nLab
Poisson-Lie T-duality

Idea

What has come to be called nonabelian T-duality (de la Ossa-Quevedo 92) or Poisson-Lie T-Duality (due to Klimcik-Ševera 95, von Unge 02) is a generalization of T-duality from fiber bundles equipped with an abelian group of isometries (torus bundles) to those with a nonabelian group of isometries.

Poisson-Lie T-duality may also be made manifest at the level of type II supergravity in the framework of double field theory on group manifolds. Using this framework both the NS/NS sector and the R/R sector are captured, and this allows to derive the transformation of the RR fields for full Poisson-Lie T-duality (Hassler 17).

As a partial duality of string theory

While ordinary abelian T-duality is supposedly a full duality in string theory, in particular in that it is an equivalence on the string perturbation series to all orders of the squared string length/Regge slope $\alpha'$ and the string coupling constant $g_s$, it has apparently been shown by Martin Roček (citation?) that there are topological obstructions at higher genus for non-abelian T-duality, letting it break down in higher orders of $g_s$; and already a genus-0 (tree level) it apparently breaks down for the open string (i.e. on punctured disks) at some order of $\alpha'$.

But see Hassler 20, Borsato-Wulff 20.

Related $n$Lab entries

• topological T-duality

• spherical T-duality

References

The original articles are

• Xenia C. de la Ossa, Fernando Quevedo, Duality Symmetries from Non–Abelian Isometries in String Theories, Nucl.Phys. B403 (1993) 377-394 (hep-th/9210021)

• Ctirad Klimcik, Pavol Ševera, Dual non-Abelian duality and the Drinfeld double, Physics Letters B, Volume 351, Issue 4, 1 June 1995, Pages 455-462 (doi:10.1016/0370-2693(95)00451-P)

• Rikard von Unge, Poisson-Lie T-plurality, Journal of High Energy Physics, Volume 2002, JHEP07 (2002) (arXiv:hep-th/0205245)

Review includes

• I. Petr, From Buscher Duality to Poisson‐Lie T‐Plurality on Supermanifolds, AIP Conference Proceedings 1307, 119 (2010) (doi:10.1063/1.3527407)

• Konstadinos Sfetsos, Recent developments in non-Abelian T-duality in string theory, Fortschr. Phys., Special Issue: Proceedings of the “Schools and Workshops on Elementary Particle Physics and Gravity” (CORFU 2010), 29 August – 12 September 2010, Corfu (Greece) Volume59, Issue11‐12 (arXiv:1105.0537)

Relation to T-folds:

• Mark Bugden, Non-abelian T-folds (arXiv:1901.03782)

• Ladislav Hlavatý, Ivo Petr, T-folds as Poisson-Lie plurals (arXiv:2004.08387)

Discussion of the duality at the level of type II supergravity equations of motion is (using Riemannian geometry of Courant algebroids) due to

• Branislav Jurco, Jan Vysoky, Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle, Journal of Geometry and Physics Volume 130, August 2018, Pages 1-26 (arXiv:1708.04079)

• Pavol Ševera, Fridrich Valach, Courant algebroids, Poisson-Lie T-duality, and type II supergravities (arXiv:1810.07763)

and in relation to double field theory:

• Falk Hassler, Poisson-Lie T-Duality in Double Field Theory (arXiv:1707.08624)

Discussion within a broader picture of dual higher gauge theories, including 4d electric-magnetic duality:

• Ján Pulmann, Pavol Ševera, Fridrich Valach, A non-abelian duality for (higher) gauge theories (arXiv:1909.06151)

Discussion of non-abelian T-duality from a comprehensive picture of higher differential geometry, relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, type II geometry, exceptional geometry, etc.:

• Luigi Alfonsi, Section 4 of Global Double Field Theory is Higher Kaluza-Klein Theory (arXiv:1912.07089)

See also

• Benjo Fraser, Dimitrios Manolopoulos, Konstantinos Sfetsos, Non-Abelian T-duality and Modular Invariance (arXiv:1805.03657)

• Francesco Bascone, Franco Pezzella, Patrizia Vitale, Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation (arXiv:2004.12858)

• Falk Hassler, Thomas B. Rochais $O(D,D)$-covariant two-loop β-functions and Poisson-Lie T-duality (arXiv:2011.15130)

Discussion in cosmology:

• Ladislav Hlavatý, Ivo Petr, Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations (arxiv:1910.08436)

Generalization to U-duality in exceptional generalized geometry:

• Emanuel Malek, Daniel C. Thompson, Energy Physics - Theory Poisson-Lie U-duality in Exceptional Field Theory (arxiv:1911.07833)

• Chris D. A. Blair, Daniel C. Thompson, Sofia Zhidkova, Exploring Exceptional Drinfeld Geometries (arxiv:2006.12452)

In the context of the BMN matrix model:

• Yolanda Lozano, Carlos Nunez, Salomon Zacarias, BMN Vacua, Superstars and Non-Abelian T-duality, JHEP 09 (2017) 008 (arXiv:1703.00417)

Discussion of $\alpha'$-corrections:

• Falk Hassler, Thomas Rochais, $\alpha'$-corrected Poisson-Lie T-duality (arXiv:2007.07897)

• Riccardo Borsato, Linus Wulff, Quantum correction to Poisson-Lie and non-abelian T-duality (arXiv:2007.07902)

See also:

• Alex S. Arvanitakis, Chris D. A. Blair, Daniel C. Thompson, A QP perspective on topology change in Poisson-Lie T-duality (arXiv:2110.08179)