Boundary and Defect Correlators and Blocks

A defect reduces the conformal group, so even a two-point function of bulk primaries contains nontrivial functions. Those functions can be expanded in ambient bulk conformal blocks or in defect conformal blocks. Cross-ratio normalization, Casimir eigenvalues, OPE asymptotics, and the treatment of shadow solutions must agree before coefficients from the two channels can be compared.

Required background. Bulk-to-defect expansion and defect operators provide the defect-channel OPE. Conformal blocks and Casimir equations provide the eigenfunction construction.

Helpful background. The embedding-space formalism linearizes the preserved subgroup and its tensor structures.

Cross-ratios for two bulk points

Let $x_k=(\mathbf x_k,y_k^i)$, $r_k=\sqrt{y_k^iy_{ki}}$, and $n_k^i=y_k^i/r_k$. Use the boundary-friendly variables

$$\xi=\frac{\lvert x_1-x_2\rvert^2}{4r_1r_2}, \qquad \eta=n_1\cdot n_2.$$

For generic $q>1$, $(\xi,\eta)$ are independent. In Euclidean kinematics, $\xi\geq0$ and $-1\leq\eta\leq1$, subject to the geometry of the parallel separation. At $q=1$ the unit normals on a chosen side are fixed, $\eta=1$, and only $\xi$ remains. Billò et al. use a first cross-ratio four times larger than this $\xi$; translating that factor is necessary when importing their block formulas Billò et al. 2016, §3.3.

For identical unit-normalized bulk scalars,

$$\langle\mathcal O(x_1)\mathcal O(x_2)\rangle_{\mathcal D} =\frac{\mathcal G(\xi,\eta)} {(4r_1r_2)^\Delta},$$

and the bulk OPE limit requires

$$\mathcal G(\xi,\eta)\sim\xi^{-\Delta} \qquad(\xi\to0,\ \text{Euclidean approach}).$$

The defect OPE limit takes $r_1,r_2\to0$ at fixed separated parallel points; in this convention $\xi\to\infty$. For $q>1$, the transverse harmonic in $\eta$ identifies the exchanged $SO(q)$ representation.

The exact image correlator

For the canonical free scalar with a boundary,

$$\Delta_\phi=\frac{d-2}{2}, \qquad \langle\phi(x_1)\phi(x_2)\rangle_\sigma =\frac{\kappa_d}{(4y_1y_2)^{\Delta_\phi}} \left[ \xi^{-\Delta_\phi} \sigma(\xi+1)^{-\Delta_\phi} \right],$$

where $\sigma=+1$ is Neumann and $\sigma=-1$ is Dirichlet. The direct separation obeys

$$\lvert x_1-x_2\rvert^2=4y_1y_2\xi,$$

while the reflected separation obeys

$$\lvert x_1-\bar x_2\rvert^2=4y_1y_2(\xi+1).$$

These identities check the factor of four and the image sign without any block technology. Near $\xi=0$, the image term is regular and belongs to nonidentity bulk data. Near $\xi=\infty$, the sum or difference selects the Neumann boundary value or Dirichlet normal derivative.

Bulk and defect blocks

The bulk channel fuses the two ambient operators. For scalar external fields, an exchanged bulk primary of dimension $\Delta'$ and spin $\ell$ solves the full conformal Casimir equation with eigenvalue

$$C_{\Delta',\ell}^{(d)} =\Delta'(\Delta'-d)+\ell(\ell+d-2).$$

Only bulk operators allowed by the ordinary $\mathcal O\times\mathcal O$ OPE appear. Their contributions are multiplied by the product

$$\lambda_{\mathcal O\mathcal O\mathcal X}\,a_{\mathcal X},$$

because the defect supplies the one-point coefficient $a_{\mathcal X}$. This product need not be nonnegative even in a unitary theory.

The defect channel expands each bulk operator into defect primaries. A primary with dimension $\widehat\Delta$, parallel spin $\widehat\ell$, and transverse symmetric-traceless rank $s$ has preserved-group Casimir eigenvalue

$$\widehat C =\widehat\Delta(\widehat\Delta-p) +\widehat\ell(\widehat\ell+p-2) +s(s+q-2).$$

For two bulk scalars, the exchanged defect primary has $\widehat\ell=0$, while $s$ is carried by an $SO(q)$ harmonic. The factorization of the preserved group makes the scalar defect block a product of a parallel hypergeometric function and a transverse Gegenbauer harmonic Billò et al. 2016, §4.1.

For a boundary, Liendo, Rastelli, and van Rees use

$$\langle\mathcal O(x_1)\mathcal O(x_2)\rangle =\frac{\xi^{-\Delta}G(\xi)} {(2y_1)^\Delta(2y_2)^\Delta}.$$

Their scalar blocks are

$$f_{\mathrm{bulk}}(\Delta';\xi) =\xi^{\Delta'/2}\, {}_2F_1\left( \frac{\Delta'}2,\frac{\Delta'}2; \Delta'+1-\frac d2; -\xi \right),$$ $$f_{\mathrm{bdy}}(\widehat\Delta;\xi) =\xi^{-\widehat\Delta}\, {}_2F_1\left( \widehat\Delta,\widehat\Delta+1-\frac d2; 2\widehat\Delta+2-d; -\frac1\xi \right).$$

The first is normalized by its $\xi\to0$ bulk-OPE behavior; the second by its $\xi\to\infty$ boundary-OPE behavior Liendo, Rastelli, and van Rees 2013, §2.2, pp. 7–9. Multiplying a block by a constant and dividing its coefficient by the same constant changes no correlator, so a block table without its limiting normalization is insufficient.

Casimir solutions, shadows, and radial variables

A second-order Casimir equation has more solutions than the physical block. In the bulk channel, the shadow dimension $d-\Delta'$ has the same quadratic eigenvalue as $\Delta'$. In the defect channel, $p-\widehat\Delta$ has the same parallel eigenvalue as $\widehat\Delta$. Select the physical solution by:

1. the primary power in the appropriate OPE limit;
2. regularity in the Euclidean domain reached from that limit;
3. the descendant spectrum in radial quantization; and
4. any shortening or conservation condition.

The variables $(\xi,\eta)$ are geometrically transparent but not uniformly efficient for series expansion. Bulk and defect radial variables map their respective OPE convergence regions into unit disks and organize the block by descendant level. The radial series converges absolutely within its OPE domain. Positivity-based estimates of the defect-channel tail require identical Hermitian external operators and a reflection-positive configuration; the bulk channel lacks an analogous sign in general because its coefficients contain one-point data Lauria, Meineri, and Trevisani 2018, §§3–4.

Boundary and defect data table

The table below is a semantic comparison of the data used on this page. Every row declares codimension, object, symmetry, normalization, channel, mathematical status, and evidence; numerical values should not be compared across rows before converting conventions.

Codimension	Boundary or defect	Preserved symmetry	Normalization	Channel	Theorem status	Evidence
$q=1$	Free scalar, Neumann	$SO(d,1)$	Canonical action; $\langle\widehat{\mathcal O}_{\mathrm N}\widehat{\mathcal O}_{\mathrm N}\rangle=\lvert x\rvert^{2-d}$	Boundary: $\widehat\Delta=(d-2)/2$	Exact separated-point solution	Image Green function and Liendo et al. 2013, §3.1
$q=1$	Free scalar, Dirichlet	$SO(d,1)$	Canonical action; $\langle\widehat{\mathcal O}_{\mathrm D}\widehat{\mathcal O}_{\mathrm D}\rangle=\lvert x\rvert^{-d}$	Boundary: $\widehat\Delta=d/2$	Exact separated-point solution	Image Green function and Liendo et al. 2013, §3.1
$q>1$	Planar conformal defect	$SO(p+1,1)\times SO(q)$	Unit bulk and defect two-point bases	Defect: $(\widehat\Delta,0,s)$	Kinematic block decomposition	Casimir derivation in Billò et al. 2016, §4
$q=1$	General unitary boundary CFT	$SO(d,1)$	Unit Hermitian boundary operators	Bulk and boundary decompositions	Crossing is exact; boundary-channel positivity is conditional on reflection positivity	Liendo et al. 2013, §2.2
$q=1$ in $d=2$	Rational conformal boundary	Diagonal Virasoro or extended chiral symmetry	Character and boundary-state conventions declared	Annulus open and closed channels	Cardy consistency under rationality and completeness hypotheses	Di Francesco et al. 1997, §11.3.2

Failure tests

Cross-ratio test. Reconstruct direct and reflected squared distances from $\xi$. If the image term contains $\xi+1/4$ rather than $\xi+1$ in the convention above, a factor-of-four translation was missed.

OPE-limit test. Expand every block at the limit that defines it. A shadow solution has the same Casimir eigenvalue but the wrong leading power.

Angular test. At $q>1$, project onto $SO(q)$ harmonics before reading coefficients. Values at a single $\eta$ do not separate transverse-spin sectors.

Positivity test. Do not impose a sign on $\lambda_{\mathcal O\mathcal O\mathcal X}a_{\mathcal X}$. Positivity belongs to an appropriate defect-channel norm, not automatically to the bulk channel.

Equality of the two decompositions is formulated on Boundary and defect bootstrap. The localized operator that controls shape dependence is normalized on The displacement operator and defect Ward identities.

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.
• Lauria, Edoardo, Marco Meineri, and Emilio Trevisani. “Radial Coordinates for Defect CFTs.” Journal of High Energy Physics 11 (2018): 148. DOI. Open PDF
• 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