Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness
Abstract
For a vertex operator algebra V and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface Σ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation ΩV(Σ;−) of the mapping class group of Σ. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of V is a not necessarily semisimple modular category 𝒜, the spaces ΩV(Σ) are equivalent to the string-net spaces for 𝒜 and hence to the modular functor for the Drinfeld center Z(𝒜)≃ 𝒜̅boxtimes𝒜. However, the construction of ΩV in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that ΩV satisfies excision, is finite-dimensional in the C2-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet 𝒲2,3 with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.
Related
- Finiteness of singularities for higher-weight axially symmetric minimizers of the relaxed Dirichlet energy
- Frobenius functors and one-sided Hopf algebras in braided monoidal categories
- Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation
- Equivalence of categories of bivariant K-theory for C*-algebras over topological spaces via reflection functors
- Representation Type and Schurian Finiteness of Tits Algebras
BibTeX
@misc{muller2026modular,
title = {{Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness}},
author = {Lukas Müller and Lukas Woike},
howpublished = {HAL},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28579},
eprint = {2608.28579},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05738673v1},
}
RIS
TY - UNPB
TI - Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness
AU - Lukas Müller
AU - Lukas Woike
PY - 2026
DA - 2026/08/28
PB - HAL
DO - 10.48550/arxiv.2608.28579
UR - https://hal.science/hal-05738673v1
AB - For a vertex operator algebra V and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface Σ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation Ω_V(Σ;−) of the mapping class group of Σ. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of V is a not necessarily semisimple modular category 𝒜, the spaces Ω_V(Σ) are equivalent to the string-net spaces for 𝒜 and hence to the modular functor for the Drinfeld center Z(𝒜)≃ 𝒜̅boxtimes𝒜. However, the construction of Ω_V in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that Ω_V satisfies excision, is finite-dimensional in the C_2-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet 𝒲_2,3 with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.
ER -