  Using Picard--Lefschetz periods for the singularity of type $A_N$, we
construct a projective representation of the Lie algebra of differential
operators on the circle with central charge $h:=N+1$. We prove that the total
descendant potential $\D_{A_N}$ of $A_N$-singularity is a highest weight
vector. It is known that $\D_{A_N}$ can be interpreted as a generating function
of a certain class of intersection numbers on the moduli space of $h$-spin
curves. In this settings our constraints provide a complete set of recursion
relations between the intersection numbers. Our methods are based entirely on
the symplectic loop space formalism of A. Givental and therefore they can be
applied to the mirror models of symplectic manifolds.
