Modular invariance of characters of quasi-lisse vertex algebras

Abstract: We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible affine vertex algebras as well as admissible W-algebras associated with nilpotent elements of standard Levi type, we prove the holonomicity of the sheaf of conformal blocks over the moduli space of bundles. Furthermore, we show that the space of flat sections of the associated Jacobi-invariant connection is spanned by trace functions on modules. This result provides a substantial generalization of the celebrated theorem of Yongchang Zhu to quasi-lisse vertex algebras. As a special case, we deduce that for affine vertex algebras at admissible level, the dimension of the space of conformal blocks coincides with the number of admissible weights at that level.

• The paper extends Zhu’s theorem to quasi-lisse vertex algebras by proving the modular invariance of trace function characters.
• It employs holonomicity and stable Zhu algebra techniques to derive explicit transformation and MLDE formulas on elliptic curves.
• The results yield practical tools for analyzing orbifolds, superalgebras, and dualities in conformal field theory and algebraic geometry.

Modular Invariance for Characters of Quasi-Lisse Vertex Algebras

Overview and Motivation

The paper "Modular invariance of characters of quasi-lisse vertex algebras" (2605.29921) develops a significant generalization of Zhu's theorem on modular invariance of vertex algebra characters, extending it from lisse rational vertex algebras to quasi-lisse vertex algebras. The principal innovation is the introduction of "charged conformal blocks" associated to elliptic curves with degree zero line bundles, and the proof that trace functions on stably rational, stably quasi-lisse vertex algebras span the space of conformal blocks and transform equivariantly under the Jacobi group. The work provides a rigorous, algebraic framework connecting vertex algebra representation theory, modular/Jacobi forms, and holonomic D-module theory.

This advances both theoretical and practical understanding of modular properties in vertex algebra representation theory. It resolves longstanding conjectures on the extension of modular invariance beyond the conventional setting, specifically including admissible affine and $W$-algebras. Concrete consequences include finiteness results for spaces of conformal blocks, and explicit prescriptions for modular differential equations (MLDEs) satisfied by characters.

Charged Conformal Blocks and Holonomic D-Modules

The authors define charged conformal blocks as linear functionals annihilating images of twisted chiral algebra sections governed by a triple $(V, \omega, h)$, with $V$ a conformal vertex algebra, $\omega$ a conformal vector, and $h$ an integer-graded current. These blocks depend on moduli $(\alpha, \tau)$—parameters for both the elliptic curve and the associated line bundle—and are assembled into a sheaf over the moduli space equipped with an integrable connection $\nabla$.

A central theorem in the paper asserts that for finitely strongly generated, stably quasi-lisse vertex algebras, the sheaf of charged conformal blocks forms a holonomic $D$-module with regular singularities on a prescribed lattice in $(\alpha, \tau)$. Regularity is established via explicit calculation and Fuchsian reduction, underpinning both analytic and algebraic finiteness statements. Away from the singular locus, the sheaf is a vector bundle with flat connection, supporting analytic continuation and monodromy analysis.

Trace Functions, Stable Rationality, and Exhaustion Theorems

For stably rational vertex algebras, the space of conformal blocks is shown to be spanned by trace functions

$$S_W(u, \alpha, \tau) = \mathrm{Tr}_W\, u_0\, y^{h_0}\, q^{L_0 - c/24},$$

where $(V, \omega, h)$0, $(V, \omega, h)$1, and $(V, \omega, h)$2 ranges over irreducible stable $(V, \omega, h)$3-modules. The stable Zhu algebra $(V, \omega, h)$4 is introduced to classify modules with appropriate charge gradings; its semisimplicity is both necessary and sufficient for stable rationality, and the paper provides detailed arguments for the finite-dimensionality and structure of $(V, \omega, h)$5 in the admissible affine and $(V, \omega, h)$6-algebra cases.

An exhaustion method, adapted from Zhu, is employed to demonstrate that all flat sections of the connection—i.e., all charged conformal blocks—are finite combinations of such trace functions, assuming stable rationality. The proof utilizes Frobenius-type expansions, inductive arguments on $(V, \omega, h)$7- and $(V, \omega, h)$8-logarithmic sectors, and application of symmetric invariants of the stable Zhu algebra.

Jacobi Group Equivariance and Modular/Jacobi Forms

A substantial result is the explicit demonstration that the connection $(V, \omega, h)$9 is equivariant under the Jacobi group action $V$0, which acts naturally on $V$1. The transformation laws for trace functions are computed, showing that they behave as vector-valued Jacobi forms with precise weight and index determined by $V$2 and $V$3 (with index $V$4 given by $V$5).

This generalizes the modular invariance of characters previously known for lisse rational and admissible modules, and demonstrates that in the admissible affine case, the dimension of the space of conformal blocks precisely matches the number of admissible weights and the explicit characters form a vector-valued Jacobi form.

Finiteness, MLDEs, and Examples

The paper proves finiteness of charged conformal block spaces under mild structural hypotheses, notably for all stably quasi-lisse, finitely strongly generated vertex algebras. The authors give explicit analysis of associated MLDEs (modular linear differential equations) for the characters in both the rational and admissible cases, confirming classical formulas and providing new explicit constraints. Examples include $V$6 and $V$7, with detailed calculations of character transformation and MLDEs.

Implications for Vertex Algebra Theory and Beyond

The work provides theoretical underpinnings for the modularity conjectures in generalized settings, including those inspired by 4d/2d duality (class S theories). The relationship between conformal block spaces, modular forms, and representation theory is clarified for a broad class of algebras. The results lay groundwork for further analysis of MLDEs for quasi-lisse and non-rational algebras, offer strategies for comparison with predicted module counts (e.g., via affine Springer fibers), and open pathways for studying orbifold constructions, spectral flows, and distributional characters.

The integration of D-module theory, Frobenius expansion methods, and explicit modular/Jacobi group actions enhances the precision of structural and analytic results in vertex algebra representation theory. Potential future directions include investigation of non-semisimple cases, extension to twisted modules and superalgebras, and connections to arithmetic invariants via modular forms.

Conclusion

The paper establishes a comprehensive generalization of modular invariance for characters of quasi-lisse vertex algebras, integrating representation theory, algebraic geometry, and analytic function theory. The modularity and finiteness of charged conformal blocks is shown to hold for a broad class of admissible and $V$8-algebras, and trace functions are given explicit significance as solutions to holonomic D-module systems with regular singularities. The work paves the way for deeper exploration of modular forms and MLDEs in the vertex algebra context, and provides robust, algebraically grounded tools for further research in conformal field theory and related areas.