Heterotic free fermionic and symmetric toroidal orbifold models

Free fermionic models and symmetric heterotic toroidal orbifolds both constitute exact backgrounds that can be used effectively for phenomenological explorations within string theory. even though it is widely believed that for $$ {mathbb{z}}_2times {mathbb{z}}_2 $$ orbifolds the two descriptions should be equivalent, a detailed dictionary between both formulations is still lacking. this paper aims to fill this gap: we give a detailed account of how the input data of both descriptions can be related to each other. in particular, we show that the generalized gso phases of the free fermionic model correspond to generalized torsion phases used in orbifold model building. we illustrate our translation methods by providing free fermionic realizations for all $$ {mathbb{z}}_2times {mathbb{z}}_2 $$ orbifold geometries in six dimensions.