Boundaries, Defects, and Interfaces
A conformal defect changes which observables and operator expansions are available. Bulk operators can approach one another or approach the defect, giving two ways to describe the same correlator. This chapter connects those expansions to preserved symmetry, shape response, and consistency conditions, with boundary and higher-codimension cases kept distinct.
Enter this chapter
Section titled “Enter this chapter”Begin with Conformal Boundaries and Defects to identify the support and the symmetries it preserves.
Helpful background. Support, codimension, and operator data supply the geometric language for extended operators. The local OPE and conformal data supply the ambient expansion. Boundaries, interfaces, and domain walls and fusion, junctions, and endpoints provide the general QFT constructions whose conformal specialization is developed here.
Topics
Section titled “Topics”- Conformal Boundaries and Defects — Determines the geometric and symmetry data of a planar defect.
- Defect Representations, Transverse Spin, and Tensor Structures — Classifies parallel and transverse representations and their tensor structures.
- Bulk-to-Defect Expansion and Defect Operators — Expands ambient operators in local operators on the defect.
- Boundary and Defect Correlators and Blocks — Constructs bulk-channel and defect-channel conformal blocks.
- Boundary and Defect Bootstrap — Equates channel expansions with their distinct sign restrictions.
- The Displacement Operator and Defect Ward Identities — Defines the operator that measures response to transverse shape changes.
- Interfaces, Folding, and Fusion — Relates interfaces to product-theory boundaries and examines composition limits.
- Boundary Entropy and Defect Monotonicity — Separates universal defect quantities from dimension-dependent flow theorems.
How the ideas fit together
Section titled “How the ideas fit together”The same bulk two-point function can be expanded by first fusing the two bulk operators, or by bringing each to the defect. Equality constrains bulk OPE coefficients, bulk one-point coefficients, and bulk-to-defect couplings together. The preserved subgroup fixes the allowed representations, while the displacement operator records the failure of transverse momentum to be conserved within the bulk alone. Billò et al. 2016, §§ 2–5.
Positivity is channel dependent. In a suitable reflection-positive identical-operator configuration, defect-channel weights are squares. Bulk-channel coefficients multiply ordinary bulk OPE coefficients by defect-dependent one-point coefficients, and need not have a common sign.
Review the chapter
Section titled “Review the chapter”Why can one not impose nonnegative coefficients in both channels merely because the ambient CFT is unitary?
Solution
The two expansions pair different data. A defect-channel norm can produce a squared bulk-to-defect coupling. A bulk-channel term contains a bulk OPE coefficient times a one-point coefficient, which can change sign with the boundary condition. Positivity must follow from the actual reflected pairing.
Where to go next
Section titled “Where to go next”Continue to Weyl Anomalies, Deformations, and Flow Constraints for defect curvature response. For computations, use Numerical Boundary and Defect Bootstrap.