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The reduction is implemented by requiring a globally defined symmetry and restricting to the sector in which the gauge connection is invariant along the symmetry flow. The resulting theory reduces to a two-dimensional BF-like model together with an induced one-dimensional boundary action. We show that the reduced theory admits two inequivalent boundary sectors, originated by two different boundary conditions for the parent 3d theory at the l"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"On the boundary subspace A_τ=Φ, the universal one-dimensional action reproduces the standard Drinfel'd--Sokolov reduction in JT gravity which captures the Schwarzian boundary dynamics. 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