Modular Linear Differential Equations (MLDEs)

• MLDEs are linear ordinary differential equations on the upper half-plane whose solution spaces are finite-dimensional representations of modular groups.
• They establish a bridge between modular forms, partition functions, and characters in conformal and gauge theories, aiding in the classification of RCFTs and SCFTs.
• Explicit constructions involve Eisenstein series, Ramanujan–Serre derivatives, and Rankin–Cohen brackets, with examples ranging from second-order hypergeometric equations to higher-order cases.

A modular linear differential equation (MLDE) is a linear ordinary differential equation on the upper half-plane whose solution space forms a finite-dimensional representation of a modular group or congruence subgroup. MLDEs play a central role in the paper of rational conformal field theories (RCFTs), vertex operator algebras (VOAs), and special functions with modular transformation properties. The algebraic and analytic structure of MLDEs tightly links them to the modular forms, partition functions, and characters of two-dimensional chiral quantum field theories and four-dimensional partition functions with modular symmetry.

1. Mathematical Definition of MLDE

Consider a holomorphic vector-valued function $f(\tau)$ defined on the upper half-plane $\mathbb{H}$, transforming as a modular form (possibly vector-valued or with multiplier system) under a discrete subgroup $\Gamma\subset SL(2,\mathbb{Z})$. Let $D_{k}$ denote the Ramanujan–Serre (or Serre) derivative acting on weight $k$ modular forms,

$$D_k f(\tau) = q\frac{d}{dq} f(\tau) - \frac{k}{12} E_2(\tau) f(\tau),\quad q=e^{2\pi i \tau},$$

where $E_2(\tau)$ is the weight-2 Eisenstein series. An order-$n$ MLDE of weight $k$ is a monic differential operator,

$$L = D_{k+2n-2}\cdots D_k + \sum_{j=1}^{n} \varphi_j(\tau) D_{k+2n-2-j} \cdots D_k,$$

where each $\varphi_j(\tau)$ is a modular (or quasi-modular) form of weight $2j$. The equation $L f = 0$ is the MLDE. The solution space is invariant under $\Gamma$ and, for generic coefficients, is spanned by functions (possibly vector-valued) with $q$-expansions whose exponents and coefficients are determined by the indicial equation and recurrence relations (Franc et al., 2015Yamashita, 2018Nagatomo et al., 2022).

2. Wronskian Index and Indicial Equation

A critical invariant of an MLDE is its Wronskian index $\ell$, defined by the zero and pole structure of the Wronskian determinant of the fundamental system of solutions. If the MLDE has order $n$, the Wronskian $W(\tau)$ transforms as a modular form of weight $n(n-1)$ under $\Gamma$, and its zeros (counted with multiplicities in the moduli space) sum to $\ell/6$ for $SL(2,\mathbb{Z})$. The Riemann–Roch (valence) formula relates the exponents $\alpha_i$ appearing in the Frobenius solutions,

$$\sum_{i=0}^{n-1} \alpha_i = \frac{n(n-1)}{12} - \frac{\ell}{6}.$$

For one-character MLDEs, admissibility requires $\ell$ even, leading to central charge quantization in meromorphic CFTs (Das, 2023).

The indicial equation probes the singularity at $q=0$ (the cusp) and determines the allowed exponents,

$$\prod_{j=0}^{n-1}(\gamma - j/6) + \sum_{k=1}^n \nu_k \prod_{r=0}^{n-k-1}(\gamma - r/6) = 0,$$

where $\nu_k$ are sums over coefficients of modular forms in the MLDE (Chandra et al., 1 Dec 2025).

3. Classification of MLDEs and Solution Spaces

The coefficients $\varphi_j(\tau)$ are constructed from modular or quasi-modular forms. For $SL(2,\mathbb{Z})$, the ring of holomorphic modular forms is $\mathbb{C}[E_4, E_6]$. Explicitly, MLDEs can always be written in terms of polynomials in Eisenstein series and Rankin–Cohen brackets; the MLDO algebra is generated by $D$, $E_4$, and $E_6$ with computable commutation relations and division algorithm (Yamashita, 2018Nagatomo et al., 2022).

For solutions with integer (or rational) exponents and appropriately constrained Fourier coefficients, the solution space can be matched with the space of RCFT characters, minimal model partition functions, or Schur indices of four-dimensional SCFTs. In many cases, the solution spaces can be mapped to vector-valued modular forms, correlators in 2d CFT, or multi-contour integral (Jastrow) representations.

Examples include:

• Second-order MLDEs with solutions identified as hypergeometric functions $_2F_1$ in a Hauptmodul variable (Mason et al., 2018Franc et al., 2015).
• Third- and higher-order MLDEs matching higher-dimensional characters, including minimal $\mathcal{W}$-algebra characters (1803.02022Nagatomo et al., 2023).
• Jastrow integral representations giving $(N+1)$ fundamental solutions for order-$(N+1)$ MLDEs with vanishing Wronskian index, as in the analysis for $USp(2N)$ with $2N+2$ hypers (Chandra et al., 1 Dec 2025).

4. MLDEs in Conformal Field Theory, RCFT, and Gauge Theories

MLDEs describe the modular properties of torus partition functions and serve as a classification tool for RCFTs by encoding the requirements of modular invariance, finiteness of conformal weights, and positivity of degeneracies (Govindarajan et al., 31 Mar 2025Gowdigere et al., 2023). The monic (minimal) MLDE associated to an RCFT or VOA typically determines the space of holomorphic or weakly holomorphic vector-valued modular forms closed under $\Gamma$.

For four-dimensional $\mathcal{N}=2$ SCFTs, generalised Schur partition functions $\mathcal{Z}_G(q;\alpha)$, appearing e.g. in $USp(2N)$ gauge theory with $2N+2$ hypermultiplets, are annihilated by monic order-$(N+1)$ MLDEs with $\ell=0$ (Chandra et al., 1 Dec 2025). This property enables a direct translation between gauge theory parameters (such as $\alpha$ or $\beta$) and critical exponents of the MLDE, providing a bridge between four-dimensional and two-dimensional quantum field theory partition functions.

A key analytic bridge is provided by contour integral (Jastrow) representations for the solutions, equating partition functions with vector-valued modular forms satisfying the MLDE (Chandra et al., 1 Dec 2025).

5. Examples and Explicit Constructions

Low-rank cases yield explicit MLDEs closely related to classical RCFTs:

• For $N=1$ (i.e., $SU(2)$ with four hypers), the MLDE reduces to a monic second-order equation (Mathur–Mukhi–Sen equation),

$$[D^2 + \mu E_4(\tau)] Z_{SU(2)}(q;\alpha) = 0,\quad \mu = -[(6\alpha - 1)(6\alpha + 1)]/144.$$

• For $N=2$ ($USp(4)$ with six hypers), an order-3, $\ell=0$ MLDE appears,

$$[D^3 + \mu_2 E_4 D + \mu_3 E_6] Z_{USp(4)}(q;\alpha) = 0.$$

Vector-valued modular forms appearing as partition functions of 2d RCFTs and Schur indices in 4d SCFTs are concrete instances of MLDE solution spaces, determined through their Frobenius expansions or contour integral representations (Arakawa et al., 2016Chandra et al., 1 Dec 2025).

6. Extensions: Two-Parameter MLDEs, Movable Poles, and Quantum Monodromy

Beyond one-parameter deformations, MLDEs with two-parameter families of solutions, as in $\mathcal{Z}_{USp(2N)}(q;\alpha,\beta)$, satisfy order-$(N+1)$, $\ell=0$ MLDEs,

$$[D^{N+1} + \sum_{k=1}^{N+1}\sum_i \mu_{k,i}(\alpha,\beta;N)\, \varphi_{2k,i}(\tau) D^{N+1-k}]\, Z_{USp(2N)}(q;\alpha,\beta) = 0,$$

with exponents $\gamma_A$ given by explicit functions of $\alpha$, $\beta$ (Chandra et al., 1 Dec 2025).

More generally, MLDEs with movable poles, controlled by accessory parameters, allow the classification of quasi-characters beyond rigid cases—critical for the "holomorphic modular bootstrap" and for exploring new RCFTs. Conditions for logarithm-freeness around poles and recursion relations as poles are sent to cusps elucidate the structure of the MLDE parameter space (Das et al., 2023).

Quantum monodromy traces $I_k(q)$ in the context of 4d $\mathcal{N}=2$ SCFTs are empirically found to solve MLDEs of fixed order and vanishing Wronskian index, aligning their spectral data with that of Schur partition functions at special values of deformation parameters (Chandra et al., 1 Dec 2025). Conjecturally, for any rank-$r$ 4d $\mathcal{N}=2$ SCFT, the $k$-th quantum monodromy trace is annihilated by an order-$(r)$ MLDE with the same Wronskian index as the $k=-1$ case.

7. Symmetry, Pullbacks, and the Schwarzian Equation

MLDEs are closely related to the modular symmetries of the solution space. The Schwarzian differential condition characterizes when the pullback of a solution under a rational function remains a solution, leading to modular equations and identities between modular forms and hypergeometric functions (Abdelaziz et al., 2016). This structure unifies the MLDE framework with the symmetry properties of special functions and the geometry of covering maps between moduli spaces.

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References:

• "Generalised 4d Partition Functions and Modular Differential Equations" (Chandra et al., 1 Dec 2025)
• "Meromorphic CFTs have central charges c = 8N: a proof based on the MLDE approach and Rademacher series" (Das, 2023)
• "Modular forms, Schwarzian conditions, and symmetries of differential equations in physics" (Abdelaziz et al., 2016)
• "On modular linear differential operators and their applications" (Yamashita, 2018)
• "Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets" (Nagatomo et al., 2022)
• "Quasi-lisse vertex algebras and modular linear differential equations" (Arakawa et al., 2016)
• Additional explicit constructions and classification: (Franc et al., 2015, Mason et al., 2018, Nagatomo et al., 2023, Govindarajan et al., 31 Mar 2025, Gowdigere et al., 2023, Das et al., 2023)