Lightnovel
Conventions
In CFT-related pages, the following terminologies are adopted without further specification:
| Term | Meaning |
|---|---|
| Virasoro CFT | CFT with Virasoro symmetry on \(\RR^{2}\) |
| (Euclidean) CFT | CFT with \(\solie(d+1,1)\) symmetry on \(\RR^{d}\) |
| Lorentzian CFT | CFT with \(\solie(d,2)\) symmetry on \(\RR^{d-1,1}\) |
| Virasoro primary/descendant | vectors in Verma modules of Virasoro algebra |
| primary/descendant | vectors in parabolic Verma modules of \(\solie(d+2,\CC)\) |
| (Euclidean) principal series | Harish-Chandra modules of \(\solie(d+1,1)\) |
| Lorentzian principal series | Harish-Chandra modules of \(\solie(d,2)\) |
| BPZ/Lorentzian conjugation | $Q_{L}^{\dagger}=-Q_{L}\, \implies\, D^{\dagger}=D, P^{\dagger}=K, \dots $ |
| dS/Euclidean conjugation | $Q_{E}^{\dagger}=-Q_{E}\, \implies\, D^{\dagger}=-D, P^{\dagger}=-P, \dots $ |
The metric signature is most positive. The conserved charges are anti-Hermitian by default. The local operators on \(\RR^{d}\) and \(\RR^{d-1,1}\) are distinguished by \(\op_{E}(x)\) and \(\op_{L}(x)\) if necessary, and \(\op_{i}\) denotes \(\op_{i}(0)\) or \(\op_{i}(x_{i})\) according to the context.
| Celestial CFT | Usual CFT |
|---|---|
| \(\solie(d+1,1)\)/Virasoro | \(\solie(d+1,1)\)/Virasoro |
| gauge symmetry | current algebra |
| supertranslation | Carrollian and its extension |
| BGG modules | BGG modules |
| HC modules | / |
| colinear singularity | OPE singularity |
| other singularities | ? |
| single-valuedness | locality (?) |
| celestial optical theorem | / |
| spacetime crossing | ? |