Integral-level semi-infinite cohomology and the Verlinde algebra

The paper considers the affine Lie algebra $\hat{\mathfrak g}$, its central element $\mathbf k$, and truncated vertex operator algebras $\hat{\mathbf F}_{k+h^\vee}$ and $\hat{\mathbb F}_{k+h^\vee}$ expected to exist for positive integral $k$. Let $\mathbf V_k(\hat{\mathfrak g})$ be the Verlinde algebra of integrable level $k$ representations, and let $\mathbb V_k(\hat{\mathfrak g})$ be its counterpart associated with big projective modules. Integral-level Verlinde conjecture. For positive integral $k$, there are commutative algebra isomorphisms

$$H^{{\frac\infty2}+0}(\hat{\mathfrak g},\mathbb C\mathbf k;\hat{\mathbf F}_{k+h^\vee})\cong\mathbf V_k(\hat{\mathfrak g}),\qquad H^{{\frac\infty2}+0}(\hat{\mathfrak g},\mathbb C\mathbf k;\hat{\mathbb F}_{k+h^\vee})\cong\mathbb V_k(\hat{\mathfrak g}).$$

This predicts that the degree-zero semi-infinite cohomology of the truncated modified regular representations encodes the fusion rules and their big-projective counterpart. The paper develops the corresponding structures for generic central charge; the positive-integral case is presented as a conjectural truncation.