The semiclassical ("light") limit $c\to \infty$ of the irreducible Virasoro representation $\varphi_{n,1}$ with highest weight $h_{n,1}\to -\frac{n-1}{2}$ is $\mathbb{C}[L_{-1},L_{-2},\dotsc]/(N_n)$ as a module over the commutative ring $R=\mathbb{C}[L_{-2},\dotsc]$, free of rank $n$, where $N_n=(L_{-1})^n+\cdots$ is the Virasoro null vector. Also there is an $\mathfrak{sl}_2$-action coming from $L_{-1},L_0,L_{1}$.
We can interprete $R$ as the (Poisson-)algebra of functions on the set of Sturm Liouville operators $d^2+q$ with regular $q=a_0+a_1z+\dotsb$. In this view we can interpret the limit of $\varphi_{n,1}$ as an $\mathfrak{sl}_2$-vector bundle over this set, whose fibre is the irreducible $\mathfrak{sl}_2$-representation $\mathbb{C}^n$ (as one can check).
I would be happy for any references or insight on this bundle and the involved Poission structure. This should have to do with the KdV hierarchy, where the second Poisson structure of $R$ appears.
- I have read Zuber: KdV and W− Flows about the "mysterious" matching formulas for $N_n$ and the $n$-th order differential $\mathcal{F}_{-\frac{n-1}{2}}\to \mathcal{F}_{\frac{n+1}{2}}$ – what is the current state explaining this match? (… I would somewhat suspect their $n$-dimensional solution space to be identified with the irreducible $\mathfrak{sl}_2$-representation $\mathbb{C}^n$ …?)
- On the other hand the limit of $\varphi_{n,1}$ can be identified with the differential operators $d^2+q$ with regular singular $q=a_{-2}z^{-2}+a_{-1}z^{-1}+a_0+a_1z+\dotsb$ and a fixed diagonal (!) monodromy around $z=0$.
To summarize, I want to understand the semiclassical limit of $\varphi_{n,1}$ in the interpretation of quadratic differentials, (Poisson-)geometry and KdV. I am comfortable with CFT but new to KdV, so I have problems putting the pieces together and any help would be greatly appreciated.
