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In QFT and String theory
The T-duality 2-group is a smooth 2-group (or rather a class of such) which controls T-duality and T-folds. It is the string 2-group for the cup product universal characteristic class on fiber products of torus-fiber bundles with their dual torus-principal bundles.
For $T$ a torus and $\tilde T$ its dual torus, there is the cup product universal characteristic class
This has a smooth refinement to morphism of smooth groupoids/moduli stacks
In fact it has furthermore a differential refinement to a universal Chern-Simons circle 3-bundle with connection
of which the above is obtained by forgetting the connections (FSS 12, section 3.2.1)
As such this is the local Lagrangian of abelian Chern-Simons theory with two abelian gauge field species (the diagonal is 3d abelian CS theory itself).
This universal class is suitably equivariant under the action of the integral T-duality group $O(n,n,\mathbb{Z})$, so that one may consider (Nikolaus 14)
As for the string 2-group, this defines an infinity-group extension (the looping of the homotopy fiber of this map) and this one may call the T-duality 2-group as it controls T-duality pairs by the discussion at T-Duality and Differential K-Theory. Indeed, according to (Nikolaus 14) the principal 2-bundles for this 2-group are the correct formalization of the concept of T-folds.
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Extended higher cup-product Chern-Simons theories, Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 (arXiv:1207.5449)
Thomas Nikolaus, T-Duality in K-theory and Elliptic Cohomology, talk at String Geometry Network Meeting, Feb 2014, ESI Vienna (website)
Thomas Nikolaus, Konrad Waldorf, Higher geometry for non-geometric T-duals, Commun. Math. Phys. 374 (2020) 317-366 [arXiv:1804.00677, doi:10.1007/s00220-019-03496-3]
See also:
Last revised on December 9, 2022 at 09:21:19. See the history of this page for a list of all contributions to it.