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

TL;DR

This paper computes chiral operators in 2D (0,2) theories, tests triality conjectures, and explores how chiral rings behave under RG flow, revealing that mismatched states become massive and are not topologically protected.

Contribution

It provides a detailed computation of chiral operators in (0,2) models and tests the Gadde-Gukov-Putrov triality, highlighting the non-topological nature of (0,2) chiral rings.

Findings

Different UV theories can have mismatched chiral operators.

Mismatched operators do not affect elliptic genera and become massive.

Chiral states in phases of GLSMs transform as 27s and 27^*s under E_6 symmetry.

Abstract

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 test 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 E_6 symmetries in one such model, verifying that (matching) chiral states in phases of corresponding GLSMs transform as 27s, 27^*s.