# Notes on CFT by Alday ## 1. Motivation: RG flow and scale invariance - reviews RG flow - Why effective field theory can be succesful: - no matter how complicated the theory is in the UV, there can be only a few relevant terms in the IR (due to dimensional analysis) - Poincare invariance + scale invariance => conformal invariance (assumption but true for all known theories) ## 2. Conformal transformations - defines transformations - obtains a rep in terms of differential operators on the fields - derives the [[0028 Conformal symmetry]] algebra - obtains action on *operators* - defines [[0029 Primary operator]] and descendents. ## 3. Consequences of conformal symmetry - For scale invariant theories, trace of stress tensor is a total derivative; - If conformal invariant: $T_{\mu}^{\mu}=0$. - 2-pt fn. (scalar field) - 0 if different conformal dimension - $\left\langle\phi_{1}\left(x_{1}\right) \phi_{2}\left(x_{2}\right)\right\rangle = \frac{C_{12}}{\left|x_{12}\right|^{2 \Delta}}$ for $\Delta_1=\Delta_2=\Delta$ - $C_{12}$ is not physical: can be rescaled - 3-pt fn. (scalar field) - $\left\langle\phi_{1}\left(x_{1}\right) \phi_{2}\left(x_{2}\right) \phi_{3}\left(x_{3}\right)\right\rangle=\frac{C_{123}}{\left|x_{12}\right|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}\left|x_{23}\right|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}\left|x_{13}\right|^{\Delta_{3}+\Delta_{1}-\Delta_{2}}}$ - $C_{123}$ physical: rescaling done at 2-pt funciton already - Correlation function of fields with Lorentz indices, i.e. not just scalar - $J^\mu$ - $T_{\mu\nu}$ stress tensor - 4-pt. fn. and higher - powers of $|x_{ij}|$ not fixed by conformal symmetry - will develop new tool in the next section ## 4. Radial quantization and the [[0030 Operator product expansion|OPE]] - radial quantization - label states by scaling dimension and spin - [[0025 Operator-state correspondence]] - [[0030 Operator product expansion]] - [[0031 Conformal block]] ## 5. Conformal invariance in 2d - conformal transformations in 2d (in complex coordinates) - its algebra - global part - correlation functions - radial quantization - deriving [[0030 Operator product expansion]] between stress tensor and primary operator - conformal [[0106 Ward identity]] (can do for higher d easily): $\left\langle T(z) \phi_{1}\left(w_{1}, \bar{w}_{1}\right) \cdots \phi_{n}\left(w_{n}, \bar{w}_{n}\right)\right\rangle=\sum_{i=1}^{j}\left(\frac{h_{j}}{\left(z-w_{j}\right)^{2}}+\frac{1}{z-w_{j}} \frac{\partial}{\partial w_{j}}\right)\left\langle\phi_{1}\left(w_{1}, \bar{w}_{1}\right) \cdots \phi_{n}\left(w_{n}, \bar{w}_{n}\right)\right\rangle$ ## 6. The Virasoro algebra - central charge - Virasoro algebra - Hilbert space - Verma module - adjoint: $\phi^\dagger$ = outgoing state - [[0034 Null states]]: need to quotient them out - correlation of descendents - confomal family $[\phi]$ for each primary $\phi$: two memebers of a conformal family are correlated only if their primaries are & if they are descendents of the same level - [[0031 Conformal block]] - harder than higher dimensional ones because Virasoro algebra is much more intricate than the global subgroup ## 7. Minimal models (d=2) - [[0035 Unitarity of CFT]] of 2d CFTs - Minimal models - Critical Ising model (simplest non-trivial unitary CFT) ## 8. Conformal bootstrap in $d > 2$ - crossing symmetry => [[0036 Conformal bootstrap|bootstrap]] equation.