In this paper we compute spaces of chiral operators in general two-dimensional (0,2)
nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in
non-twistable theories, and apply them to check recent duality conjectures. The fact that
in a nonlinear sigma model, the Fock vacuum can act as a section of a line bundle on the
target space plays a crucial role in our (0,2) computations, so we begin with a review of
this property. We also take this opportunity to show how even in (2,2) theories, the Fock
vacuum encodes in this way choices of target space spin structures, and discuss how such
choices enter the A and B model topological field theories. We then compute chiral operators
in general (0,2) nonlinear sigma models, and apply them to the recent Gadde-Gukov-Putrov
triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial
IR fixed points. We find that different UV theories in the same proposed universality class
do not necessarily have the same space of chiral operators – but, the mismatched operators
do not contribute to elliptic genera and are in non-integrable representations of the proposed
IR affine symmetry groups, suggesting that the mismatched states become massive along RG
flow. We find this state matching in examples not only of different geometric phases of the
same GLSMs, but also in phases of different GLSMs, indirectly verifying the triality proposal,
and giving a clean demonstration that (0,2) chiral rings are not topologically protected. We
also check proposals for enhanced IR affine E6 symmetries in one such model, verifying that
(matching) chiral states in phases of corresponding GLSMs transform as 27’s, \bar{27}’s.