Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -110,6 +110,23 @@ The residue selects $n=j$. Since $\omega_{(j)}=L_{j-1}$, this is the claimed formula; lower truncation makes the sum finite even for $m<-1$. The three special cases use $Y(L_{-1}a,z)=\partial_zY(a,z)$. ::: +::: remark [Global conformal transformations] {@rem:global-conformal}+The modes $L_{-1},L_0,L_1$ span an $\mathfrak{sl}_2$ subalgebra with no central term. Their actions on fields describe infinitesimal translations, dilations, and special conformal transformations, respectively. The corresponding coordinate flows are++$$+z\longmapsto z+t,\qquad z\longmapsto e^t z,+\qquad z\longmapsto\frac{z}{1-tz}.+$$++Together they generate the Möbius transformations, the global holomorphic conformal transformations of the Riemann sphere. For $a\in V_h$ with $L_1a=0$ (a quasiprimary state), the three commutators act on $Y(a,z)$ as++$$+\partial_z,\qquad z\partial_z+h,\qquad z^2\partial_z+2hz.+$$++The terms involving $h$ account for the conformal weight of the field. For a general homogeneous state, the additional term $Y(L_1a,z)$ describes mixing with another field. See [Keller, §7.3](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf#page=25).+:::+ ## Weights and dilations  ::: proposition [Weight compatibility] {@prop:voa-weights}