  In this note we show that abstract planar algebras are algebras over the
topological operad of moduli spaces of stable maps with Lagrangian boundary
conditions, which in the case of the projective line are described in terms of
real rational functions. These moduli spaces appear naturally in the
formulation of open string theory on the projective line. We also show two
geometric ways to obtain planar algebras from real algebraic geometry, one
based on string topology and one on Gromov-Witten theory. In particular,
through the well known relation between planar algebras and subfactors, these
results establish a connection between open string theory, real algebraic
geometry, and subfactors of von Neumann algebras.
