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In this article, we study conditions of continuous emergent symmetries ingapless states, either as topological quantum critical points (TQCPs) or astable phase with protecting symmetries and connections to smooth deformationsof the gapped states around. We illustrate that for a wide class of gaplessstates that can be associated with fully-isolated scale invariant fixed points,there shall always be emergent continuous symmetries that are directly relatedto smooth deformations of gapped states with symmetries lower than theprotecting ones $G_p$. For a 3D TQCP in DIII classes with $G_p=Z^T_2$,$U_{EM}=U(1)$ and $N_f=\frac{1}{2}$ fermions but without charge $U(1)$symmetry, we explicitly construct a corresponding boundary representation basedon a $4D$ topological state with lattice symmetry $H=Z^T_2 \ltimes U(1)$ and$N_f={1}$ fermions. Although emergent continuous symmetries appear to be robustat weakly interacting TQCPs, we further show the breakdown of such one-to-onecorrespondence between deformations of gapped states and emergent continuoussymmetries when gapless states become strongly interacting. In a stronglyinteracting limit, gapless states can be represented by a smooth manifold ofconformal-field-theory fixed points rather than a fully isolated one. A smoothmanifold of strong coupling fixed points hinders emergence of a continuousemergent symmetry in the strongly interacting gapless limit, as deformations nolonger leave a gapless state or a TQCP invariant, unlike in the moreconventional weakly interacting case. This typically reduces continuousemergent symmetries to a discrete symmetry originating from dualitytransformations under the protection symmetry $G_p$.

Emergent symmetries and Interactions: An isolated fixed point Vs a manifold of strongly interacting fixed points