Generalized conformal modules over the Virasoro conformal algebra
Abstract
This paper investigates generalized conformal modules over the Virasoro conformal algebra Vir, a notion originally defined by V.~Kac but largely unexplored in the literature. First, we present a twisted construction that produces new generalized conformal modules over any Lie conformal algebra from existing conformal modules. Then we give a complete classification of non-trivial generalized conformal modules over Vir that are free of rank one over ℂ[∂], which in turn yields a classification of modules over the Lie algebra W1=⨁i≥ −1C Li of vector fields on a line that are free of rank one over ℂ[L−1] with explicit actions expressed in terms of Bell polynomials. For arbitrary finite rank n, we construct a class of generalized conformal modules Va,b,C(∂) and establish a complete irreducibility criterion in terms of the differential operator ℒ=d/d∂+C(∂) acting on ℂ(∂)n. In the case of n=2, this criterion reduces to the absence of rational solutions of a Riccati equation. Then we prove the existence of irreducible generalized conformal modules of arbitrary finite rank, revealing a striking contrast with the conformal setting. We also construct and classify a family of infinite torsion generalized conformal modules over Vir, which are non-trivial and irreducible under suitable conditions.
Related
- simple cartan-free modules of finite rank over the super-virasoro algebras
- Classification of U(h)-Free Modules over the Heisenberg-Virasoro type Lie superalgebra via Polynomial Shift Operators
- Generalized Castelnuovo-Mumford Regularity for Representations of the Virasoro Algebra
- Restricted conformal blocks and the braiding on Zhu algebra modules
- Geometry of holonomic modules over the first Weyl algebra
BibTeX
@misc{wu2026generalized,
title = {{Generalized conformal modules over the Virasoro conformal algebra}},
author = {Henan Wu and Yanyong Hong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18184},
eprint = {2609.18184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18184},
note = {CC0},
}
RIS
TY - UNPB
TI - Generalized conformal modules over the Virasoro conformal algebra
AU - Henan Wu
AU - Yanyong Hong
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18184
UR - https://arxiv.org/abs/2609.18184
AB - This paper investigates generalized conformal modules over the Virasoro conformal algebra Vir, a notion originally defined by V.~Kac but largely unexplored in the literature. First, we present a twisted construction that produces new generalized conformal modules over any Lie conformal algebra from existing conformal modules. Then we give a complete classification of non-trivial generalized conformal modules over Vir that are free of rank one over ℂ[∂], which in turn yields a classification of modules over the Lie algebra W_1=⨁_i≥ −1C L_i of vector fields on a line that are free of rank one over ℂ[L_−1] with explicit actions expressed in terms of Bell polynomials. For arbitrary finite rank n, we construct a class of generalized conformal modules V_a,b,C(∂) and establish a complete irreducibility criterion in terms of the differential operator ℒ=d/d∂+C(∂) acting on ℂ(∂)^n. In the case of n=2, this criterion reduces to the absence of rational solutions of a Riccati equation. Then we prove the existence of irreducible generalized conformal modules of arbitrary finite rank, revealing a striking contrast with the conformal setting. We also construct and classify a family of infinite torsion generalized conformal modules over Vir, which are non-trivial and irreducible under suitable conditions.
ER -