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group theory

duality

# Contents

## 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.