SymTFT construction of gapless exotic-foliated dual models

We construct Symmetry Topological Field Theories (SymTFTs) for continuous subsystem symmetries, which are inherently non-Lorentz-invariant. Our framework produces dual bulk descriptions – gapped foliated and exotic SymTFTs – that generate gapless boundary theories with spontaneous subsystem symmetry breaking via interval compactification. In analogy with the sandwich construction of SymTFT, we call this Mille-feuille. This is done by specifying gapped and symmetry-breaking boundary conditions. In this way we obtain the foliated dual realizations of various models, including the XY plaquette, XYZ cube, and $\phi$, $\widehat{\phi }$ theories. This also captures self-duality symmetries as condensation defects and provides a systematic method for generating free theories that non-linearly realize subsystem symmetries.