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  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 (6,095 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 ]

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