Enhanced Gauge Symmetry in Three-Moduli Models of Type-II String and Hypergeometric Series

TL;DR

This paper explores conifold singularities in three-moduli Type-II string models, showing they can be associated with enhanced gauge symmetries and expressed through hypergeometric series, particularly Appell functions, revealing model-dependent behaviors.

Contribution

It demonstrates that periods in these models can be written as hypergeometric series and relates Appell functions to Riemann surfaces, highlighting model-dependent features of gauge symmetry enhancements.

Findings

Conifold singularities correspond to points of enhanced gauge symmetry.

Periods are expressed as hypergeometric series around singular points.

Enhanced gauge symmetry of SU(2)×SU(2) observed at one singular point.

Abstract

The conifold singularities in the type-II string are considered as the points of phase transition. In some cases, these singularities can be understood in the framework of the conventional fields theores as the points of enhanced gauge symmetry. We consider a class of three moduli Type-II strings. It is shown the periods can be written in the form of hypergeometric series around the singular points in these models. The leading expansion around the conifold locus turns out to be described by Appell functions. In one singular point, we observe the enhanced gauge symmetry of $SU(2)\times SU(2)$ independent of the models. Around another conifold locus, however, the resulting expression of the Appell functions depends on the models. We examine the result by considering a relation between these Appell functions and underlying Riemann surfaces.