Boundaries, Defects, and Interfaces

A conformal boundary, defect, or interface preserves enough spacetime symmetry to organize local observables, but it changes what counts as conformal data. Bulk operators can acquire one-point functions, local defect operators carry transverse as well as parallel quantum numbers, and a bulk two-point function has inequivalent bulk and defect OPE decompositions. This chapter develops those structures with codimension, normalization, channel, and theorem hypotheses visible at every step.

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.

Enter this chapter

Write a flat Euclidean defect as

$$\mathcal D=\{(x^a,y^i):y^i=0\}, \qquad a=1,\ldots,p,\quad i=1,\ldots,q,\quad p+q=d.$$

The connected subgroup that preserves the plane is

$$SO(p+1,1)\times SO(q).$$

The first factor acts conformally along $\mathcal D$; the second rotates its normal bundle. A boundary is the special case $q=1$, for which there is no continuous transverse-rotation group. A two-sided codimension-one interface can have an additional reflection exchanging its sides, but that discrete operation is extra data rather than part of the connected conformal group. These distinctions control the operator labels, tensor structures, and positivity statements below Billò et al. 2016, §§1–3.

Question	Route	Information that must be declared
Which transformations and local data survive?	Conformal boundaries and defects	$d$, $p$, $q$, orientation, preserved group, and intrinsic versus ambient operators
How are spinning defect operators classified?	Defect representations and transverse spin	Parallel representation, $SO(q)$ representation, parity, orientation, and tensor basis
What replaces an ordinary OPE near the support?	Bulk-to-defect expansion	Distance convention, operator normalization, identity term, and convergence region
Which variables and blocks describe correlators?	Boundary and defect correlators and blocks	Cross-ratios, channel, Casimir eigenvalue, OPE asymptotic, and shadow exclusion
What does equality of channels imply?	Boundary and defect bootstrap	Complete spectra, one-point data, positivity domain, gaps, and independent benchmark
How does the defect respond to a change of shape?	The displacement operator	Delta-function convention, normal orientation, stress-tensor improvement, and $C_D$
How are interfaces folded or composed?	Interfaces, folding, and fusion	Product theory, orientation reversal, regulator, anomaly matching, and fusion limit
Which flow quantities are monotone?	Boundary entropy and defect monotonicity	Universal subtraction, endpoint assumptions, unitarity, locality, and theorem domain

One normalized benchmark throughout

The common exact example is a real massless scalar on the half-space $y\equiv x_\perp\geq0$, with $d>2$ and canonical Euclidean action

$$S=\frac12\int_{y\geq0}d^d x\,(\partial\phi)^2.$$

Let

$$S_d=\frac{2\pi^{d/2}}{\Gamma(d/2)}, \qquad \kappa_d=\frac{1}{(d-2)S_d}.$$

The full-space Green function is the distribution

$$G_0(x,x')=\frac{\kappa_d}{\lvert x-x'\rvert^{d-2}}, \qquad -\partial_x^2G_0(x,x')=\delta^{(d)}(x-x').$$

Reflect $x'=(\mathbf x',y')$ to $\bar x'=(\mathbf x',-y')$. The two conformal image solutions are

$$G_{\mathrm N,\mathrm D}(x,x') =\kappa_d\left[ \frac{1}{\lvert x-x'\rvert^{d-2}} \mathbin{\pm} \frac{1}{\lvert x-\bar x'\rvert^{d-2}} \right],$$

where $+$ is Neumann, $\partial_y\phi|_{y=0}=0$, and $-$ is Dirichlet, $\phi|_{y=0}=0$. The normal derivative or boundary value verifies the condition term by term. The formula is distributional away from the coincident source; at coincidence it has the same bulk delta function as $G_0$. On a compact region, a massless Neumann problem also has a constant zero mode, so its inverse exists only after the zero mode is fixed or projected out. The noncompact half-space correlator is understood with the usual decay and distributional prescription.

Three normalizations extracted from this one formula will recur:

$$\begin{aligned} \langle\phi(\mathbf x,0)\phi(\mathbf0,0)\rangle_{\mathrm N} &=\frac{2\kappa_d}{\lvert\mathbf x\rvert^{d-2}},\\ \langle\partial_y\phi(\mathbf x,0)\partial_y\phi(\mathbf0,0)\rangle_{\mathrm D} &=\frac{2/S_d}{\lvert\mathbf x\rvert^d},\\ \langle\phi^2(\mathbf x,y)\rangle_{\mathrm{ren}} &=\mathbin{\pm}\frac{\kappa_d}{(2y)^{d-2}}. \end{aligned}$$

The last line subtracts the full-space coincident singularity and retains the image contribution. Thus the image sign changes boundary CFT data; it is not a harmless convention. The same solutions appear as one-block crossing solutions in Liendo, Rastelli, and van Rees 2013, §§2–3, pp. 6–12.

Data, equations, and conclusions

A defect CFT problem separates into three layers:

1. Kinematics: the preserved group, representation labels, invariant tensor structures, cross-ratios, and conformal blocks.
2. CFT data: bulk dimensions and OPE coefficients, bulk one-point coefficients, defect dimensions and OPE coefficients, interface data, and the displacement normalization.
3. Evidence: exact Ward identities and solvable models, conditional positivity arguments, numerical certificates, or dimension-specific monotonicity theorems.

Crossing equates two complete decompositions, but it does not by itself establish that a proposed spectrum comes from a local CFT. Positivity applies only after choosing a reflection-positive configuration and a compatible Hermitian operator basis. A finite truncation can test an ansatz, not replace channel completeness.

The two-dimensional specialization is bounded

In two-dimensional rational CFT, a conformal boundary admits boundary states, Ishibashi states, and an annulus amplitude that can be read in open and closed channels. Cardy consistency requires the resulting open-channel multiplicities to be nonnegative integers. This is a powerful specialization, reviewed in Di Francesco, Mathieu, and Sénéchal 1997, §11.3.2, pp. 422–426, but it is not a general formalism for codimension-$q$ defects. The interface page states the rationality, modular, gluing, and completeness hypotheses before using it.

Boundary and defect Weyl anomalies are developed in the next chapter. Executable optimization and Lorentzian inversion require the specialized later chapters; this chapter supplies their normalized input rather than anticipating their conclusions.

Review the chapter

After completing the leaves, you should be able to:

• derive $SO(p+1,1)\times SO(q)$ for a flat codimension-$q$ defect;
• enumerate parallel and transverse representation labels without importing codimension-one identities;
• normalize a bulk-to-defect OPE and state where it converges;
• derive the scalar boundary crossing equation in bulk and boundary channels;
• identify exactly which channel coefficients are nonnegative in a unitary problem;
• obtain the displacement dimension and normalization from the localized Ward identity;
• fold an interface with the correct orientation and explain why generic fusion needs a regulator;
• match rational two-dimensional annulus channels under Cardy’s hypotheses; and
• apply an entropy or monotonicity theorem only inside its proven dimensional and dynamical domain.

The free Dirichlet and Neumann scalar should reproduce every convention used in those answers.

References

• Billò, Marco, Vasco Gonçalves, Edoardo Lauria, and Marco Meineri. “Defects in Conformal Field Theory.” Journal of High Energy Physics 04 (2016): 091. DOI. Open PDF
• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Liendo, Pedro, Leonardo Rastelli, and Balt C. van Rees. “The Bootstrap Program for Boundary CFT$_d$.” Journal of High Energy Physics 07 (2013): 113. DOI. Open PDF