# Conformal bootstrap The goal of the bootstrap program is to constrain the possible CFT data, $\left\{\Delta_i, J_i, C_{i j k}\right\}$, which, together with the [[0033 Central charge|central charge]], fully determine all correlations functions on arbitrary Riemann surfaces (in 2d). As argued in [[1989#Moore, Seiberg]], in 2d, modular covariance of one-point function on the torus and four-point crossing on the plane are believed to be enough to define a consistent CFT. This has been proven for rational CFTs. ## Refs - reviews - [[2016#Simmons-Duffin (Lectures)]] - [[2024#Ribault (Review)]] - minimal set of consistency conditions - [[1987#Friedan, Shenker]] - [[1988#Moore, Seiberg]] - [[1989#Moore, Seiberg]] - different approach and more rigorous: [[1998#Bakalov, Kirillov]] ## Conformal bootstrap equation - Obtained by requiring crossing symmetry: $\frac{G(u, v)}{x_{12}^{2 \Delta_{\phi}} x_{34}^{2 \Delta_{\phi}}}=\frac{G(v, u)}{x_{23}^{2 \Delta_{\phi}} x_{14}^{2 \Delta_{\phi}}}$ - => $\sum_{\Delta, \ell} c_{\Delta, \ell}^{2}\left(\frac{v^{\Delta_{\phi}} G_{\Delta, \ell}(u, v)-u^{\Delta_{\phi}} G_{\Delta, \ell}(v, u)}{u^{\Delta_{\phi}}-v^{\Delta_{\phi}}}\right)=1$ - usually combined with [[0035 Unitarity of CFT|unitarity]] to give further constraints ## Higher-point bootstrap equations There are analogous "crossing symmetries" for higher point functions, e.g. ![[FortinMaSkiba2020_8.png|300]] but they are not independent! > Although crossing relations for the four-point function are sufficient for the bootstrap if one incorporates all operators, including spinning operators, it may be that if one uses n-point functions, then external scalars are sufficient. -- [[Rosenhaus2018]] ## Related topics - [[0123 BMS bootstrap]]