  In this work we attempt to bridge the gap between heterotic orbifold models
and Calabi-Yau compactifications using gauged linear sigma models (GLSMs) with
(2,0) worldsheet supersymmetry. We associate a specific GLSM to a heterotic
orbifold model with twisted states that have non-vanishing vacuum expectation
values (VEVs): The charges of the GLSM superfields are essentially determined
by the shifted momenta of these states. When a twisted state contains an
oscillator excitation, a fermionic gauging is introduced on the worldsheet
inducing a non-Abelian gauge bundle, e.g. the standard embedding. However,
irrespectively of whether the twisted states contain oscillators or not, they
can be interpreted as blow-up modes, as their VEVs are correlated with sizes of
exceptional cycles in the resolved geometry. By considering marginal
deformations of a GLSM in the large volume limit we are able to directly
determine its effective four dimensional spectrum. In the cases considered the
spectra coincide with those computed via index theorems.
