Modular invariance of characters of quasi-lisse vertex algebras

This paper generalizes Zhu's theorem on modular invariance to quasi-lisse vertex algebras by proving the holonomicity of conformal blocks over the moduli space of bundles and showing that their flat sections are spanned by trace functions, thereby establishing that the dimension of the space of conformal blocks for affine vertex algebras at admissible levels equals the number of admissible weights.

The Gist

The Big Picture: A Symphony of Shapes and Numbers

Imagine you are a musician trying to understand a complex piece of music. In the world of mathematics and physics, this "music" is a Vertex Algebra. Think of a vertex algebra as a massive, intricate library of rules that describe how tiny particles interact and transform.

For a long time, mathematicians had a famous rule (discovered by Yongchang Zhu) that worked perfectly for "perfectly tuned" libraries. This rule said: If you take the "notes" (called trace functions) played by the different instruments (modules) in this library, they will always form a beautiful, repeating pattern called a Modular Form.

A Modular Form is like a musical phrase that sounds exactly the same even if you change the tempo or the key of the song in a specific, symmetrical way. This symmetry is crucial because it helps physicists and mathematicians understand the deep structure of the universe (specifically, Conformal Field Theory).

The Problem: The Library Got Messy

The problem is that many interesting libraries are not "perfectly tuned." They are what the authors call Quasi-Lisse. These libraries are a bit messy; they have "non-ordinary" instruments that don't play by the standard rules. Because of this messiness, the old rule (Zhu's theorem) broke down. The notes didn't seem to form a perfect pattern anymore.

The authors of this paper asked: Can we fix the rule so it works for these messy libraries too?

The Solution: Adding a "Flavor" Knob

The authors' brilliant idea was to add a new ingredient to the mix. Imagine the library is a recipe for a cake. The old rule only worked if you baked the cake with a specific amount of sugar. But for the messy libraries, the cake tastes wrong.

So, the authors introduced a new variable: a line bundle.

• The Analogy: Think of the "line bundle" as a special flavor knob or a seasoning dial you can turn on the cake.
• In the math, this knob is represented by a parameter called $\alpha$ (alpha).
• By turning this knob, they changed the way they measured the "notes" (the trace functions). Instead of just measuring the raw sound, they measured the sound with the flavor knob turned.

They call these new measurements Charged Conformal Blocks.

The Three Main Discoveries

The paper proves three major things about this new approach:

1. The Pattern Exists (Holonomicity)
Even though the library is messy, if you turn the flavor knob correctly, the notes do form a pattern. The authors proved that these new "Charged Conformal Blocks" behave like a holonomic system.

• The Metaphor: Imagine a maze. In the old messy library, the path was a tangled knot. But with the flavor knob, the path straightens out into a clear, predictable road. The notes follow a specific set of rules (differential equations) that allow them to be solved, even if the library is complex.

2. The Notes Fill the Room (Spanning the Space)
The authors showed that if you take all the possible "flavor settings" (the trace functions on different modules), they are enough to describe every single possible sound in this new system.

• The Metaphor: Imagine a room full of empty chairs (the space of all possible sounds). The authors proved that if you bring in the specific chairs made from the "stable modules" (the good instruments), they perfectly fill every seat in the room. You don't need any other chairs; these specific ones are enough to describe the whole room.

3. The Pattern is Super-Symmetrical (Jacobi Invariance)
This is the most exciting part. The old rule said the notes were symmetrical under "Modular" transformations (changing the shape of the time/space grid). The new rule says they are symmetrical under Jacobi transformations.

• The Metaphor: Think of a kaleidoscope.
  • Modular symmetry is like rotating the kaleidoscope. The pattern looks the same.
  • Jacobi symmetry is like rotating it and sliding the mirrors around at the same time.
  • The authors proved that even when you rotate and slide the kaleidoscope (changing the time, space, and the flavor knob $\alpha$), the pattern of the notes remains perfectly consistent. They call these Jacobi Forms.

Why This Matters (According to the Paper)

The paper focuses on two specific types of "messy libraries" that are very important in physics:

1. Admissible Affine Vertex Algebras: These are related to simple Lie algebras (mathematical structures describing symmetries).
2. Admissible W-algebras: These are more complex structures derived from the first ones.

The authors prove that for these specific libraries, the number of distinct "notes" (the dimension of the space) is exactly equal to the number of "admissible weights" (a specific list of allowed settings).

In simple terms: They took a broken rule, added a flavor knob to fix it, and proved that the resulting music is not only harmonious but follows a super-symmetrical pattern (Jacobi forms) that holds true for a huge class of complex mathematical objects.

Summary

• Old Rule: Works for perfect libraries. Notes = Modular Forms.
• New Rule: Works for messy (quasi-lisse) libraries. Notes = Charged Conformal Blocks.
• The Trick: Add a "flavor knob" (line bundle/parameter $\alpha$).
• The Result: The notes form a perfect, super-symmetrical pattern called Jacobi Forms, and the specific instruments (stable modules) are enough to describe the entire system.

The paper is a mathematical proof that this "flavor knob" method successfully generalizes a famous theorem, allowing us to understand the symmetries of complex, messy mathematical structures that were previously out of reach.

Technical Summary

Problem Statement
The paper addresses the modular invariance of characters for a broad class of vertex algebras known as quasi-lisse vertex algebras. While Yongchang Zhu's seminal theorem established that trace functions on irreducible modules of lisse (C2-cofinite) and rational vertex algebras span the space of genus-one conformal blocks and form vector-valued modular forms, this result does not directly apply to quasi-lisse algebras. Quasi-lisse algebras, which include admissible affine vertex algebras and admissible W-algebras, often possess non-rational representation categories and non-ordinary modules, rendering standard trace functions ill-defined or divergent. Furthermore, the standard Zhu theorem does not account for the "flavour" parameters associated with line bundles over elliptic curves, which are crucial for understanding the characters of these algebras in the context of 4d/2d dualities and Schur indices.

Methodology
The authors generalize Zhu's framework by shifting from ordinary conformal blocks on elliptic curves to charged conformal blocks associated with pairs $(E, L)$, where $E$ is an elliptic curve and $L$ is a holomorphic line bundle of degree zero. The methodology proceeds through several key steps:

1. Geometric Framework: They construct a sheaf of charged conformal blocks over the moduli space of pairs $(E, L)$, parametrized by $(\alpha, \tau) \in \mathbb{C} \times \mathbb{H}$. This sheaf is equipped with a connection $\nabla$ derived from Ward identities involving the Virasoro field $L(t)$ and a current $h(t)$ (a vector of conformal weight 1).
2. Stable Quasi-Lisse Condition: They introduce a "stably quasi-lisse" condition. This involves defining a relative Zhu Poisson algebra $R^h_V$ by quotienting Zhu's Poisson algebra $R_V$ by the ideal generated by vectors of non-zero charge. The condition requires the spectrum of $R^h_V$ to have finitely many symplectic leaves.
3. Stable Rationality and Modules: To handle the divergence of standard traces, the authors define stable V-modules. These are weight modules with bounded conformal weights and bounded charge eigenvalues in specific directions. They introduce the stable Zhu algebra $A_{stab}(V)$, a quotient of the standard Zhu algebra, which governs the structure of these stable modules.
4. Holonomicity and Isospectrality: Using the integrability of the connection $\nabla$, they prove that the sheaf of charged conformal blocks is a holonomic D-module with regular singularities. They establish an isospectrality result showing that flat sections of $\nabla$ admit series expansions of Frobenius type in the variables $y = e^{2\pi i \alpha}$ and $q = e^{2\pi i \tau}$.
5. Exhaustion by Trace Functions: By assuming the stable Zhu algebra is semisimple (a property termed stable rationality), they demonstrate that the space of charged conformal blocks is exhausted by trace functions on irreducible stable modules. These trace functions are defined as $S_W(u, \alpha, \tau) = \text{Tr}_W u_0 y^{h_0} q^{L_0 - c/24}$.

Key Contributions and Results

• Theorem 1.1 (Holonomicity): For a finitely strongly generated and stably quasi-lisse charged conformal vertex algebra, the sheaf of charged conformal blocks carries the structure of a holonomic D-module with regular singularities on the lattice $N\alpha \in \mathbb{Z} + \mathbb{Z}\tau$. Away from this lattice, it is a vector bundle with an integrable connection.
• Theorem 1.2 (Exhaustion): If the vertex algebra is also stably rational (the category of stable modules is semisimple), the space of charged conformal blocks at any point in the domain of convergence is spanned by the trace functions on the irreducible stable modules. These trace functions are flat sections of the connection $\nabla$.
• Theorem 1.3 (Jacobi Invariance): The connection $\nabla$ is equivariant under the action of the Jacobi group $SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$. Consequently, the span of the trace functions forms a vector-valued Jacobi form of a specific weight and index $\kappa$ (defined by $h_{[1]}h = 2\kappa |0\rangle$).
• Applications to Affine and W-Algebras:
  • The authors prove that simple affine vertex algebras $L_k(\mathfrak{g})$ at admissible levels are stably quasi-lisse and stably rational when the current $h$ is chosen as the Weyl vector $\rho$.
  • They extend this to admissible W-algebras $W_k(\mathfrak{g}, f)$ associated with nilpotent elements of standard Levi type.
  • As a corollary, for admissible affine vertex algebras, the dimension of the space of conformal blocks coincides with the number of admissible weights at that level.
• Explicit MLDEs: The paper derives explicit Modular Linear Differential Equations (MLDEs) for the characters of specific examples, such as $L_1(\mathfrak{sl}_2)$ and $L_{-4/3}(\mathfrak{sl}_2)$, demonstrating the practical utility of the connection $\nabla$.

Significance
The paper claims to provide a substantial generalization of Zhu's theorem to the class of quasi-lisse vertex algebras. By incorporating the moduli of line bundles (the "flavour" parameter $\alpha$), the authors render the trace functions convergent for a much larger class of modules than previously possible. This establishes a rigorous mathematical foundation for the modular and Jacobi properties of characters in non-rational settings, which are central to the representation theory of admissible affine and W-algebras. The results confirm that the space of conformal blocks is finite-dimensional and spanned by trace functions even when the vertex algebra is not rational, provided it satisfies the stable quasi-lisse and stable rationality conditions. This work bridges the gap between the geometric theory of conformal blocks and the representation theory of quasi-lisse algebras, offering a unified framework for understanding their modular properties.