Flat Gauging of Continuous (Non-invertible) Symmetries and Non-compact BF SymTFT for Compact Boson

Abstract: We study flat gauging of continuous symmetries by summing over flat gauge-field configurations. We focus on the two-dimensional compact boson and construct the torus partition function with general flat $U(1)_M\times U(1)_W$ backgrounds. We show that flat gauging either $U(1)_M$ or $U(1)_W$ decompactifies the theory to the non-compact free boson, and that the dual $\mathbb{Z}$ background combines with the remaining $U(1)$ background into a non-compact $\mathbb{R}$ symmetry background due to the mixed anomaly. We also revisit the self-dual radius, where flat gauging the diagonal $SO(3)\subset (SU(2)_L\times SU(2)_R)/\mathbb{Z}_2$, first pointed out by Gaberdiel and Suchanek, gives the continuous orbifold which lies outside the usual $c=1$ moduli space. On the orbifold branch, we study finite and continuous non-invertible flat gaugings and explain why the continuous case requires a prescription for zero-measure fixed loci on the moduli space. Finally, we formulate the SymTFT of torus sigma models as a non-compact BF theory, whose topological boundary states encode the Narain moduli space and the $O(D,D;\mathbb{Z})$ T-duality action.