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In this paper we compute spaces of chiral operators in generaltwo-dimensional (0,2) nonlinear sigma models, both in theories twistable to theA/2 or B/2 model, as well as in non-twistable theories, and apply them to checkrecent duality conjectures. The fact that in a nonlinear sigma model, the Fockvacuum can act as a section of a line bundle on the target space plays acrucial role in our (0,2) computations, so we begin with a review of thisproperty. We also take this opportunity to show how even in (2,2) theories, theFock vacuum encodes in this way choices of target space spin structures, anddiscuss how such choices enter the A and B model topological field theories. Wethen compute chiral operators in general (0,2) nonlinear sigma models, andapply them to test the recent Gadde-Gukov-Putrov triality proposal, which saysthat certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixedpoints. We find that different UV theories in the same proposed universalityclass do not necessarily have the same space of chiral operators -- but, themismatched operators do not contribute to elliptic genera and are innon-integrable representations of the proposed IR affine symmetry groups,suggesting that the mismatched states become massive along RG flow. We findthis state matching in examples not only of different geometric phases of thesame GLSMs, but also in phases of different GLSMs, indirectly verifying thetriality proposal, and giving a clean demonstration that (0,2) chiral rings arenot topologically protected. We also check proposals for enhanced IR affine E_6symmetries in one such model, verifying that (matching) chiral states in phasesof corresponding GLSMs transform as 27s, 27^*s.

Chiral operators in two-dimensional (0,2) theories and a test of triality