  The expectation value of a smooth conformal line defect in a CFT is a
conformal invariant functional of its path in space-time. For example, in large
$N$ holographic theories, these fundamental observables are dual to the open
string partition function in AdS. In this paper, we develop a bootstrap method
for studying them and apply it to conformal line defects in Chern-Simons matter
theories. In these cases, the line bootstrap is based on three minimal
assumptions -- conformal invariance of the line defect, large $N$
factorization, and the spectrum of the two lowest-lying operators at the end of
the line. On the basis of these assumptions, we solve the one-dimensional CFT
on the line and systematically compute the defect expectation value in an
expansion around the straight line. We find that the conformal symmetry of a
straight defect is insufficient to fix the answer. Instead, imposing the
conformal symmetry of the defect along an arbitrary curved line leads to a
functional bootstrap constraint. The solution to this constraint is found to be
unique.
