We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

We consider the polynomial representation S(V*) of the rational Cherednik algebra Hc(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and m∈ℕ the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d−1+hm, where h is the Coxeter number of W; these polynomials generate an Hc (W) submodule with the parameter c=(d−1)/h+m. We express these singular polynomials through the Saito polynomials which are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.

We introduce the notion of an alternate product of Frobenius manifolds and we give, after Ciocan-Fontanine et al., an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom–Sebastiani product of Laurent polynomials.