SL (2, ℤ) action on quantum field theories with U(1) subsystem symmetry

We consider SL$(2,Z)$ action on quantum field theories with U(1) subsystem symmetry in five dimensions. This is an analog of the SL$(2,Z)$ action considered by E. Witten [From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, p. 1173 (2003) [arXiv:hep-th/0307041]]. We show that the exotic level-one BF theory and the exotic level-one Chern-Simons theories are trivial and almost trivial, respectively. By using this fact, we define the S operation and T operation. These operations give the SL$(2,Z)$ group up to a possible invertible phase that is unity within the spacetimes treated in this paper. We also demonstrate SL$(2,Z)$ action on the φ theory as an example.