# Conformal symmetry A conformal symmetry transformation changes the metric up to an overall factor, i.e., under $x^\mu \to x^\mu+\epsilon^\mu$, $\delta g_{\mu\nu}=\left(\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu\right)=c(x)g_{\mu\nu}.$Dimensions higher than 2 are quite different from 2-dimensional CFT. In this note, let us work in $d$ dimensions with $d$ higher than 2. The $d=2$ case is discussed in a [[0003 2D CFT|separate note]]. ## Refs - [[Rsc0002 Alday Notes on CFT]] (Euclidean signature used) - [[Rsc0006 Tong String theory]] - [[Rsc0051 The big yellow CFT book by Di Francesco et al]] - Polchinski ## Generators - ($d$) Translation: $x^{\prime \mu} = x^\mu+a^\mu$ - on Riemann sphere ($R^d\cup \{\infty\}$), moves the origin - ($\frac{d(d-1)}{2}$) Rigid rotations: $x^{\prime \mu}={M^{\mu}}_{\nu} x^{\nu}$ - (1) Dilatations: $x^{\prime \mu} =\alpha x^\mu$ - ($d$) Special CT: $x^{\prime \mu}=\frac{x^{\mu}-b^{\mu} x^{2}}{1-2 b \cdot x+b^{2} x^{2}}$ - can be decomposed as $I\circ T(a)\circ I$, where $I: x^\mu\rightarrow x^\mu/x^2$ is the inversion, and $T(a)$ is a translation - on Riemann sphere ($R^d\cup \{\infty\}$), moves the infinity - Total number of generators: $\frac{(d+1)(d+2)}{2}$ ## Representation of the conformal generators as differential operators - Definition - $x^{\prime \mu}=x^{\mu}+\omega_{a} \frac{\delta x^{\mu}}{\delta \omega_{a}}$ - $\delta_{\omega} \Phi(x)=\Phi^{\prime}(x)-\Phi(x)=-i \omega_{a} G_{a} \Phi(x)$ - -> $i G_{a} \Phi(x)=\frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} \Phi(x)$ - Reps - Translations: $P_{\mu}=-i \partial_{\mu}$ - Rigid rotations: $L_{\mu \nu}=i\left(x_{\mu} \partial_{\nu}-x_{\nu} \partial_{\mu}\right)$ - Dilatations: $D=-i x^{\mu} \partial_{\mu}$ - Special CT: $K_{\mu}=-i\left(2 x_{\mu} x^{\nu} \partial_{\nu}-x^{2} \partial_{\mu}\right)$ ## Algebra - $\left[D, P_{\mu}\right]=i P_{\mu}$ - $\left[D, K_{\mu}\right]=-i K_{\mu}$ - $\left[K_{\mu}, P_{\nu}\right]=2 i\left(\eta_{\mu \nu} D-L_{\mu \nu}\right)$ - $\left[L_{\mu \nu}, P_{\rho}\right]=-i\left(\eta_{\mu \rho} P_{\nu}-\eta_{\nu \rho} P_{\mu}\right)$ - $\left[L_{\mu \nu}, K_{\rho}\right]=-i\left(\eta_{\mu \rho} K_{\nu}-\eta_{\nu \rho} K_{\mu}\right)$ - $\left[L_{\mu \nu}, L_{\rho \sigma}\right]=-i\left(L_{\mu \rho} \eta_{\nu \sigma}-L_{\mu \sigma} \eta_{\nu \rho}-L_{\nu \rho} \eta_{\mu \sigma}+L_{\nu \sigma} \eta_{\mu \rho}\right)$ - $\left[D, L_{\mu \nu}\right]=0$ - $\left[P_{\mu}, P_{\nu}\right]=0$ - $\left[K_{\mu}, K_{\nu}\right]=0$ - $[D, D]=0$ - They satisfy $SO(d+1,1)$ or $SO(d,2)$ (in Minkowski space) algebra. ## Action on operators - Full group - $\left[P_{\mu}, \phi_{\alpha}(x)\right]=i \partial_{\mu} \phi_{\alpha}(x)$ - $\left[D, \phi_{\alpha}(x)\right]=i\left(\Delta+x^{\mu} \partial_{\mu}\right) \phi_{\alpha}(x)$ - $\left[L_{\mu \nu}, \phi_{\alpha}(x)\right]=-i\left(x_{\mu} \partial_{\nu}-x_{\nu} \partial_{\mu}\right) \phi_{\alpha}(x)+i\left(S_{\mu \nu}\right)_{\alpha \beta} \phi_{\beta}(x)$ - $\left[K_{\mu}, \phi_{\alpha}(x)\right]=2 i x_{\mu} \Delta \phi_{\alpha}(x)+i\left(2 x_{\mu} x^{\nu} \partial_{\nu}-x^{2} \partial_{\mu}\right) \phi_{\alpha}(x)+2 i x^{\rho}\left(S_{\rho \mu}\right)_{\alpha \beta} \phi_{\beta}(x)$ - Stability subgroup (leaves origin invariant) - $\left[D, \Phi_{\alpha}(0)\right] =i \Delta \Phi_{\alpha}(0)$ - $\left[L_{\mu \nu}, \Phi_{\alpha}(0)\right] =i\left(S_{\mu \nu}\right)_{\alpha}^{\beta} \Phi_{\beta}(0)$ - $\left[K_{\mu}, \Phi_{\alpha}(0)\right] =0$ - $\Delta$ = scaling dimension - $\left(S_{\mu \nu}\right)_{\alpha}^{\beta}$ depends on spin - defines [[0029 Primary operator|primary operators]] - n.b. $\alpha$ is an index for any Lorentz representation e.g. spinor index