nLab T-duality 2-group

Idea

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.

Definition

For $T$ a torus and $\tilde T$ its dual torus, there is the cup product universal characteristic class

$$\langle -\cup -\rangle \;\colon\;B T \times B \tilde T \longrightarrow K(\mathbb{Z}, 4) \,.$$

This has a smooth refinement to morphism of smooth groupoids/moduli stacks

$$\langle -\cup -\rangle \;\colon\;\mathbf{B} T \times \mathbf{B} \tilde T \longrightarrow \mathbf{B}^3 U(1) \,.$$

In fact it has furthermore a differential refinement to a universal Chern-Simons circle 3-bundle with connection

$$\langle -\cup -\rangle_{conn} \;\colon\;(\mathbf{B} T \times \mathbf{B} \tilde T)_{conn} \longrightarrow \mathbf{B}^3 U(1)_{conn}$$

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)

$$\langle -\cup -\rangle \;\colon\;\mathbf{B} T \times \mathbf{B} \tilde T // O(n,n,\mathbb{Z}) \longrightarrow \mathbf{B}^3 U(1) \,.$$

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.

Related concepts

• T-duality, topological T-duality

• string 2-group

References

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §3.2.1 in: Extended higher cup-product Chern-Simons theories, Journal of Geometry and Physics 74 (2013) 130-163 [arXiv:1207.5449, doi:10.1016/j.geomphys.2013.07.011]

• Thomas Nikolaus, T-Duality in K-theory and Elliptic Cohomology, talk at String Geometry Network Meeting ESI Vienna (Feb 2014) [website]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rem. 7.2 in: 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]

• 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:

• Nora Ganter, Categorical Tori, SIGMA 14 (2018), 014, 18 (arXiv:1406.7046)