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A characteristic property of a gapless liquid state is its emergent symmetryand dual symmetry, associated with the conservation laws of symmetry chargesand symmetry defects respectively. These conservation laws, considered on anequal footing, can't be described simply by the representation theory of agroup (or a higher group). They are best described in terms of a topologicalorder (TO) with gappable boundary in one higher dimension; we call this thesymTO of the gapless state. The symTO can thus be considered a fingerprint ofthe gapless state. We propose that a largely complete characterization of agapless state, up to local-low-energy equivalence, can be obtained in terms ofits maximal emergent symTO. In this paper, we review thesymmetry/topological-order (Symm/TO) correspondence and propose a precisedefinition of maximal symTO. We discuss various examples to illustrate theseideas. We find that the 1+1D Ising critical point has a maximal symTO describedby the 2+1D double-Ising topological order. We provide a derivation of thisresult using symmetry twists in an exactly solvable model of the Ising criticalpoint. The critical point in the 3-state Potts model has a maximal symTO ofdouble (6,5)-minimal-model topological order. As an example of a noninvertiblesymmetry in 1+1D, we study the possible gapless states of a Fibonacci anyonchain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chainwithout translation symmetry has a critical point with unbrokendouble-Fibonacci symTO. In fact, such a critical theory has a maximal symTO ofdouble (5,4)-minimal-model topological order. We argue that, in the presence oftranslation symmetry, the above critical point becomes a stable gapless phasewith no symmetric relevant operator.

Emergent generalized symmetry and maximal symmetry-topological-order