Stepwise Projection: Toward Brane Setups for Generic Orbifold Singularities

TL;DR

This paper introduces a stepwise projection method to construct brane setups for complex orbifold singularities, extending techniques to exceptional E6, E7, E8 series, and suggesting new generalizations in string theory.

Contribution

The paper presents an algorithmic stepwise projection approach for deriving brane configurations for complex orbifolds, including exceptional series, using transformation rules and representation theory.

Findings

Applied the method to D-series and E6 orbifolds.

Demonstrated the generality via Frobenius' Induced Representations.

Suggested potential generalizations of orientifold planes.

Abstract

The construction of brane setups for the exceptional series E6,E7,E8 of SU(2) orbifolds remains an ever-haunting conundrum. Motivated by techniques in some works by Muto on non-Abelian SU(3) orbifolds, we here provide an algorithmic outlook, a method which we call stepwise projection, that may shed some light on this puzzle. We exemplify this method, consisting of transformation rules for obtaining complex quivers and brane setups from more elementary ones, to the cases of the D-series and E6 finite subgroups of SU(2). Furthermore, we demonstrate the generality of the stepwise procedure by appealing to Frobenius' theory of Induced Representations. Our algorithm suggests the existence of generalisations of the orientifold plane in string theory.