2025/03/31 by Gui, Bin, Zhang, Hao · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2503.23995
Let \mathbb V=\bigoplusn∈\mathbb N\mathbb V(n) be a C2-cofinite VOA, not necessarily rational or self-dual. In this paper, we establish various versions of the sewing-factorization (SF) theorems for conformal blocks associated to grading-restricted generalized modules of \mathbb V⊗ N (where N∈\mathbb N). In addition to the versions announced in the Introduction of [GZ23], we prove the following coend version of the SF theorem: Let \mathfrak F be a compact Riemann surface with N incoming and R outgoing marked points, and let \mathfrak G be another compact Riemann surface with K incoming and R outgoing marked points. Assign \mathbb W\inMod(\mathbb V⊗ N) and \mathbb X\inMod(\mathbb V⊗ K) to the incoming marked points of \mathfrak F and \mathfrak G respectively. For each \mathbbM ∈ Mod(\mathbbV⊗ R), assign \mathbbM and its contragredient \mathbb M' to the outgoing marked points of \mathfrak F and \mathfrak G respectively. Denote the corresponding spaces of conformal blocks by \mathscr T\mathfrak F^*(\mathbb M⊗\mathbb W) and \mathscr T_\mathfrakG^*(\mathbb M'⊗\mathbb X). Let the \mathfrak X be the (N+K)-pointed surface obtained by sewing \mathfrak F, \mathfrak G along their outgoing marked points. Then the sewing of conformal blocks-proved to be convergent in [GZ24]-yields an isomorphism of vector spaces ∫^\mathbbM\inMod(\mathbb V⊗ R)\mathscr T\mathfrak F^*(\mathbb M⊗\mathbbW)⊗\mathbb C \mathscr T\mathfrak G^*(\mathbb M'⊗ \mathbb X)≃\mathscr T\mathfrak X^*(\mathbb W⊗ \mathbb X) We also discuss the relation between conformal blocks and the modular functors defined using Lyubashenko's coend in the case where \mathbb V is strongly finite and rigid.