The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices

Abstract

We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large $AdS$ radius limit of gapped theories in $AdS_2$. In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large $AdS_2$. In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.

Figures (5)

• Figure 1: The physical region where the 1D crossing equation holds is $z\in(0,1)$. Allowing complex $z$ shows the full domain of validity of the equation is the entire blue region $\mathcal{R}\equiv\mathbb{C}\backslash((-\infty,0]\cup[1,\infty))$. The equation stops holding after an analytic continuation from this region through one of the branch cuts.
• Figure 2: The representation of a general functional in terms of a pair of weight functions $f(z)$ and $g(z)$. The branch cuts of $f(z)$ are shown in green. The branch cuts of $g(z)$ coincide with those of the bootstrap vectors $F_{\Delta}^{\Delta_\phi}(z)$ and are shown in red. The integration contours approach the boundary of $\mathcal{R}$ at $z=\infty$ and $z=1$, and consequently $f(z)$ and $g(z)$ respectively must have appropriate fall-off there.
• Figure 3: The contour deformation we can use to go from the representation of a general functional \ref{['eq:fh']} in terms of $h(z)$ to the more convenient representation \ref{['eq:ffg']} in terms of $f(z)$ and $g(z)$. The branch cuts of $F_{\Delta}^{\Delta_\phi}(z)$ are shown in red and the branch cuts of $h(z)$ are shown in green. Whenever we have two integration contours running in opposite directions above and below a branch cut, we are really integrating the discontinuity across the branch cut.
• Figure 4: Gap maximization functional at $\Delta_\phi=1/\pi$. The three dashed curves are numerical results obtained using JuliBootS and the flow method Paulos:2014vyaEl-Showk2016 with $N=104,184$ and 264 derivatives. As the number of components is increased, the functional action converges to the red curve, with a simple zero at $\Delta=2\pi^{-1}+1$ and double zeros for $\Delta=2\pi^{-1}+2n+1$ with $n\geq 1$. The red curve in turn was obtained by acting with the analytic normal functional \ref{['eq:allaf']} on the $F_\Delta^{\Delta_\phi}$ vectors as in \ref{['eq:ffg']}.
• Figure 5: OPE maximization functional at $\Delta_\phi=1/\pi$. The three dashed curves which are almost entirely overlapping are the numerical results obtained using JuliBootSPaulos:2014vyaEl-Showk2016 with $N=104,184$ and 264 derivatives. As the number of components is increased, the functional action rapidly converges to the red curve, with double zeros for $\Delta=2\pi^{-1}+2n+1$ with $n\geq 1$. This curve in turn was obtained by acting with the combination of logarithmic and normal functionals that decays as $\sim z^{-4}$ as $z\rightarrow\infty$.