Block Approximations and Semidefinite Programs

Semidefinite programming becomes useful when the action of a functional on every allowed block can be reduced to scalar or matrix polynomial positivity on a continuous half-line. There are two distinct questions: whether the polynomial positivity encoding is exact for the finite problem, and whether the finite block representation encloses the exact conformal blocks accurately enough for the intended CFT claim. This page makes the first conversion explicit and keeps the second as a separate error problem.

Required background. Linear Functionals and Positivity fixes the dual signs and PSD sectors. Conformal blocks and Casimir equations fix block normalization and boundary conditions. Helpful background. Asymptotic scales, remainders, and uniformity supply the language for nonuniform tails.

Radial blocks and sign-definite prefactors

For a scalar cross-ratio, define

$$\rho(z)= \frac{z}{\bigl(1+\sqrt{1-z}\bigr)^2}.$$

At the crossing-symmetric point,

$$\rho\!\left(\frac12\right)=3-2\sqrt2\approx0.1716.$$

The small value of $|\rho|$ explains why descendant-level radial expansions converge rapidly away from their boundary. The convergence and positivity properties of the radial expansion are developed in Hogervorst and Rychkov 2013, §3.1, pp. 14–16.

For numerical optimization, derivatives of a crossing vector at a fixed point are commonly represented schematically as

$$\partial_z^m\partial_{\bar z}^{,n} \mathbf F_{\Delta,\ell} \approx \chi_\ell(\Delta) \mathbf P_{\ell}^{mn}(\Delta), \qquad \chi_\ell(\Delta)>0$$

on the declared dimension interval. After extracting a sign-definite rational prefactor, $\mathbf P$ is polynomial in $\Delta$. A typical prefactor contains an exponential radial factor and retained exchanged-dimension poles, schematically

$$\chi_\ell(\Delta) \propto \frac{\rho_*^{\Delta}} {\prod_i(\Delta-\Delta_{*,i})}.$$

Its sign is removable only when every retained denominator has a fixed sign on the allowed domain. A pole inside the domain, or a denominator whose sign changes there, must be canceled, treated explicitly, or separated into intervals; simply discarding it corrupts positivity.

The exchanged-dimension singularities and recursion relations are derived by Penedones, Trevisani, and Yamazaki 2016. Their use in rational block approximations is reviewed in Poland, Rychkov, and Vichi 2019, §§III.F.4–III.F.5, pp. 16–18, and the polynomial-matrix program used by SDPB is given in Simmons-Duffin 2015, §3.2 and Appendix A.

The approximation sign matters. A truncated pole or radial expression can be refined to high accuracy, but it does not become an exact block identity merely because the associated polynomial SDP is solved exactly. A rigorous CFT exclusion additionally needs an enclosure or perturbation argument showing that block error cannot reverse the functional signs. Without that step, refinement is strong convergence evidence rather than a theorem about exact blocks. Current block software and the polynomial preprocessing used in practice are surveyed in Rychkov and Su 2024, §2.1, pp. 2–3.

From half-line positivity to PSD Gram matrices

For spin $\ell$, shift the allowed dimension to

$$x=\Delta-\Delta_{\min}(\ell)\geq0.$$

After the positive prefactor is removed, a scalar functional condition becomes $p_\ell(x)\geq0$ for every $x\geq0$. A univariate polynomial nonnegative on the half-line admits a representation

$$p(x)=s_0(x)+x\,s_1(x),$$

where $s_0$ and $s_1$ are sums of squares. With the monomial vector $q_d(x)=(1,x,\ldots,x^d)^{\mathsf T}$, write

$$p(x)=q_d(x)^{\mathsf T}Y_0q_d(x) +x\,q_{d-1}(x)^{\mathsf T}Y_1q_{d-1}(x), \qquad Y_0,Y_1\succeq0.$$

Matching polynomial coefficients gives linear equations in the entries of $Y_0$ and $Y_1$; PSD is the semidefinite constraint. This is the bridge from infinitely many inequalities in $x$ to a finite conic problem.

For a matrix polynomial $M(x)$, replace $q_d$ by the block basis $\mathcal Q_d=q_d\otimes I$:

$$M(x)=\mathcal Q_d(x)^{\mathsf T}Y_0\mathcal Q_d(x) +x\,\mathcal Q_{d-1}(x)^{\mathsf T}Y_1\mathcal Q_{d-1}(x), \qquad Y_0,Y_1\succeq0.$$

Coefficient matching remains linear, while the PSD Gram matrices guarantee $u^{\mathsf T}M(x)u\geq0$ for every vector $u$ and every $x\geq0$. Orthogonal polynomial bases and congruence scalings can improve conditioning, but their definitions and inverse maps must travel with the solver input.

Worked fixture, stage 3: an exact Gram certificate

For the half-line vector $v(x)=(1,x,x^2)$ and functional $a=(1,-2,1)$,

$$p(x)=a^{\mathsf T}v(x)=(x-1)^2.$$

With $q=(1,x)^{\mathsf T}$,

$$p(x)=q^{\mathsf T} \begin{pmatrix} 1&-1\\ -1&1 \end{pmatrix}q.$$

The Gram matrix is the outer product $(1,-1)^{\mathsf T}(1,-1)$ and has eigenvalues $0$ and $2$. It is PSD even though the polynomial touches zero at $x=1$. The zero is important: a verifier that demands strict positivity would incorrectly reject a valid boundary certificate.

The one-dimensional CFT lower-envelope program

The normalized functional from the preceding page can also be converted into an exact polynomial positivity problem. Set

$$x=\Delta-\frac92\geq0, \qquad r_\Delta=\frac{g_\Delta'(1/2)}{g_\Delta(1/2)}.$$

Using the Casimir equation and $r_\Delta\geq8\Delta/3$, its action can be organized exactly as

$$\widehat\alpha(F_\Delta) =2g_\Delta\!\left(\frac12\right) \left\{ K(x)\left(r_\Delta-\frac{8\Delta}{3}\right) +P(x) \right\},$$

where

$$K(x)=\frac{133}{8}+8x+x^2,$$

and

$$P(x)=11+\frac{133}{3}x +\frac{64}{3}x^2+\frac{8}{3}x^3.$$

Both polynomials have transparent half-line Gram representations:

$$K(x)=q^{\mathsf T} \begin{pmatrix}133/8&0\\0&1\end{pmatrix}q +x\,(8),$$ $$P(x)=q^{\mathsf T} \begin{pmatrix}11&0\\0&64/3\end{pmatrix}q +x\,q^{\mathsf T} \begin{pmatrix}133/3&0\\0&8/3\end{pmatrix}q.$$

Every displayed Gram matrix is positive definite. Since $g_\Delta(1/2)>0$ and $r_\Delta-8\Delta/3\geq0$, these exact rational matrices prove $\widehat\alpha(F_\Delta)>0$ for $\Delta\geq9/2$. This is an analytic lower-envelope polynomial-matrix program: it supplies a sufficient cone for this particular functional. It is not an equality replacing a generic conformal block by $\chi P$, and it does not validate a production pole table, spin truncation, or rational-approximation error.

The machine-readable exact certificate stores both the elementary half-line Gram matrix and these CFT lower-envelope matrices. The verification record records zero coefficient residuals and exact rational PSD checks.

Independent approximation axes

Axis	Typical finite choice	What it can miss	Required check
Functional space	derivatives through $\Lambda$	a separator outside the chosen span	compare several $\Lambda$ values
Radial order	truncate powers of $\rho$	omitted descendants	compare orders on the full evaluation domain
Pole set	retain finitely many recursion poles	a nearby pole or accumulated remainder	enlarge the set and bound or enclose the remainder
Polynomial degree	finite numerator basis	shape error between comparison nodes	interval or high-precision direct-block comparison
Spin coverage	explicit spins plus a tail	a negative omitted spin sector	analytic or enclosed large-spin positivity
Arithmetic	fixed bit precision	cancellation and false PSD signs	precision ladder and independently rounded bounds
Matrix scaling	congruence and row or column factors	misleading residuals after scaling	reconstruct the unscaled problem

No single “block error” represents all these effects. State a norm and domain for each comparison. Relative error alone is uninformative near a zero; an absolute enclosure is needed there.

Direct block regression and solver interchange

In one dimension, a direct reference block is

$$g_\Delta(z)=z^\Delta,{}_2F_1(\Delta,\Delta;2\Delta;z), \qquad 0<z<1.$$

A generalized-free-boson correlator with fixed external dimension provides a useful regression fixture because its spectrum and nonnegative squared OPE coefficients are known. It is not the excluded-gap example above: at $\Delta_\phi=1$ its low spectrum lies below $9/2$, so it is perfectly compatible with the bound. A future production fixture should generate blocks independently from the hypergeometric definition, compare derivative tables over a declared domain, and bound the omitted OPE tail. Two tables produced by the same truncated recurrence are repeatability checks, not independent validation.

A portable polynomial-matrix input records the ordered crossing components, sector and spin labels, $\Delta_{\min}$ maps, polynomial basis, sample points, positive prefactors, pole list, matrix block sizes, objective and feasibility signs, scaling matrices, arithmetic precision, and cryptographic hashes of every generated table. Printed precision must be high enough that parsing error stays below the certificate margin.

Continue to Automated Crossing-System Generation for canonical system construction and Precision, Convergence, and Numerical Error Budgets for refinement design before interpreting solver output.

Common pitfalls

Stripping a denominator without checking its sign. A pole or zero inside the allowed interval can reverse positivity. Split the domain or retain the factor explicitly.

Calling an approximate block identity exact. The SOS representation can be exact for the polynomial surrogate while the surrogate still differs from the conformal block.

Testing positivity only at nodes. Polynomial minima can lie between all stored points. Coefficient-level SOS or an independent interval proof must cover the continuum.

Exercises

Expand the Gram representation of $(x-1)^2$ and verify its eigenvalues. Why is the zero eigenvalue acceptable?

Solution

Multiplication gives $q^{\mathsf T}Yq=1-2x+x^2$. The trace and determinant of $Y$ are $2$ and $0$, so its eigenvalues are $2$ and $0$. PSD allows zero eigenvalues; they correspond here to the legitimate boundary zero $p(1)=0$.

Let $\widetilde p(x)=x^2-3x+1$. Show that checking only $x=0$ and $x=3$ misses a negative interval.

Solution

$\widetilde p(0)=\widetilde p(3)=1$, but $\widetilde p(1)=-1$. Its two roots are $(3\pm\sqrt5)/2$, so the polynomial is negative between them. No half-line SOS representation of the displayed form can exist for $\widetilde p$.

Evaluate $\rho(1/2)$ exactly. Explain why agreement between two block tables generated from the same truncated recurrence does not validate the truncation remainder.

Solution

Substitution gives

$$\rho\!\left(\frac12\right) =\frac{1/2}{(1+1/\sqrt2)^2} =3-2\sqrt2.$$

The two tables share the same omitted terms and can therefore agree while carrying the same bias. An independent direct block representation or a proved remainder enclosure is needed.

References

• Hogervorst, Matthijs, and Slava Rychkov. “Radial Coordinates for Conformal Blocks.” Physical Review D 87 (2013): 106004. DOI. Open PDF
• Penedones, João, Emilio Trevisani, and Masahito Yamazaki. “Recursion Relations for Conformal Blocks.” Journal of High Energy Physics 09 (2016): 070. DOI. Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF
• Rychkov, Slava, and Ning Su. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96 (2024): 045004. DOI. Open PDF
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 06 (2015): 174. DOI. Open PDF