  We describe the role conformal nets, a mathematical model for conformal field
theory, could play in a geometric definition of the generalized cohomology
theory TMF of topological modular forms. Inspired by work of Segal and
Stolz-Teichner, we speculate that bundles of boundary conditions for the net of
free fermions will be the basic underlying objects representing TMF-cohomology
classes. String structures, which are the fundamental orientations for
TMF-cohomology, can be encoded by defects between free fermions, and we
construct the bundle of fermionic boundary conditions for the TMF-Euler class
of a string vector bundle. We conjecture that the free fermion net exhibits an
algebraic periodicity corresponding to the 576-fold cohomological periodicity
of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we
establish a lower bound of 24 on this periodicity of the free fermions.
