Spherical Virasoro conformal blocks from Shapovalov matrices

Use a Virasoro representation generated by a highest-weight vector $v_h$ with $L_n{v}_h=0$ for every $n>0$ and $L_0{v}_h=h{v}_h$, as in the cited highest-weight module definition. For Liouville, $c_L=1+6Q^2$ and $h={\mathrm{\Delta }}_{\alpha }=\frac{\alpha }{2}(Q-\frac{\alpha }{2})$ for a charge $\alpha \in \mathbb{C}$. Fix four external charge parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\in \mathbb{C}$ and write ${\mathrm{\Delta }}_{{\alpha }_j}=\frac{{\alpha }_j}{2}(Q-\frac{{\alpha }_j}{2})$; this formal block definition imposes no Seiberg or reality conditions on these external parameters. A partition of $n\in \mathbb{N}$ is a finite nonincreasing sequence $\nu =({\nu }_1,\dots ,{\nu }_k)$ of positive integers with $\mathrm{\mid }\nu \mathrm{\mid }={\sum }_j{\nu }_j=n$; set $T_0=\{\mathrm{\varnothing }\}$ and let $T_n$ denote the partitions of $n$ for $n>0$. For the block data below, require that the descendant words indexed by $T_n$ form a basis at each level, and choose the real form generated over $\mathbb{R}$ by $v_h$ and the Virasoro modes when $c_L,h\in \mathbb{R}$; the complex module is its complexification. Write $L_{-\nu }=L_{-{\nu }_k}\cdots L_{-{\nu }_1}$ and $L_{-\mathrm{\varnothing }}=I$.

The Young diagram for $\nu =(3,1)$ uses the English convention: its first and second rows have respectively $3$ and $1$ left-aligned unit boxes. The boxes record the parts of the partition only; the order of the corresponding Virasoro operators is fixed separately by $L_{-\nu }$ above.

For a highest-weight vector $v_h$, use the real-valued contravariant bilinear form normalized by $⟨v_h,v_h⟩=1$ and $⟨L_{-m}u,v⟩=⟨u,L_{m}v⟩$ for $m>0$, then extend it complex-bilinearly. At level $n$, let $F_{c,h}^{(n)}(\nu ,{\nu }^{\mathrm{\prime }})=⟨L_{-\nu }{v}_h,L_{-{\nu }^{\mathrm{\prime }}}{v}_h⟩$ for $\nu ,{\nu }^{\mathrm{\prime }}\in T_n$. For real $c,h$ every matrix entry is real because it is the value of the form on vectors in the chosen real module; therefore an inverse, when it exists, is real as well. Only when this finite matrix is invertible is $F_{c,h}^{(n)}{}^{-1}$ used. This definition does not assert nondegeneracy at every weight.

For three weights $\mathrm{\Delta },{\mathrm{\Delta }}^{\mathrm{\prime }},{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}$ and a partition $\nu =({\nu }_1,\dots ,{\nu }_k)$, put

$v(\mathrm{\Delta },{\mathrm{\Delta }}^{\mathrm{\prime }},{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }},\nu )={\prod }_{j=1}^k({\nu }_j{\mathrm{\Delta }}^{\mathrm{\prime }}-\mathrm{\Delta }+{\mathrm{\Delta }}^{\mathrm{\prime }\mathrm{\prime }}+{\sum }_{u<j}{\nu }_u)\mathrm{.}$

For $P>0$ such that every required internal Gram matrix at $c=c_L$ and $h={\mathrm{\Delta }}_{Q+iP}$ is invertible, define the formal spherical block

$F_P(z)={\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n(P)z^n,{\beta }_n(P)={\sum }_{\nu ,{\nu }^{\mathrm{\prime }}\in T_n}v({\mathrm{\Delta }}_{{\alpha }_1},{\mathrm{\Delta }}_{{\alpha }_2},{\mathrm{\Delta }}_{Q+iP},\nu )[F_{c_L,{\mathrm{\Delta }}_{Q+iP}}^{(n)}{}^{-1}{]}_{\nu ,{\nu }^{\mathrm{\prime }}}v({\mathrm{\Delta }}_{{\alpha }_4},{\mathrm{\Delta }}_{{\alpha }_3},{\mathrm{\Delta }}_{Q+iP},{\nu }^{\mathrm{\prime }})\mathrm{.}$

The series is initially formal. Analytic convergence on $\mathrm{\mid }z\mathrm{\mid }<1$, nondegeneracy of all matrices for $P>0$, and convergence of the spectral integral are theorem obligations, not consequences of this definition.

Highest-weight vector and module for Virasoro

Probabilistic Liouville CFT on the sphere