Overview of an explanation :
Jones-Wassermann subfactors for the loop algebra :
Let $\mathfrak{g} = \mathfrak{sl}_{2}$ be the Lie algebra, $L\mathfrak{g}$ its loop algebra and $\mathcal{L}\mathfrak{g} = L\mathfrak{g} \oplus \mathbb{C}\mathcal{L}$ the central extension :
$$[X^{a}_{n},X^{b}_{m}] = [X^{a},X^{b}]_{m+n} + m\delta_{ab}\delta_{m+n}\mathcal{L}$$ with $(X^{a})$ the basis of $\mathfrak{g}$.
The unitary highest weight representations of $\mathcal{L}\mathfrak{g}$ are $(H_{i}^{\ell},\pi_{i}^{\ell})$ with :
$\mathcal{L} \Omega = \ell \Omega$ with $\ell \in \mathbb{N}$ the level, and $\Omega$ the vacuum vector.
$i \in \frac{1}{2}\mathbb{N}$ and $i \le \frac{\ell}{2}$, the spin (related to the irreducible representation $V_{i}$ of $\mathfrak{g}$)
Let $I \subset \mathbb{S}^{1}$ an interval, and $\mathcal{L}_{I}\mathfrak{g}$ the local Lie algebra generated by $(X^{a}_{f})$ with :
- $f(\theta) = \sum \alpha_{n}e^{in\theta}$ and $f \in C^{\infty}_{I}(\mathbb{S}^{1})$
- $X^{a}_{f} = \sum \alpha_{n}X^{a}_{n}$
Let $\mathcal{M}_{i}^{\ell}(I)$ be the von Neumann algebra generated by $\pi_{i}^{\ell}(\mathcal{L}_{I}\mathfrak{g})$.
We obtain the Jones-Wassermann subfactor :
$$\mathcal{M}_{i}^{\ell}(I) \subset \mathcal{M}_{i}^{\ell}(I^{c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}$ with $m=\ell + 2$ and $p=2i+1$.
Its principal graph is given by the fusion rules :
$$H_{i}^{\ell} \boxtimes H_{j}^{\ell} = \bigoplus_{k \in \langle i,j \rangle_{\ell}}H_{k}^{\ell}$$ with $\langle a,b \rangle_{n} = \{c=\vert a-b \vert, \vert a-b \vert+1,... \vert c \le a+b , a+b+c \le n \}$
Let $\mathcal{R}_{\ell}$ be the fusion ring generated.
Temperley-Lieb case (with $\ell \ge 1$) :
If $i=1/2$ then index=$\frac{sin^{2}(2\pi/(\ell+2))}{sin^{2}(\pi/(\ell+2))} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{\ell+2})$ and the principal graph is $A_{\ell+1}$.
In this case, the subfactors are known to be completely classified by their principal graph.
The subfactor planar algebra it generates is the Temperley-Lieb planar algebra $TL_{\delta}$.
Jones-Wassermann subfactors for the Virasoro algebra :
Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension:
$$
[L_m,L_n]=(m-n)L_{m+n}+\frac{C}{12}(m^3-m)\delta_{m+n,0},
$$
Its discrete series representations are $(H_{pq}^{m})$ with :
- $C\Omega = c_{m} \Omega$ with $c_{m}= 1-\frac{6}{m(m+1)}$ for $m=2,3,...$
- $L_{0} \Omega = h^{pq}_{m} \Omega$ with $h^{pq}_{m} = \frac{[(m+1)p-mq]^{2}-1}{4m(m+1)}$ with $1 \le p \le m-1$ and $1 \le q \le p $
As for the loop algebra, there are $\mathfrak{Vir}_{I}$ and $\mathcal{N}_{pq}^{m}(I)$ generated by $\pi_{pq}^{m}(\mathfrak{Vir}_{I})$.
We obtain the Jones-Wassermann subfactor :
$$\mathcal{N}_{pq}^{m}(I) \subset \mathcal{N}_{pq}^{m}(I^{c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}.\frac{sin^{2}(q\pi/(m+1))}{sin^{2}(\pi/(m+1))}$.
Its principal graph is given by the fusion rules :
$$H_{pq}^{m} \boxtimes H_{p'q'}^{m} = \bigoplus_{(i'',j'') \in \langle i,i' \rangle_{\ell} \times \langle j,j' \rangle_{\ell + 1} }H_{p''q''}^{m}$$ with $p=2i+1, q=2j+1, p'=2i'+1, ..., m=\ell+2$
Let $\mathcal{T}_{m}$ be the fusion ring they generate, it's an easy quotient of $\mathcal{R}_{\ell} \otimes_{\mathbb{Z}} \mathcal{R}_{\ell+1}$, with $\mathcal{R}_{\ell}$ the fusion ring obtained above for the loop algebra.
Temperley-Lieb case (with $m \ge 3$) :
If $(p,q) = (2,1)$, index$=\frac{sin^{2}(2\pi/m)}{sin^{2}(\pi/m)} = \delta^{2}$ with $\delta = 2cos(\frac{\pi}{m})$ and the principal graph is $A_{m-1}$.
As above, the subfactor planar algebra is Temperley-Lieb $TL_{\delta}$.
$\rightarrow$ We obtain the natural maps $c \leftrightarrow \delta$
and $\mathfrak{Vir}_{c} \leftrightarrow TL_{\delta}$ that you
expected.
Generalizations for similar phenomenon :
Here is a list of possibilities :
- take $i$ other than $1/2$ or $(p,q)$ other than $(2,1)$
- take $\mathfrak{g}$ other than $\mathfrak{sl}_{2}$
- take the continuous series
- take a $N$-super-symmetric extension of $\mathfrak{Vir}$ : $N=1$ for the Neveu-Schwarz and Ramond algebras.
References :
- V.F.R. Jones, Fusion en algèbres de von Neumann et groupes de lacets (d'après A. Wassermann), Séminaire Bourbaki, Vol. 1994/95. Astérisque No. 237 (1996), Exp. No. 800, 5, 251--273.
- T. Loke, Operator algebras and conformal field theory for the discrete series representations of $\textrm{Diff}(\mathbb{S}^{1})$, thesis, Cambridge 1994.
- S. Palcoux, Neveu-Schwarz and operators algebras I : Vertex operators superalgebras, arXiv:1010.0078 (2010)
- S. Palcoux, Neveu-Schwarz and operators algebras II : Unitary series and characters, arXiv:1010.0077 (2010)
- S. Palcoux, Neveu-Schwarz and operators algebras III : Subfactors and Connes fusion, arXiv:1010.0076 (2010)
- V. Toledano Laredo, Fusion of Positive Energy Representations of LSpin(2n), thesis, Cambridge 1997, arXiv:math/0409044 (2004)
- R. W. Verrill, Positive energy representations of $L^{\sigma}SU(2r)$ and orbifold fusion. thesis, Cambridge 2001.
- A. J. Wassermann, Operator algebras and conformal field theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 966--979, Birkhuser, Basel, 1995.
- A. J. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of ${\rm LSU}(N)$ using bounded operators. Invent. Math. 133 (1998), no. 3, 467--538.
- A. J. Wassermann, Kac-Moody and Virasoro algebras, 1998, arXiv:1004.1287 (2010)
- A. J. Wassermann, Subfactors and Connes fusion for twisted loop groups, arXiv:1003.2292 (2010)
