Gauging

1. gauge a finte group/discrete symmetry
   ~discrete Fourier transformation of the partition function $Z(\mathcal{T},J)$: theory $\mathcal{T}$ with global p-form symmetry $G^{(p)}$ and associated BG currents $J_{p+1}$,
   $Z(\mathcal{T}/G^{(p)},{\hat{J}}_{d-p-1})\sim \sum _{J_{p+1}\in H^{p+1}(M_d,G^{(p)})}{e}^{2\pi i(\hat{J},J)}Z(\mathcal{T},J_{p+1})$
   where $e^{2\pi i(\hat{B},B)}$ is coupling the dual theory $\mathcal{T}/G^{(p)}$ to the current of $J$ (or a pairing between $J\in H^{p+1}(M_d,G^{(p)})$ and $\hat{J}=\star J\in H^{d-p-1}(M_d,{\hat{G}}^{(d-p-2)})\simeq H_{p+1}$, Poincare duality) and is the weight in the Fourier transform.
   The dual symmetry ${\hat{G}}^{(d-p-2)}$ is a $d-p-2$-form symmetry, the Pontryagin dual of $G$: $\hat{G}=Hom(G,U(1))$. It provides a explicit expression for the $p$-form symmetry generator: (just charges for the $p$-form one)
   $D_{d-p-1}=exp(i{\int }_{M_{d-p-1}}{\hat{J}}_{d-p-1})$
   which acts on/links with $p$-dim operators. Its own generator is given by $(M_{p+1},J_{p+1})$.
2. gauge a continuous symmetry
   eg. Maxwell in 4d
   $\star J_e^{(2)}=\frac{i}{e^2}\star F^{(2)}=\frac{i}{e^2}D{A}_D,\star J_m^{(2)}=\frac{1}{2\pi }{F}^{(2)}$

ref: [2305.18296] ICTP Lectures on (Non-)Invertible Generalized SymmetriesP24