Block Approximations and Semidefinite Programs

Semidefinite programming becomes available when the functional action on every allowed conformal block can be represented by scalar or matrix polynomials that remain positive on a continuous half-line. This conversion is approximate unless the block representation, poles, interpolation error, spin coverage, and asymptotic tail are all controlled.

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 supplies the language for nonuniform tails.

From blocks to positive polynomials

Near the crossing-symmetric point, radial coordinates make block expansions rapidly convergent because $|\rho|<1$. Derivatives can be represented schematically as

$$\partial^m g_{\Delta,\ell} \approx \chi_{\ell}(\Delta)\,P_{m,\ell}(\Delta), \qquad \chi_{\ell}(\Delta)>0$$

on an allowed interval. The positive prefactor contains known exponentials and retained poles; $P$ is a polynomial or rational numerator. Conformal-block poles and recursion relations determine the approximation rather than a blind fit Hogervorst and Rychkov 2013, §§3–4.

With $x=\Delta-\Delta_{\min}(\ell)\geq0$, the functional condition becomes

$$M_j(x)=M_{0j}(x)+\sum_n y_nM_{nj}(x)\succeq0 \quad\text{for all }x\geq0.$$

Polynomial-matrix positivity on a half-line is encoded by sums of squares and positive-semidefinite Gram matrices. This is the problem class solved by SDPB Simmons-Duffin 2015, §2. A basis such as orthogonal polynomials at chosen sample points can improve conditioning, but its normalization must be serialized.

Independent approximation axes

Axis	Typical choice	What it can miss	Required check
Radial order	truncate powers of $\rho$	omitted descendants	compare orders on the full evaluation domain
Pole set	retain a finite recursion set	nearby or accumulated poles	enlarge the set and bound the remainder
Polynomial degree	finite numerator basis	shape between nodes	interval or high-precision comparison to direct blocks
Spin coverage	explicit $\ell\leq\ell_{\max}$ plus a tail	a negative high-spin sector	analytic or interval-controlled large-spin positivity
Arithmetic	fixed bit precision	cancellation and false PSD eigenvalues	precision ladder and conditioning diagnostics
Matrix scaling	congruence and row/column factors	distorted residual interpretation	invert scaling in the verifier

No single “block error” represents all these effects. State the norm and domain of each comparison; a small relative error is meaningless near a zero unless an absolute bound is also given.

Exact and synthetic checks

In one dimension, use

$$k_h(z)=z^h\,{}_2F_1(h,h;2h;z)$$

with $0<z<1$ as a direct reference. A generalized-free-boson four-point function with fixed external dimension has a known even spectrum and nonnegative squared OPE coefficients. Generate the crossing vector from the hypergeometric definition, independently from the rational approximation, and compare derivatives at nonsingular points. The direct sum needs a declared tail bound; agreement of two truncations that share the same omitted tail is not independent.

For a polynomial test, $p(x)=(x-1)^2+x$ is nonnegative on $x\geq0$. Encode it in the same polynomial basis used for the solver, recover its Gram representation, and check the minimum independently. Replacing $+x$ by $-x$ creates a negative interval even though selected sample points can remain positive.

Interchange specification

A portable SDP 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, and a cryptographic hash of every generated table. Text precision must be high enough that parsing does not dominate the declared residual.

Failure tests

Pole test. Move a retained pole across the approximation domain or omit one nearby. Direct block evaluation must expose the discrepancy.

Tail test. Increase $\ell_{\max}$ while leaving the claimed tail unchanged. If the bound moves outside its envelope, the old tail statement was insufficient.

Zero test. Compare only relative errors at a crossing-vector zero. Add an absolute error bound.

Continue to Automated Crossing-System Generation for reproducible system construction and Precision, Convergence, and Numerical Error Budgets for refinement design.

References

• Hogervorst, Matthijs, and Slava Rychkov. “Radial Coordinates for Conformal Blocks.” Physical Review D 87 (2013): 106004. 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