Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?

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)

The Cherednik algebra has a similar classification into discrete and unitary series: see arXiv:1106.5094 and arXiv:0901.4595. Strictly speaking, these papers classify the unitary irreducibles in category O. I don't know whether there is a larger category in which contravariant forms will exist, but anyway for the symmetric group category O will be closely tied to affine Lie algebras (thus to Virasoro) by the Arakawa-Suzuki functor, and to Hecke (thus TL algebras) by the Knizhnik-Zamolodchikov functor (which actually identifies O with the category of q-Schur modules for most values of the parameter). Maybe the Cherednik algebra can serve as a bridge between them: Etingof conjectures (true by case by case check for the symmetric group) that KZ of a unitary module is unitary, and it is true (again case by case) that via Arakawa-Suzuki the unitary modules (i.e. integrable modules) for affine $gl_n$ correspond to unitary modules for the Cherednik algebra.

At least for the symmetric group, the question of when there is a faithful unitary module in O is not very interesting: there is always one (either $L_c(triv)$ or $L_c(sign)$ will work). But if one is to make the connection to TL and the Virasoro algebra work probably one needs more detail.

Every Cherednik algebra module is in particular a module over a ring C[V] of polynomial functions on a vector space V, and its support is a subvariety of V. The faithful unitaries should be the unitaries with full support (I have not checked this, though one direction is obvious).

In the (much simpler) case of the Cherednik algebra of the symmetric group $S_n$, the algebra depends on one parameter c, which we may assume positive. The irreducibles in O are indexed by irreducible $S_n$-modules, and therefore by partitions of n. Writing $a(\lambda)$ for the largest hook length of the partition $\lambda$ and $b(\lambda)$ for a certain smaller hook length (see the paper of Etingof/Stoica for the precise def'ns), the corresponding irreducible $L_c(\lambda)$ is unitary iff $\lambda=(1^n)$ (corresponding to the sign representation), or $c \leq a(\lambda)$ or $c=1/m$ for a positive integer $m$ with $m \leq b(\lambda)$. The continuous part of the unitary set is precisely the closure of the set where the corresponding standard module is irreducible and unitary (this much is not surprising: the condition for the contravariant form to be positive definite on the standard module is open, and it's obviously pos. def. at $0$).

The module $L_c(\lambda)$ has full support iff: $c$ is not rational or $c=k/m$ and the partition is $m$-regular: the differences $\lambda_i-\lambda_{i+1}$ are strictly less than $m$. Thus $L_c(\lambda)$ is unitary of full support iff (1) $\lambda=(1^n)$, (2) $\lambda=(n)$ and $0 \leq c < 1/n$, (3) $\lambda \neq (n),(1^n)$ is a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with $m<b(\lambda)$, (4) $\lambda$ is not a rectangle and $c \in [0,1/a(\lambda)]$ or $c=1/m$ for a positive integer $m$ with $m \leq b(\lambda)$.

Taking the $n \rightarrow \infty$ limit of all this should be possible; I am running out of time again.

I think it is not a coincidence, although the only relationship I can think of is a bit distant. Roughly, it goes: From positive energy representations of affine Kac-Moody algebras one gets certain values of $c$ in the discrete series. The TL algebras appear as centralizer algebras for quantum $\mathfrak{sl}_2$ (i.e. $End(V^{\otimes n})$ for $V$ the "vector" representation). At roots of unity one gets the TL discrete series (here $\delta$ is the $q$-dimension of $V$. On the other hand, the level-preserving tensor product on reps. of the affine Kac-Moody algebra of type $A$ gives a fusion category equivalent to the category one obtains from the quantum group situation (due to Finkelberg, although Lepowsky tells me there is a small gap that can be fixed using VOAs).

So I guess I am saying that there is a sort of Schur-Weyl duality relationship. This is not restricted to the type $A$ situation, for example, BMW algebras exhibit similar behavior which corresponds to the type $BCD$ quantum groups (or affine Kac-Moody algebras).

Probably the appropriate language to use is that of tensor categories associated with quantum groups. At roots of unity one gets unitary reps (see Wenzl or Xu's work on this) giving a discrete series which (at least combinatorially) corresponds to level-preserving fusion products for Kac-Moody algebras, which then are responsible for the Virasoro algebra situation. For non-roots of unity one still gets a continuous series of unitary reps.