Open Saito Theory for A and D Singularities

A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types $A$ and $D$ that gives a solution of the open WDVV equations. For the $A$-singularity, the resulting solution describes the intersection numbers on the moduli space of $r$-spin disks, introduced recently in a work of the 2nd author, E. Clader and R. Tessler. In the 2nd part of the paper, we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups.

The 2-Component BKP Grassmannian and Simple Singularities of Type D

It was proved in 2010 that the principal Kac–Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac–Wakimoto hierarchy. We find explicitly the point in the Grassmannian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3

We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field$\mathbb{Q}$of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type$E_{6}$. We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.

-constraints for the total descendant potential of a simple singularity

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.