Speaker
Description
Numerical conformal bootstrap bounds are commonly described as “rigorous,” yet as published they are not theorems: they hold at sampled points rather than over regions of parameter space, for a truncated rational approximation of the conformal blocks rather than the blocks themselves, for a finite list of spins rather than all spins, and only up to floating-point rounding. We address all four gaps.
I show how to efficiently search for functionals that are positive over regions, including regions in external-dimension space; how positivity on the large-spin crossing vectors can be imposed as a region constraint; how new analytic bounds on conformal-block recursion relations allow us to efficiently find functionals guaranteed to be positive on the true, untruncated crossing vectors; and how to mathematically certify the positivity of the resulting functionals using exact arithmetic.
As a flagship application, we certify that any Z₂-symmetric CFT with a Z₂-odd scalar of dimension Δσ ∈ [0.505, 0.5405], at most one relevant Z₂-even scalar ε, a stress tensor, and a gap ΔT′ ≥ 11/2 in the spin-2 sector must satisfy (Δσ, Δε) ∈ (0.5055, 0.540) × (1.23, 1.55)
The exact certification pipeline is based on Bernstein-basis positivity certificates in up to three variables; exact arithmetic over the number field ℚ(√2); explicit bounds on the scalar block truncation errors.
Finally, I emphasize that the innovations introduced in this talk are useful not only for formalizing bounds. They can prevent genuine practical failure modes routinely encountered in the numerical conformal bootstrap and can, perhaps surprisingly, also speed up numerical computations.