R$.

A first rough look

A quick way to get an indication for this is to consider the center-of-mass energy of the string in such a circle-bundle background. In the simplified setup we mentioned before, a string on a circle of radius $R$ has quantized momentum $p = \frac{\ell \in \mathbb{Z}}{R}$. In a state in which the string winds around the circle $m$ times and has $\ell$ quanta of kinetic momentum for propagation around the circle, its energy is

$$E_0 = \sqrt{p^2 + M^2} = \sqrt{(\ell/R)^2 + (R m)^2 } \,,$$

This energy is clearly invariant under exchanging

$$(R, (\ell, m)) \mapsto (\frac{1}{R}, (m,\ell)) \,.$$

This is of course far from being a proof that the corresponding two QFTs are equivalent, but it does already capture a good deal of the essence of what T-duality does and why it works.

In slightly more detail, but still at a very rough level, if we denote by

$$X : \Sigma \to S^1_R$$

the $\sigma$-model field on the worldsheet $\Sigma = \mathbb{R}^2$ with values in target space $S^1_R$ then T-duality with respect to this circle may be thought of as exchanging worldsheet momentum $\partial_t X$ with worldsheet winding $\partial_\sigma X$.

This then also means that for the open string it exchanges von Neumann boundary conditions $\partial_\sigma X|_{\sigma = 0} = 0$ with Dirichlet boundary conditions $\partial_t X|_{\sigma = 0} = 0$. The first boundary condition is that describing an open string whose endpoints are free to propagate in worldsheet time, whereas the second boundary condition describes a situation where the endpoint of the string is fixed at some point in target space. In terms of the language of geometric target space data, a sigma-model with such a constraint is said to describe a D-brane in target space: the locus where the endpoints of the string are fixed. This is a first indication that the T-duality operation on geometric background also involves the RR-field.

The path integral

We follow Kentaro Hori‘s path integral discussion of T-duality. Here the strategy is to consider a path integral over a certain space of auxiliary fields and show or argue that by “algebraically integrating out” some of these in two different ways, the path integral is equivalent to that over two different action functionals, which describe two T-dual geometric backgrounds.

Let the boundary components of the worldsheet $\Sigma$ be labeled by $\partial \Sigma_{(1)}$.

We consider the following fields on the worldsheet:

• $\tilde X : \Sigma \to \mathbb{R}/(2\pi/R)\mathbb{Z} = S^1_{1/R}$ – a circle-valued function; this is the standard $\sigma$-model field describing propagation of the string on the circle;

• $X_{i} : \partial \Sigma_{(i)} \to S^1_R$ – the boundary values of this field;

• $b \in \Omega^1(\Sigma, \mathbb{R})$ – a 1-form; this is the auxiliary field that will not contribute to the dynamics but serves to make the T-duality manifest.

Consider then the following action functional on this collection of fields given by the assignment

$$S'_E(\tilde X,b) = \frac{1}{2 \pi} \int_\Sigma \left( \frac{1}{2} b \wedge \star b - b \wedge d \tilde X \right) - \frac{i}{2 \pi} \sum_{i = 1}^s \int_{\partial \Sigma_{(i)}} (\tilde X -a_i) d X_i \,,$$

where the $(a_i)$ are a collection of real numbers.

We now want to formally perform the path integral over the fields in two different orders, which should give the same quantum field theories but in terms of different effective action functionals.

If we do first the path integral over the field $b$ then by the general formal rule of “algebraically integrating out a non-dynamical field” which says that we can evaluate this path integral that formally looks like a Gaussian integral by the usual formulas for Gaussian integrals, we obtain the action functional

$$\tilde S_E = \frac{1}{4 \pi} \int_\Sigma d \tilde X \wedge \star d \tilde X + \frac{i}{2\pi} \int_{\partial \Sigma_{(i)}} (\tilde X - a_i) d X_i$$

then doing the integral over the boundary values $X_i$ yields

$$\tilde X|_{\partial \Sigma_i} = a_i$$

This is the action functional for a $\sigma$-model on $S^1_{1/R}$ with a D-brane at $\tilde X = a_i$.

Now we evaluate the original path integral in a different way, this way first integrating over components of $\tilde X$. To do so, we imagine that we may re-encode the field $\tilde X$ in terms of its de Rham differential

$$d \tilde X = d f + \frac{2\pi}{R}\sum_A \eta_A \omega_A$$

where $\eta_A$ are integers and

$$\{\omega_A\} \subset H^1(\Sigma, \mathbb{Z}) \simeq \mathbb{Z}^{\oplus 2g + s - 1} \,.$$

Then formally performing the path integral over $f$ yields $d b = 0$ and $b|_{\partial \Sigma} = d X_i$. It follows that $b = d X$ for some other field $X : \Sigma \to S^1_R$.

So we get the action

$$S_E = \frac{1}{4\pi} \int_\Sigma d X \wedge \star d X + \frac{i}{2 \pi} \sum_{i = 1}^s \int_{\partial \Sigma_i} a_i d X$$

in terms of the field $X$. This is the $\sigma$-model for string propagation on $S^1_R$. with D-brane wrapped on $S^1_R$ that carries on its worldvolume a gauge field given by a constant connection 1-form $a_i$.

Topological T-duality

It turns out to be possible and useful to discuss just the topological aspects of T-duality, meaning all the aspects that depend on the $X$ as a topological space, on the topological class of the gerbe and of its 3-form curvature, but not on the Riemannian metric and not on the precise connection on the gerbe (there may be several inequivalent ones for a given curvature)!

This sub-phenomenon is discussed in more detail at topological T-duality.

Geometric T-duality

In terms of generalized differential cohomology

Gauge fields are cocycles in differential cohomology. The Kalb-Ramond field is given by degree-3 ordinary differential cohomology, the differential refinement on degree-3 integral cohomology. The RR-field is given by differential K-theory.

Induced by the morphisms $\mathbf{c}(n)$ in the fiber sequences

$$\mathbf{B}U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{c}_n}{\to} \mathbf{B}^2 U(1)$$

is induced a notion of twisted cohomology which makes the Kalb-Ramond field act as a twist for twisted K-theory.

In these terms, the setup of T-duality is a correspondence of Kalb-Ramond fields over spacetime torus-bundles $P \to X$ and $\hat P \to X$ that induces an integral transform

$$K_{diff}^{\bullet + \tau}(P) \to K_{diff}^{\bullet + \tau -1}(\hat P)$$

of twisted differential K-theory classes.

This is an isomorphism – the action of the T-duality isomorphism on the Kalb-Ramond field and the RR-field.

See (KahleValentino).

In generalized complex geometry

Another approach to the study of T-duality takes a somewhat complementary point of view and ignores the integral cohomology class in $H^3(X,\mathbb{Z})$ of the gerbe but does consider the Riemannian metric.

In this context T-duality is described as an isomorphism of standard Courant algebroids. This picture emerged in the study of generalized complex geometry.

Examples of T-dual pairs

In Mirror symmetry

One special case of T-duality is mirror symmetry.

• Andrew Strominger, Shing-Tung Yau, Eric Zaslow, Mirror Symmetry is T-Duality, Nucl. Phys. B 479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) hep-th/9606040

In Langlands duality

In some cases the passage to the Langlands dual group in the geometric Langlands correspondence can be understood as T-duality. (Daenzer-vanErp)

Related concepts

duality in physics, duality in string theory

• T-duality

  • topological T-duality

    • spherical T-duality

  • Poisson-Lie T-duality

  • differential T-duality

  • T-duality 2-group

• T-fold, double field theory

• duality in string theory

  • U-duality, S-duality

  • AdS/CFT correspondence

The geometry of the fiber product of two torus fiber bundles with a circle 2-bundle on it is sometimes referred to as Bn-geometry.

References

In string theory

The observation of T-duality is attributed to

• Thomas Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59 (doi:10.1016/0370-2693(87)90769-6)

• Thomas Buscher, Path integral derivation of quantum duality in nonlinear sigma models, Phys. Lett. B 201 (1988) 466 (doi:10.1016/0370-2693(88)90602-8)

Precursors include (according to Schwarz 96, first paragraph):

• Keiji Kikkawa, Masami Yamasaki, Casimir effects in superstring theories, Physics Letters B Volume 149, Issues 4–5, 20 December 1984, Pages 357-360 (doi:10.1016/0370-2693(84)90423-4)

Review:

• Amit Giveon, Massimo Porrati, Eliezer Rabinovici: Target space duality in string theory, Physics Reports 244 2–3 (1994) 77-202 [doi:10.1016/0370-1573(94)90070-1]

• E. Alvarez, Luis Alvarez-Gaumé, Yolanda Lozano: An Introduction to T-Duality in String Theory, Nucl. Phys. Proc. Suppl. 41 (1995) 1-20 [arXiv:hep-th/9410237, doi:10.1016/0920-5632(95)00429-D]

• Jonathan Rosenberg, Topology, $C^*$-algebras, and string duality, Regional Conference Series in Mathematics 111, Amer. Math. Soc. (2009) [doi:10.1090/cbms/111, ZMATH]

  (focus on topological T-duality)

• Peter West, section 17.1 of Introduction to Strings and Branes, Cambridge University Press (2012) [doi:10.1017/CBO9781139045926]

• Mathai Varghese, T-duality: A basic introduction, 10th Geometry and Physics Conference Quantum Geometry, Anogia 2012 (pdf)

• Jnanadeva Maharana, The Worldsheet Perspective of T-duality Symmetry in String Theory (arXiv:1302.1719)

• Mark Bugden, A Tour of T-duality: Geometric and Topological Aspects of T-dualities [arXiv:1904.03583]

Discussion at strong coupling (“F-theory”):

• John Schwarz: M Theory Extensions of T Duality, in: Frontiers in Quantum Field Theory, World Scientific (1996) 3-14 [arXiv:hep-th/9601077, doi:10.1142/3245]

• Jorge Russo, T-duality in M-theory and supermembranes, Phys.Lett. B400 (1997) 37-42 (arXiv:hep-th/9701188)

On the worldsheet formulation of T-duality in the presence of background RR-fields

via the pure spinor superstring:

• Raphael Benichou, Giuseppe Policastro, Jan Troost, T-duality in Ramond-Ramond backgrounds, Phys. Lett. B 661 (2008) 192-195 [arXiv:0801.1785, doi:10.1016/j.physletb.2008.01.059]

Dscussion 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 of geometric T-duality in terms of some form of differential cohomology:

as an operation on twisted differential K-theory:

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory

using adjusted principal 2-connections:

• Hyungrok Kim, Christian Saemann: Non-Geometric T-Duality as Higher Groupoid Bundles with Connections [arXiv:2204.01783]

• Christian Saemann: Non-Geometric T-duality from

  Higher Groupoid Bundles with Connections, talk atESI Workshop on Higher Structures* (2022) [pdf]

• Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections, Proceedings of CORFU2022, PoS 436 [arXiv:2303.16162, doi:10.22323/1.436.0336]

• 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]

More physically oriented discussion of this is in

• Katrin Becker, Aaron Bergman, Geometric Aspects of D-branes and T-duality (arXiv:0908.2249)

Geometric T-duality is identified as an isomorphism of standard Courant algebroids (generalized complex geometry) in section 4 of

• Gil Cavalcanti, Marco Gualtieri, Generalized complex geometry and T-duality (arXiv:1106.1747)

Discussion of the sigma-model description of T-duality in this context includes

• Ulf Lindstöm, Martin Rocek, Itai Ryb, Rikard von Unge, Maxim Zabzine, T-duality and Generalized Kähler Geometry, JHEP 0802:056,2008 (arXiv:0707.1696)

• Jonas Persson, T-duality and Generalized Complex Geometry (arXiv:hep-th/0612034)

On Poisson-Lie T-duality:

• Peggy Kao, T-duality and Poisson-Lie T-duality in generalized geometry, PhD thesis, Australian National University (2008) [pdf]

Discussion of the infinitesimal T-duality geometry, replacing gerbes on torus-fiber bundles with the corresponding dg-manifolds is in

• Ernesto Lupercio, Camilo Rengifo, Bernardo Uribe, T-duality and exceptional generalized geometry through symmetries of dg-manifolds (arXiv:1208.6048)

For references on topological T-duality see there.

A relation to Langlands dual groups:

• Calder Daenzer, Erik Van Erp, T-Duality for Langlands Dual Groups, Advances in Theoretical and Mathematical Physics 18 6 (2014) 1267–1285 [arXiv:1211.0763, doi:10.4310/ATMP.2014.v18.n6.a2]

For RR-fields:

• Kentaro Hori, D-branes, T-duality and Index theory, Adv. Theor. Math. Phys. 3 (1999) 281 (arXiv:hep-th/9902102)

Discussion of T-duality that takes into account the super p-brane charges (i.e. the fermionic components of the RR-fields) on super spacetime, hence also of Green-Schwarz action functionals, includes the following:

• Mirjam Cvetic, H. Lu, Christopher Pope, Kellogg Stelle, T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map, Nucl.Phys.B573:149-176,2000 (arXiv:hep-th/9907202)

• Bogdan Kulik, Radu Roiban, T-duality of the Green-Schwarz superstring, JHEP 0209 (2002) 007 (arXiv:hep-th/0012010)

• Igor Bandos, Bernard Julia, Superfield T-duality rules, JHEP 0308 (2003) 032 (arXiv:hep-th/0303075)

  reviewed in: Superfield T-duality rules in ten dimensions with one isometry [arXiv:hep-th/0312299]

Lift of T-duality from string theory to a SL(2,Z)-U-duality acting on the M2-brane-sigma-model:

• Maria P. Garcia del Moral, I. Martin, Alvaro Restuccia, Nonperturbative $SL(2,\mathbb{Z})$ $(p,q)$-strings manifestly realized on the quantum M2 [arXiv:0802.0573]

• Maria P. Garcia del Moral, J. M. Pena, Alvaro Restuccia, Aspects of the T-duality construction for the Supermembrane theory, J. Phys.: Conf. Ser. 720 (2016) 012025 [arXiv:1504.06907, doi:10.1088/1742-6596/720/1/012025]

T-duality formulated on superspace (see also references at double supergeometry):

• M. Abou-Zeid, Chris Hull, Ulf Lindström, Martin Roček: T-Duality in $(2,1)$ Superspace, J. High Energ. Phys. 2019 138 (2019) [arXiv:1901.00662, doi:10.1007/JHEP06(2019)138]

• Willie Carl Merrell: Application of superspace techniques to effective actions, complex geometry and T-duality in String theory, PhD thesis, University of Maryland (2007) [hdl:1903/6865]

• Igor Bandos: Superstring in doubled superspace, Physics Letters B 751 (2015) 408-412 [doi:10.1016/j.physletb.2015.10.081, arXiv:1507.07779]

• 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]

• Chris D. A. Blair, Ondrej Hulik, Alexander Sevrin, Daniel C. Thompson: Doubled space and extended supersymmetry, J. High Energ. Phys. 2022 119 (2022) [doi:10.1007/JHEP08(2022)119]

• Daniel Butter, Falk Hassler, Christopher N. Pope, Haoyu Zhang: Generalized Dualities and Supergroups, J. High Energ. Phys. 2023 52 (2023) [arXiv:2307.05665, doi:10.1007/JHEP12(2023)052]

• Grigorios Giotopoulos, Hisham Sati, Urs Schreiber: Super $L_\infty$ T-Duality and M-theory [arXiv:2411.10260]

In topological phases of matter

On T-duality in the K-theory classification of topological phases of matter, related to the Fourier transform between crystals and their Brillouin torus:

• Varghese Mathai, Guo Chuan Thiang, T-Duality of Topological Insulators, J. Phys. A: Math. Theor. 48 (2015) 42FT02 [doi:10.1088/1751-8113/48/42/42FT02, arXiv:1503.01206]

• Varghese Mathai, Guo Chuan Thiang: T-Duality Simplifies Bulk-Boundary Correspondence, Commun. Math. Phys. 345 (2016) 675–701 [doi:10.1007/s00220-016-2619-6, arXiv:1505.05250]

• Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: some higher dimensional cases, Annales Henri Poincaré 17 12 (2016) 3399-3424 [doi:10.1007/s00023-016-0505-6, arXiv:1506.04492]

• Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality trivializes bulk-boundary correspondence: the parametrised case, Adv. Theor. Math. Phys. 20 (2016) 1193-1226 [doi:10.4310/ATMP.2016.v20.n5.a8, arXiv:1510.04785]

• Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: the noncommutative case, Lett. Math. Phys. 108 5 (2018) 1163-1201 [doi:10.1007/s11005-017-1028-x, arXiv:1603.00116]

• Kiyonori Gomi, Guo Chuan Thiang, Crystallographic T-duality. J. Geom. Phys 139 (2019) 50-77 [doi:10.1016/j.geomphys.2019.01.002, arXiv:1806.11385]

Review:

• Guo Chuan Thiang, K-theory and T-duality of topological phases, Adelaide (2018) [ pdf]

• Keith C. Hannabuss, T-duality and the bulk-boundary correspondence, Journal of Geometry and Physics

  124 (2018) 421-435 [doi:10.1016/j.geomphys.2017.11.016, arXiv:1704.00278]