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 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.

References

• 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 (arXiv:1804.00677)