  Topological quantum and classical materials can exhibit robust properties
that are protected against disorder, for example for noninteracting particles
and linear waves. Here, we demonstrate how to construct topologically protected
states that arise from the combination of strong interactions and thermal
fluctuations inherent to soft materials or miniaturized mechanical structures.
Specifically, we consider fluctuating lines under tension (e.g., polymer or
vortex lines), subject to a class of spatially modulated substrate potentials.
At equilibrium, the lines acquire a collective tilt proportional to an integer
topological invariant called the Chern number. This quantized tilt is robust
against substrate disorder, as verified by classical Langevin dynamics
simulations. This robustness arises because excitations in this system of
thermally fluctuating lines are gapped by virtue of inter-line interactions. We
establish the topological underpinning of this pattern via a mapping that we
develop between the interacting-lines system and a hitherto unexplored
generalization of Thouless pumping to imaginary time. Our work points to a new
class of classical topological phenomena in which the topological signature
manifests itself in a structural property observed at finite temperature rather
than a transport measurement.
