# which letter to use for a CFT?

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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?

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

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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?

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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.

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Ahhh I see, thanks!

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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.

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answered Oct 14, 2011 by (415 points)
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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.

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answered Oct 14, 2011 by (5,900 points)
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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 $k$.

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answered Oct 15, 2011 by (300 points)

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