Strange Duality, Mirror Symmetry, and the Leech Lattice

Summary

Dedicated to Terry Wall.

Abstract

We give a survey of old and new results concerning Arnold's strange duality. We show that most of the features of this duality continue to hold for the extension discovered by C.T.C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.

Introduction

More than 20 years ago, V.I. Arnold [Ar] discovered a strange duality among the 14 exceptional unimodal hypersurface singularities. A beautiful interpretation of this duality was given by H. Pinkham [P1] and independently by I.V. Dolgachev and V.V. Nikulin [DN, D3]. I. Nakamura related this duality to the Hirzebruch-Zagier duality of cusp singularities [Na1, Na2].

In independent work in early 1982, C.T.C. Wall and the author discovered an extension of this duality embracing on the one hand series of bimodal singularities and on the other, complete intersection surface singularities in ℂ⁴ [EW]. We showed that this duality also corresponds to Hirzebruch-Zagier duality of cusp singularities.

Recent work has aroused new interest in Arnold's strange duality. It was observed by several authors (see [D4] and the references there) that Pinkham's interpretation of Arnold's original strange duality can be considered as part of a two-dimensional analogue of the mirror symmetry of families of Calabi-Yau threefolds. Two years ago, K. Saito [S] discovered a new feature of Arnold's strange duality involving the characteristic polynomials of the monodromy operators of the singularities and he found a connection with the characteristic polynomials of automorphisms of the famous Leech lattice.