The space of Frobenius manifolds has a natural involutive symmetry on it:
there exists a map I which send a Frobenius manifold to another Frobenius
manifold. Also, from a Frobenius manifold one may construct a so-called almost
dual Frobenius manifold which satisfies almost all of the axioms of a Frobenius
manifold. The action of I on the almost dual manifolds is studied, and the
action of I on objects such as periods, twisted periods and flows is studied.
A distinguished class of Frobenius manifolds sit at the fixed point of this
involutive symmetry, and this is made manifest in certain modular properties of
the various structures. In particular, up to a simple reciprocal
transformation, for this class of modular Frobenius manifolds, the flows are
invariant under the action of $I\,.
