A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, cannot be described simply by the representation theory of a group (or a higher group). They are best described in terms of a topological order (TO) with a gappable boundary in one higher dimension; we call this the symTO of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local low-energy equivalence, can be obtained in terms of its maximal emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a definition of maximal symTO. We discuss various examples to illustrate these ideas. We find that the $1+1\text{D}$ Ising critical point has a maximal symTO described by the $2+1\text{D}$ double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the three-state Potts model has a maximal symTO of double-(6,5)-minimal-model topological order. As an example of a noninvertible symmetry in $1+1\text{D}$, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double-(5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator.

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• Symmetry protected topological states
• Topological order
• Topological phases of matter