• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

132 submissions , 111 unreviewed
3,777 questions , 1,329 unanswered
4,721 answers , 19,948 comments
1,470 users with positive rep
466 active unimported users
More ...

which letter to use for a CFT?

+ 2 like - 0 dislike

In math, one says "let $G$ be a group", "let $A$ be an algebra", ...

For groups, the typical letters are $G$, $H$, $K$, ... For algebras, the typical letters are $A$, $B$, ...

I want to say things such as "let xxx be a conformal field theory"
and "let xxx $\subset$ xxx be a conformal inclusion".

Which letters should I use?
What is the usual way people go about this?

Here, I'm mostly thinking about chiral CFTs, but the question is also relevant for full (modular invariant) CFTs.

This post has been migrated from (A51.SE)
asked Oct 14, 2011 in Theoretical Physics by André_1 (215 points) [ no revision ]
I'm a bit confused, I thought conformal inclusions apply to groups (like $SU(2)\subset SO(5)$) and not entire CFT's? Or is this a seperate definition?

This post has been migrated from (A51.SE)
In VOA language, I would call a conformal inclusion a map $V\to W$ of VOAs that sends the Virasoro element of $V$ to the Virasoro element of $W$. But you're right, I've only seen the terminology used for the VOAs that correspond to loop groups.

This post has been migrated from (A51.SE)
Ahhh I see, thanks!

This post has been migrated from (A51.SE)

3 Answers

+ 7 like - 0 dislike

Ben-Zvi & Frenkel denote vertex algebras $V$,$W$,... They're using the labels specifically for the spaces of states, but one could also use them to refer the whole package.

Alternately, one sometimes sees all caps abbreviations: $YM_2$, $SYM_{4,G}$,...

There is not to my knowledge any conventional notation for morphisms of field theories.

This post has been migrated from (A51.SE)
answered Oct 14, 2011 by user388027 (415 points) [ no revision ]
+ 7 like - 0 dislike

There is, I think, no really standard symbol for the generic (chiral) CFT used universally, but there is within the different formalizations.

  • When chiral CFTs are modeled by vertex operator algebras, the standard symbol is usually "$V$" (for obvious reasons) as user388027 notes in his reply..

  • When chiral CFTs are modeled as conformal nets, then (as you know), the standard symbol is usually "$\mathcal{A}$" or "$\mathfrak{A}$" (for A lgebra of observables)

    A randomly picked standard reference with this usage is Gabbiani,Fröhlich, Operator algebras and CFT

It seems to me that most authors who need and use the notion of CFT more abstractly tend to write things like

$$ CFT_1 \to CFT_2 $$

For instance so here.

This post has been migrated from (A51.SE)
answered Oct 14, 2011 by Urs Schreiber (5,455 points) [ no revision ]
+ 6 like - 0 dislike

For conformal nets $\mathcal A,\mathcal B,\ldots$ or $A,B,\ldots$ is typical. For Virasoro nets $\mathrm{Vir}_{c=\frac 12}$ is normally used and for loop group nets $\mathcal A_{G_k}$. In VOA it seems to be common to use $V$ for a generic VOA. Kac uses in "VOA for Beginners" $V_Q$ for the lattice VOA associated with a lattice $Q$ and $V^k(\mathfrak g)$ for the affice VOA of $\mathfrak g$ at level $k$.

This post has been migrated from (A51.SE)
answered Oct 15, 2011 by Marcel (300 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights