Protected Operators, BPS Defects, and Exact Correspondences
Protected sectors retain exact information because a shortening condition, cohomology, holomorphy, topology, or anomaly prevents a specified deformation. The word “protected” is not a universal guarantee: an operator may have a fixed dimension while its normalization mixes, a correlator may be exact only on a submanifold, or an index contribution may survive only as an alternating recombination class. This chapter identifies the mechanism and the information discarded in every reduction.
Enter this chapter
Section titled “Enter this chapter”Begin with Protection, Q-Cohomology, Recombination, and Operator Mixing to identify the mechanism protecting the particular operator or observable.
Helpful background. Genuine Lines, Screening, and Charge Lattices; Boundaries, Interfaces, and Domain Walls; From the Local OPE to Conformal Data provide the genuine defect charges, interfaces, and conformal operator data used here.
Topics
Section titled “Topics”- Protection, Q-Cohomology, Recombination, and Operator Mixing — Distinguishes shortening, cohomology, and mixing.
- Chiral Rings and Exact Quantum Relations — Extracts quantum ring relations without assuming a full metric.
- Protected Correlators and Operator Algebras — Derives operator algebras on protected subspaces.
- Four-Dimensional Chiral-Algebra Sectors and Lost Information — Identifies the meromorphic sector retained from a four-dimensional theory.
- BPS Wilson, ‘t Hooft, and Dyonic Line Observables — Specifies both charge and supersymmetry data for lines.
- BPS Boundaries, Surface Defects, Interfaces, and Fusion — Treats defects and interfaces as coupled lower-dimensional systems.
- Topological Twists and the Geometric–Langlands Interface — States the sector matched by the geometric-Langlands construction.
- AGT and Exact-Correspondence Dictionaries: Status and Limits — Matches normalized objects in the AGT correspondence.
- Versioned Protected-Data Exports and Cross-Volume Round Trips — Preserves the definitions needed to reuse protected results.
How the ideas fit together
Section titled “How the ideas fit together”Taking supercharge cohomology discards Q-exact operators and can make translations in some directions trivial. The surviving correlators then form a lower-dimensional algebra, as in the meromorphic sector of a four-dimensional N=2 SCFT. That algebra can contain exact operator products while losing other spacetime and dynamical information. Operator normalization, mixing, and recombination must still be tracked before comparing sectors or importing their data into a full crossing equation. Beem et al. 2015, §§ 2–3.
Where to go next
Section titled “Where to go next”Continue to Exact Partition Functions, Indices, and Instanton Counting to compare what complementary protected observables retain and discard.