# 2D CFT Conformal field theories in 2D are special, both in terms of their mathematical structure and their special roles in areas of physics like string theory. To begin with, the [[0028 Conformal symmetry|conformal group]] in 2D is infinite-dimensional. ## Conformal transformations - $z \rightarrow z^{\prime}=f(z), \quad \bar{z} \rightarrow \bar{z}^{\prime}=\bar{f}(\bar{z})$ - Expand in basis: $z^{\prime}=z+\epsilon(z), \quad \epsilon_{n}(z)=\sum_{n} c_{n} z^{n+1}$ - For a spinless and dimensionless field: write $\delta \phi=\sum_{n}\left(c_{n} \ell_{n}+\bar{c}_{n} \bar{\ell}_{n}\right) \phi(z, \bar{z})$ with $\ell_{n}=-z^{n+1} \partial_{z}, \quad \bar{\ell}_{n}=-\bar{z}^{n+1} \partial_{\bar{z}}$ ## Classical algebra (2 Witt algebras) - $\left[\ell_{m}, \ell_{n}\right]=(m-n) \ell_{m+n}$ - $\left[\bar{\ell}_{m}, \bar{\ell}_{n}\right]=(m-n) \bar{\ell}_{m+n}$ - $\left[\ell_{m}, \bar{\ell}_{n}\right]=0$ - There are quantum corrections to these! -> [[0032 Virasoro algebra|Virasoro algebra]]: it gets a central extension ## Global part - $\left\{\ell_{1}, \ell_{0}, \ell_{-1}\right\} \cup\left\{\bar{\ell}_{1}, \bar{\ell}_{0}, \bar{\ell}_{-1}\right\}$ - $\ell_{-1}$ and $\bar{\ell}_{-1}$ generate translations - $\ell_{0}+\bar{\ell}_{0}$ generate dilatations - $i(\ell_{0}-\bar{\ell}_{0})$ generate rotations - $\ell_{1}$ and $\bar{\ell}_{-1}$ generate special CT - Group is $SL(2,\mathbb{C}) \simeq SO(3,1)$ - Define $h$ and $\bar{h}$ = eigenvalues of $\ell_{0}$ and $\bar{\ell}_{0}$ - => $\Delta=h+\bar{h}, \quad s=h-\bar{h}$ ## Mode expansion of the current (which is stress tensor in this case) - $T(z)=\sum_{m=-\infty}^{\infty} \frac{L_{m}}{z^{m+2}}$ - inverted: $L_{n}=\frac{1}{2 \pi i} \oint d z\, z^{n+1} T(z)$ - this is like integrating current $j_n(z)=z^{n+1}T(z)$ over a constant time slice (just a constant radius $S^1$ here), and it is an independent current for each $n$ ## Partition function on the torus The partition function on the torus labelled by $\tau=\frac{1}{2 \pi}(\theta+\mathrm{i} \beta)$ is given by$Z(\tau)=\operatorname{Tr} e^{-\beta H + \mathrm{i} \theta J}=\operatorname{Tr} q^{L_0} \bar{q}^{\bar{L}_0},$where $q=e^{2 \pi \mathrm{i} \tau}$. ## Refs - review/notes - [[2024#Kusuki (Notes)]] ## Related - [[0599 Minimal models]] - [[0612 Modular invariance]]