GSO PROJECTIONS OF MODULAR-INVARIANT APERIODIC STRINGS

Abstract

Closed strings with aperiodic (twisted) boundary conditions and one-loop modular-invariant partition functions are studied. A systematic enumeration and prescription for obtaining all allowed GSO projections are given, thereby generalizing the known results for spin structures to orbifold and other strings whose boundary conditions are given by a product of q cyclic groups G = Zn1 × ⋯ × Znq. Before taking into account the required connection between spin and statistics, the number of allowed GSO projections is [Formula: see text], where Dij is the largest common divisor of ni and nj. After taking spin-statistics into account, the group structure must factorize into G=G0×G1, where G0 governs the spin structure and is a product of a number of Z2 groups, and G1 = Zm1 × ⋯ × Zmr now controls the orbifold and other compactification structures. The number of different GSO projections per spin structure is now given by [Formula: see text], where s is the number of even mA’s. There explicit formulas are used to derive systematically all the known ten-dimensional type II and rank-16 heterotic strings and some four-dimensional strings.