Boundary and Defect Bootstrap

Boundary and defect bootstrap equations express one correlator in complete ambient and defect OPE bases. Their power comes from combining crossing with representation theory, unitarity, Ward identities, gaps, or independently known bulk data. Their main subtlety is sign: the defect-channel coefficients of identical Hermitian operators can be nonnegative in a reflection-positive setup, whereas bulk-channel coefficients contain one-point functions and generally have no fixed sign.

Required background. Boundary and defect correlators and blocks fix the variables and block normalizations. Crossing equations and positivity provide the reflection-positive functional framework.

Scalar boundary crossing

For an identical unit-normalized bulk scalar of dimension $\Delta$, write

$$\langle\mathcal O(x_1)\mathcal O(x_2)\rangle =\frac{\xi^{-\Delta}G(\xi)} {(2y_1)^\Delta(2y_2)^\Delta}, \qquad \xi=\frac{\lvert x_1-x_2\rvert^2}{4y_1y_2}.$$

With the block conventions on the previous page, crossing is

$$\boxed{ 1+\sum_{\mathcal X\neq\mathbf1} \lambda_{\mathcal O\mathcal O\mathcal X} a_{\mathcal X} f_{\mathrm{bulk}}(\Delta_{\mathcal X};\xi) = \xi^\Delta\left[ a_{\mathcal O}^2 +\sum_{\widehat{\mathcal X}\neq\widehat{\mathbf1}} \mu_{\mathcal O\widehat{\mathcal X}}^2 f_{\mathrm{bdy}}(\widehat\Delta_{\widehat{\mathcal X}};\xi) \right]. }$$

The left identity is fixed by the bulk two-point normalization. The boundary identity has coefficient $a_{\mathcal O}^2$. For degenerate boundary primaries, $\mu^2$ means the positive quadratic form obtained after orthonormalizing their two-point matrix. This equation is derived, including its hypergeometric blocks, in Liendo, Rastelli, and van Rees 2013, §2.2, pp. 7–9.

For nonidentical external operators, the right-hand coefficient becomes

$$\mu_{1\widehat{\mathcal X}}\mu_{2\widehat{\mathcal X}},$$

which has no fixed sign. For spinning operators or $q>1$, crossing is vector- or matrix-valued in tensor structures and transverse-spin sectors. Positivity then applies to the complete reflected matrix, not to arbitrarily chosen component functions.

Exact free-scalar solution

Use the unit-normalized bulk field

$$\mathcal O=\frac{\phi}{\sqrt{\kappa_d}}, \qquad \Delta_\phi=\frac{d-2}{2}.$$

The image correlator gives

$$G_\sigma(\xi) =1+\sigma\left(\frac{\xi}{1+\xi}\right)^{\Delta_\phi}, \qquad \sigma= \begin{cases} +1,&\mathrm N,\\ -1,&\mathrm D. \end{cases}$$

The bulk channel consists of the identity plus the scalar $\phi^2$ family at dimension $d-2$, with coefficient product

$$\lambda_{\phi\phi\phi^2}a_{\phi^2}=\sigma$$

in the block normalization of the crossing equation. Its sign flips because $a_{\phi^2}$ flips. This explicitly disproves any attempt to impose positivity term by term in the bulk channel.

The boundary channel contains one primary family:

$$\begin{array}{c|c|c} \text{boundary condition}&\widehat\Delta&\mu^2\\ \hline \mathrm{Neumann}&(d-2)/2&2\\ \mathrm{Dirichlet}&d/2&(d-2)/2 \end{array}$$

with unit-normalized boundary primaries as defined on the bulk-to-defect page. Substitution into the exact hypergeometric blocks reproduces $G_\sigma(\xi)$ for every $\xi>0$ Liendo, Rastelli, and van Rees 2013, §3.1, pp. 10–12. The check is stronger than matching a finite Taylor series: it matches the closed functions and both OPE limits.

Positivity: what is and is not nonnegative

Assume a unitary Euclidean theory, a planar defect invariant under the reflection used for radial quantization, identical Hermitian external operators placed in a reflected configuration, and positive-normalized defect states. Then the defect-channel spectral weights are nonnegative:

$$\mu_{\mathcal O\widehat{\mathcal X}}^2\geq0.$$

For general codimension, the statement is sectorwise after decomposing into real $SO(q)$ representations and a positive two-point basis. Complex $SO(2)$ charge $s$ is paired with its conjugate $-s$. If external operators mix, the coefficients form positive-semidefinite matrices only for quadratic combinations compatible with reflection.

The following quantities need not be nonnegative:

• $\lambda_{\mathcal O\mathcal O\mathcal X}a_{\mathcal X}$ in the bulk channel;
• $\mu_{1\widehat{\mathcal X}}\mu_{2\widehat{\mathcal X}}$ for distinct external operators;
• coefficients after a nonorthogonal change of tensor basis;
• Lorentzian discontinuities before their ordering and kernel are specified; or
• data of a nonunitary defect or a boundary condition that violates reflection positivity.

The numerical-bootstrap review summarizes why defect problems often have weaker bulk-channel positivity than ordinary four-point bootstrap systems in Poland, Simmons-Duffin, and Vichi 2019, §V.B.6, pp. 38–39.

A complete crossing specification

Before applying a functional or a truncation, record:

Field	Required content	Free-scalar value
Geometry	$d$, $p$, $q$, side, orientation	$p=d-1$, $q=1$, $y\geq0$
External data	Representations, dimensions, two-point matrix	$\Delta_\phi=(d-2)/2$, $\mathcal O=\phi/\sqrt{\kappa_d}$
Cross-ratios	Definition and Euclidean domain	$\xi=\lvert x_{12}\rvert^2/(4y_1y_2)>0$
Bulk channel	Spectrum and $\lambda a$ normalization	$\mathbf1$ and $\phi^2$ family, sign $\sigma$
Defect channel	$(\widehat\Delta,\widehat\rho,s)$ and $\mu$ normalization	One scalar family, Neumann or Dirichlet
Identity terms	Both channels, including one-point data	Bulk coefficient $1$; $a_\phi=0$
Positivity	Reflection and Hermiticity assumptions	Defect $\mu^2\geq0$
Gaps or truncations	Exact hypothesis, not inferred absence	Exact one-family solvable case
Independent check	Ward identity, exact model, or separate implementation	Image Green function and boundary limit

For a mixed system, replace scalar coefficients with matrices and include all correlators required for closure. A gap is an input assumption unless independently proved. Failure to find a solution at finite derivative order excludes only the stated finite approximation until a valid certificate and convergence argument are supplied.

Reproducible checks

A reproducible calculation should begin with a free three-dimensional scalar using both image signs, fixed nonsingular bulk configurations and boundary limits, the two channel decompositions, and the declared displacement convention. Numerical certificates and inversion require the additional controls developed in their specialized chapters.

Failure tests

Image-sign swap. Replace $\sigma$ in the bulk channel without changing the leading boundary family. Crossing must fail: Neumann and Dirichlet data cannot be mixed.

Missing identity. Remove either channel identity term. The $\xi\to0$ or $\xi\to\infty$ asymptotic immediately disagrees with the correlator.

False positivity. Demand $\lambda a\geq0$ for every bulk scalar. The exact Dirichlet solution fails even though the theory is unitary.

Incomplete transverse sector. At $q>1$, omit one allowed $SO(q)$ harmonic. Equality at a special angle can survive while the full function of $\eta$ fails.

Finite-grid overclaim. Matching crossing at finitely many $\xi$ values is not an existence proof for a complete positive spectrum. Check the functional identity or supply a controlled approximation with an error bound.

The Ward-normalized datum required for shape deformations is developed on The displacement operator and defect Ward identities. Certified numerical specialization begins at Numerical boundary and defect bootstrap, while defect inversion begins at Defect Lorentzian inversion and lightcone expansions.

References

• 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
• Poland, David, David Simmons-Duffin, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF