Let $\Lambda$ be a Lagrangian submanifold of $T^{*}X$ for some closed manifold X. Let $S(x,\xi )$ be a generating function for $\Lambda$ which is quadratic at infinity, and let W(x) be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_0\subset X$ of measure zero such that W is Lipschitz continuous on X, smooth on $X\setminus X_0$ and $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let H(x,p)=0 for $(x,p)\in \Lambda$. Then W is a classical solution to $H(x,\partial W/\partial x(x\left)\right)=0$ on $X\setminus X_0$ and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational...

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.