The phase factors in singularity theory

TL;DR

This paper provides explicit formulas for phase factors in singularity theory and proves their analyticity on the monodromy covering space, advancing the understanding of vertex algebra representations related to singularities.

Contribution

It introduces explicit formulas for unperturbed phase factors and establishes their analyticity in deformation parameters, extending previous constructions to arbitrary isolated singularities.

Findings

Explicit formula for unperturbed phase factors in terms of monodromy and polylogarithm functions.

Proof of analyticity of phase factors on the monodromy covering space.

Extension of vertex algebra constructions to arbitrary isolated singularities.

Abstract

The paper \cite{BM} proposed a construction of a twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularity. The main difficulty in extending the above construction to an arbitrary isolated singularity is in the so called {\em phase factors} -- the scalar functions produced by composing two vertex operators. They are certain family of multivalued analytic functions on the space of miniversal deformations. The first result in this paper is an explicit formula for the unperturbed phase factors in terms of the classical monodromy operator and the polylogorithm functions. Our second result is that with respect to the deformation parameters the phase factors are analytic functions on the monodromy covering space.