Linear Pentapods with a Simple Singularity Variety – Part I: Determination and Redundant Designs

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 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.

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The complement of the bifurcation variety of a simple singularity

Inventiones mathematicae ◽

10.1007/bf01405164 ◽

1974 ◽

Vol 23 (2) ◽

pp. 105-116 ◽

Cited By ~ 58

Author(s):

Eduard Looijenga

Keyword(s):

Simple Singularity

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

Forum of Mathematics Pi ◽

10.1017/fmp.2014.2 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 2

Author(s):

JACK A. THORNE

Keyword(s):

Rational Numbers ◽

Hyperelliptic Curves ◽

Simple Singularity ◽

Integral Points ◽

Average Size

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.

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-constraints for the total descendant potential of a simple singularity

Compositio Mathematica ◽

10.1112/s0010437x12000668 ◽

2013 ◽

Vol 149 (5) ◽

pp. 840-888 ◽

Cited By ~ 16

Author(s):

Bojko Bakalov ◽

Todor Milanov

Keyword(s):

Analytic Continuation ◽

Vertex Algebra ◽

Weight Vector ◽

Simple Singularity ◽

Moody Algebra ◽

Basic Representation ◽

Highest Weight ◽

Finite Dimensional ◽

Dynkin Diagrams ◽

Twisted Representation

AbstractSimple, 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.

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