nLab T-duality 2-group



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

:BT×BT˜K(,4). \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

:BT×BT˜B 3U(1). \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

conn:(BT×BT˜) connB 3U(1) conn \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,)O(n,n,\mathbb{Z}), so that one may consider (Nikolaus 14)

:BT×BT˜//O(n,n,)B 3U(1). \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.


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.