SL$(2,\mathbb{Z})$ action on quantum field theories with U(1) subsystem symmetry

We consider SL$(2,\mathbb{Z})$ action on quantum field theories with U(1) subsystem symmetry in five dimensions. This is an analog of the SL$(2,\mathbb{Z})$ action considered in arXiv:hep-th/0307041. We show that the exotic level 1 BF theory and the exotic level 1 Chern-Simons theories are trivial and almost trivial, respectively. By using this fact, we define S operation and T operation. These operations make SL$(2,\mathbb{Z})$ group up to a possible invertible phase that is unity within the space-times treated in this paper. We also demonstrate SL$(2,\mathbb{Z})$ action on the $φ$ theory as an example.

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