---
abstract: '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 second author, E. Clader and R. Tessler. In the second 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.'
address:
- 'A. Basalaev:Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str., 6, 119048 Moscow, Russian Federation, and Skolkovo Institute of Science and Technology, Nobelya str., 3, 121205 Moscow, Russian Federation'
- 'A. Buryak:School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom'
author:
- Alexey Basalaev
- Alexandr Buryak
title: Open Saito theory for $A$ and $D$ singularities
---

Introduction
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Frobenius manifolds, introduced by B. Dubrovin in the early 90s, gave a geometric approach to study solutions 