  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, can't 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 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 precise
definition of maximal symTO. We discuss various examples to illustrate these
ideas. We find that the 1+1D Ising critical point has a maximal symTO described
by the 2+1D 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 3-state Potts model has a maximal symTO of
double (6,5)-minimal-model topological order. As an example of a noninvertible
symmetry in 1+1D, 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.
