Exact dimension formula for vertex operator algebra bundles of general signature

Let $\Gamma$ be a cocompact Fuchsian group of signature $(g;m_1,\dots,m_r;0)$, and let $V$ be a vertex operator algebra with $k=2n$. Write $M_k(V,\Gamma)$ for the space of vector-valued automorphic forms of weight $k$, $M_{k-2j}(\Gamma)$ for scalar automorphic forms, $V_j$ for the conformal-degree-$j$ subspace of $V$, and $QP_n(V)$ for the quotient space in conformal degree $n$. The dimensions $\dim M_{2i}(\Gamma)$ are given by the full Riemann–Roch formula, including elliptic corrections.

Exact dimension formula. The dimension is

$$\dim M_k(V,\Gamma)=\sum_{j=0}^{n-1}\dim M_{k-2j}(\Gamma)\,\dim V_j+\dim QP_n(V), \qquad k=2n.$$

This extends the exact dimension formula from the torsion-free cocompact case to cocompact Fuchsian groups of arbitrary signature; the scalar-form dimensions must account for elliptic points through the full Riemann–Roch formula.