The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation

TL;DR

This work develops an analytic conformal bootstrap framework based on extremal functionals that act on the crossing equation in a holomorphic two-variable space, focusing on the restricted kinematics $z=ar{z}$ and $SL(2)$ blocks. It constructs two complete functional bases—fermionic and bosonic—dual to the double-trace spectra $ riangle_n^{ ext{F}}=2{ riangle_ ext{φ}}+2n+1$ and $ riangle_n^{ ext{B}}=2{ riangle_ ext{φ}}+2n$, enabling Regge-bounded sum rules that fix OPE data and provide a direct Polyakov-Mellin bootstrap interpretation via Regge-improved Witten diagrams in AdS$_2$. The paper derives universal large-$ riangle$ OPE bounds (both upper and lower) and proves completeness: the functional bootstrap equations are equivalent to crossing for unitary theories. It also applies the framework to compute scalar and fermionic Witten diagrams up to two loops in AdS$_2$, including explicit results at specific $ riangle_ ext{φ}$ and demonstrates how bulk contact terms are fixed by the functionals. Overall, the approach bridges analytic and numerical bootstrap, fixes contact-term ambiguities, and suggests broad generalizations to higher dimensions and modular bootstrap contexts.

Abstract

We clarify the relationships between different approaches to the conformal bootstrap. A central role is played by the so-called extremal functionals. They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap. We explain in detail that the extremal functionals probe the Regge limit. We construct two complete sets of extremal functionals for the crossing equation specialized to $z=\bar{z}$, associated to the generalized free boson and fermion theories. These functionals lead to non-perturbative sum rules on the CFT data which automatically incorporate Regge boundedness of physical correlators. The sum rules imply universal properties of the OPE at large $Δ$ in every unitary solution of SL(2) crossing. In particular, we prove an upper and lower bound on a weighted sum of OPE coefficients present between consecutive generalized free field dimensions. The lower bound implies the $φ\timesφ$ OPE must contain at least one primary in the interval $[2Δ_φ+2n,2Δ_φ+2n+4]$ for all sufficiently large integer $n$. The functionals directly compute the OPE decomposition of crossing-symmetrized Witten exchange diagrams in $AdS_2$. Therefore, they provide a derivation of the Polyakov bootstrap for SL(2), in particular fixing the so-called contact-term ambiguity. We also use the resulting sum rules to bootstrap several Witten diagrams in $AdS_2$ up to two loops.

Figures (6)

• Figure 1: Left: The limit when $z$ and $\bar{z}$ go to infinity in opposite half-planes is controlled by the u-channel OPE. Right: When $z$ and $\bar{z}$ approach infinity in the same half-plane with $z/\bar{z}$ fixed, we get the Regge limit of the u-channel.
• Figure 2: The action of the fermionic $\alpha_n$ and $\beta_n$ functionals for ${\Delta_\phi}=\frac{3}{2}$ and $n=0,1$. The functionals have double zeros at $\Delta=2{\Delta_\phi}+2m+1$ for $m\neq n$ and $m,n\in\mathbb{N}_{\geq 0}$. The action of $\alpha_n$ on the identity ($\Delta=0$) is equal to $-a^{\textrm{free}}_{2{\Delta_\phi}+2n+1}$, while $\beta_n$ vanishes there. $\alpha_n(\Delta)$ and $\beta_n(\Delta)$ also describe the OPE decomposition of the crossing-symmetric sum of Witten exchange diagrams with exchanged dimension $\Delta$.
• Figure 3: Functional $\widetilde{\alpha}_n$ used to prove the upper bound on the OPE coefficients \ref{['eq:upperbound']}, here shown in the fermionic case for ${\Delta_\phi}=\frac{5}{2}$, $n=2$. The contribution to the sum rule from operators in $[\Delta_n-1,\Delta_n+1]$ (shown in red) must be bounded above by minus the contribution from where $\widetilde{\alpha}_n(\Delta)<0$ (shown in green). The negative region stays finite in extent as $n\rightarrow\infty$ and is dominated by the identity in this limit, giving rise to the RHS of \ref{['eq:upperbound']}.
• Figure 4: Functional $\cup_n$ used to prove the lower bound on the OPE coefficients \ref{['eq:lowerbound']}, here shown in the fermionic case for ${\Delta_\phi}=\frac{5}{2}$, $n=2$. The contribution to the sum rule from operators in $[\Delta_n-2,\Delta_n+2]$ (shown in red) must compensate the contribution from where $\cup_n(\Delta)>0$ (shown in green). The latter always contains the identity, which gives rise to the RHS of \ref{['eq:lowerbound']}. There could be an additional negative region (shown in blue), but its extent stays finite as $n\rightarrow\infty$ and its contribution sub-leading compared to that of the identity in this limit.
• Figure 5: The Witten diagrams contributing to the four-point function at increasing orders in perturbation theory. "$+\textrm{perms.}$" means that diagrams obtained by permuting the external legs should be included.