Modular Differential Equations (MDEs)

• Modular Differential Equations (MDEs) are differential equations structured by modular, quasimodular, or Jacobi-form properties, each adapted to various automorphic frameworks.
• They are applied to compute characters in RCFT, VOAs, and elliptic genera, employing tools such as the Ramanujan–Serre derivative and heat operators.
• Extensions to noncommutative operators and higher-genus connections unify scalar, vector-valued, and partial differential systems under a broad invariant calculus.

Searching arXiv for papers on modular differential equations and related frameworks. Searching arXiv for “modular differential equations” and related recent variants. Modular differential equations (MDEs) are differential equations whose coefficients and transformation laws are organized by modular, quasimodular, or Jacobi-form structures. In the cited literature, the term covers several closely related but not identical frameworks: modular linear differential equations for scalar or vector-valued modular forms, heat-operator equations for Jacobi forms and elliptic genera, flavored or refined Jacobi-type partial differential equations for characters with continuous flavor fugacities, Schwarzian and equivariant formulations of second-order equations, nonlinear invariant ODEs written in terms of differential invariants, and noncommutative modular differential operators built from Ore algebras and modular connections (Yamashita, 2018, Adler et al., 2022, Pan et al., 2023, Jafari, 8 Mar 2026). This variety is structurally important: “MDE” does not denote a single canonical formalism, but a family of modularly constrained differential systems adapted to different automorphic settings.

1. Linear MLDEs and operator-algebraic structure

A standard holomorphic MLDE for a scalar modular object of weight $k$ is built from the Ramanujan–Serre operator

$D_k f = f' - \frac{k}{12} E_2 f,$

where $f'(z)=(1/(2\pi i))\,df/dz$, together with its iterates

$$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$

In Yamashita’s formulation, a weight-$k$ MLDE of order $n$ has the form

$$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$

with $g_i\in M(l+2i)$; it is monic when $l=0$ and $g_0=1$ (Yamashita, 2018).

This formalism is encoded in the graded algebra $D_k f = f' - \frac{k}{12} E_2 f,$0 of modular linear differential operators. The generators are the weight-raising operator corresponding to $D_k f = f' - \frac{k}{12} E_2 f,$1, together with multiplication by $D_k f = f' - \frac{k}{12} E_2 f,$2 and $D_k f = f' - \frac{k}{12} E_2 f,$3, and they satisfy commutation relations mirroring Ramanujan’s identities:

$D_k f = f' - \frac{k}{12} E_2 f,$4

Yamashita proves that $D_k f = f' - \frac{k}{12} E_2 f,$5 is isomorphic to the skew polynomial ring $D_k f = f' - \frac{k}{12} E_2 f,$6, that $D_k f = f' - \frac{k}{12} E_2 f,$7 and $D_k f = f' - \frac{k}{12} E_2 f,$8 are $D_k f = f' - \frac{k}{12} E_2 f,$9-bases, and that monic operators admit left and right division with remainder (Yamashita, 2018).

The same framework yields several structural statements about solution spaces. Monic MLDEs have a regular singularity only at $f'(z)=(1/(2\pi i))\,df/dz$0, and the modular Wronskian controls both the order of the equation and the leading exponents. Theorem 4.3 of Yamashita further shows that every quasimodular form of weight $f'(z)=(1/(2\pi i))\,df/dz$1 and depth $f'(z)=(1/(2\pi i))\,df/dz$2 satisfies a monic MLDE of weight $f'(z)=(1/(2\pi i))\,df/dz$3; in particular, quasimodularity is not external to MLDE theory but is intrinsic to it (Yamashita, 2018).

2. Null vectors, characters, and vector-valued modularity

In RCFT and VOA theory, MDEs arise from null-vector relations in the vacuum module. Gaberdiel and Keller prove that every modular differential equation of a rational conformal field theory comes from a null vector in the vacuum Verma module, using Zhu’s torus recursion to convert identities of the form

$f'(z)=(1/(2\pi i))\,df/dz$4

into a common differential equation for all characters,

$f'(z)=(1/(2\pi i))\,df/dz$5

(0804.0489). Their examples include the Yang–Lee model and the Monster CFT.

Bantay recasts the same problem in the language of vector-valued modular forms. If $f'(z)=(1/(2\pi i))\,df/dz$6 is a character vector transforming under a representation $f'(z)=(1/(2\pi i))\,df/dz$7, then the annihilator ideal in the universal ring of invariant differential operators determines all MLDEs satisfied by $f'(z)=(1/(2\pi i))\,df/dz$8. The Ising model provides a concrete case: its three characters obey a unique holomorphic third-order MLDE of the form

$f'(z)=(1/(2\pi i))\,df/dz$9

(Bantay, 2010).

The same null-vector mechanism extends beyond ordinary characters. In the 4d $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$0/VOA correspondence, the Schur index equals the vacuum character of the associated VOA, and a stress-tensor null of the form

$$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$1

suggests that the Schur index obeys a finite-order MLDE (Beem et al., 2017). This reinterpretation leads to the formula

$$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$2

where $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$3 is the smallest indicial exponent of the relevant MLDE, or of its twisted conjugate in half-integer-graded cases (Beem et al., 2017).

A related but distinct construction appears for Virasoro four-point sphere conformal blocks. By using the elliptic map $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$4, crossing transformations in the cross-ratio become modular transformations in $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$5, and the blocks satisfy second-order MLDEs for subgroups of the modular group. For all identical operators, the MLDE reduces to

$$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$6

and in that two-block case the resulting solutions lie on the level-two BPZ locus (Mahanta et al., 2022).

3. Jacobi-form MDEs and elliptic genera

For Jacobi forms, the natural differential operator is the heat operator. For fixed index $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$7, one sets

$$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$8

and the modularly corrected operator acting on $$\#2{0}{k}=1,\qquad \#2{n}{k}=D_{k+2n-2}\cdots D_{k+2}D_k.$$9 is

$k$0

An order-$k$1 Jacobi MDE is then an equation

$k$2

with modular coefficients $k$3 (Adler et al., 30 Sep 2025).

Cléry, Gritsenko, and Nikulin show that the basic weak Jacobi forms satisfy low-degree heat-operator equations. In particular, the elliptic genus of any $k$4 satisfies a degree-one equation; the elliptic genus of a $k$5 surface or any $k$6 satisfies a degree-three equation; and a general $k$7 elliptic genus satisfies a degree-five equation (Adler et al., 2022). Their framework also gives Kaneko–Zagier-type Jacobi equations, such as

$k$8

and

$k$9

(Adler et al., 2022).

A later classification at weight $n$0 and index $n$1 sharpens this picture. Theorem 3.1 and Theorem 3.2 imply stringent restrictions on third-order heat-operator equations, and in particular there are no weight-$n$2, index-$n$3 weak Jacobi forms satisfying a third-order MDE (Adler et al., 30 Sep 2025). For strict Calabi–Yau sixfolds, the minimal possible order is four, and exactly ten weak Jacobi forms in $n$4 satisfy order-four equations. For hyperkähler sixfolds of types $n$5, $n$6, and $n$7, the elliptic genera are generic points in coefficient space and satisfy explicit seventh-order equations (Adler et al., 30 Sep 2025).

This literature also shows that minimal order depends sensitively on geometry. The paper on six-dimensional elliptic genera distinguishes strict $n$8 and hyperkähler $n$9-folds by their $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$0 profiles and by the order of the smallest compatible MDE, and it identifies non-generic loci in the coefficient space $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$1 by two cubic plane curves, $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$2 and $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$3 (Adler et al., 30 Sep 2025). A plausible implication is that, in Jacobi settings, order is not merely a function of weight and index, but of the precise Fourier–Jacobi structure.

4. Flavored and refined modular differential equations

When characters depend on continuous flavor fugacities, the relevant equations are no longer ordinary differential equations in $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$4 but quasi-Jacobi systems in $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$5. Pan and Wang derive such flavored MDEs for affine Kac–Moody VOAs from a level-four null state

$$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$6

together with the Sugawara identity (Pan et al., 2023). The resulting equations involve the Ramanujan–Serre derivative in $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$7, flavor derivatives $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$8, and quasi-Jacobi Eisenstein series $$(g_0 \#2{n}{k}+g_1 \#2{n-1}{k}+\cdots+g_{n-1}D_k+g_n)f=0,$$9.

A central feature is “almost covariance” under modular $g_i\in M(l+2i)$0. For $g_i\in M(l+2i)$1, the weight-four flavored equation transforms as

$g_i\in M(l+2i)$2

so the $g_i\in M(l+2i)$3-orbit closes on the system of nulls and descendants rather than on a single equation (Pan et al., 2023). Requiring closure constrains the Lie algebra to the Deligne–Cvitanović exceptional series and the level to $g_i\in M(l+2i)$4 or $g_i\in M(l+2i)$5.

The same Jacobi-type structure appears in class-$g_i\in M(l+2i)$6 examples with surface defects. For $g_i\in M(l+2i)$7 theories $g_i\in M(l+2i)$8, flavored MDEs constructed from VOA nulls are satisfied not only by the refined Schur index but also by three families of defect indices: vortex defects, Gukov–Witten defects, and modular-transformed flavor defects (Zheng et al., 2022). The equations transform almost covariantly under modular transformations, and the solution spaces contain logarithmic solutions that the paper interprets as candidates for logarithmic VOA modules. In the $g_i\in M(l+2i)$9 $l=0$0 case, for instance, the unflavored limit of the weight-four system reduces to

$l=0$1

with indicial exponents $l=0$2 and $l=0$3 (Zheng et al., 2022).

A common misconception is that flavor refinement only perturbs the coefficients of an ordinary MLDE. The flavored literature shows a stronger statement: the equations become genuinely partial differential systems with Jacobi-group behavior, and modularity acts on the entire hierarchy of equations rather than on a single scalar operator (Pan et al., 2023, Zheng et al., 2022).

5. Schwarzian, invariant, and arithmetic-geometric formulations

A large branch of the subject studies second-order equations

$l=0$4

through the Schwarzian derivative. If $l=0$5 are linearly independent solutions, then $l=0$6 satisfies

$l=0$7

and conversely $l=0$8, $l=0$9 solve the linear equation when $g_0=1$0 is locally univalent (Besrour et al., 2024, Sebbar et al., 2011). In level-$g_0=1$1 theory, with $g_0=1$2, Besrour and Sebbar classify modular solutions in terms of $g_0=1$3-equivariant functions and finite monodromy images in $g_0=1$4, and prove a genus-$g_0=1$5 classification theorem expressed by explicit cusp-width conditions on the constants in

$g_0=1$6

(Besrour et al., 2024).

Sebbar and Sebbar treat the special family of equations with Eisenstein-series coefficients. They solve Riccati and Schwarzian forms associated with

$g_0=1$7

for several values of the parameter, using modular forms, modular functions, and equivariant forms. Their explicit equivariant example

$g_0=1$8

satisfies

$g_0=1$9

and the transformation law $D_k f = f' - \frac{k}{12} E_2 f,$00 (Sebbar et al., 2011).

Recent work pushes this approach into a complete classification in the reducible case. For

$D_k f = f' - \frac{k}{12} E_2 f,$01

reducible monodromy occurs precisely for $D_k f = f' - \frac{k}{12} E_2 f,$02 with $D_k f = f' - \frac{k}{12} E_2 f,$03. Saber and Sebbar construct all such solutions from the ansatz

$D_k f = f' - \frac{k}{12} E_2 f,$04

reduce residue cancellation to an algebraic system in the scaled $D_k f = f' - \frac{k}{12} E_2 f,$05-variables, and show that the roots are encoded by monic Jacobi-type orthogonal polynomials satisfying a Fuchsian equation (Besrour et al., 14 Aug 2025). A parallel paper shows how explicit algebraic systems

$D_k f = f' - \frac{k}{12} E_2 f,$06

generate infinite families of Schwarzian equations

$D_k f = f' - \frac{k}{12} E_2 f,$07

and therefore infinite families of second-order modular differential equations (Saber et al., 2023).

There is also a nonlinear invariant formulation. Opanasenko and Ferapontov show that every modular form satisfies an $D_k f = f' - \frac{k}{12} E_2 f,$08-invariant third-order nonlinear ODE of the form

$D_k f = f' - \frac{k}{12} E_2 f,$09

and a $D_k f = f' - \frac{k}{12} E_2 f,$10-invariant fourth-order consequence

$D_k f = f' - \frac{k}{12} E_2 f,$11

where $D_k f = f' - \frac{k}{12} E_2 f,$12 are differential invariants expressible via Rankin–Cohen brackets (Opanasenko et al., 2023). In their Jacobi extension, Jacobi forms satisfy involutive third-order PDE systems invariant under $D_k f = f' - \frac{k}{12} E_2 f,$13 (Opanasenko et al., 2023).

Arithmetic-geometric formulations supply a further variant. Golyshev and Vlasenko study modular determinantal equations of type $D_k f = f' - \frac{k}{12} E_2 f,$14 and $D_k f = f' - \frac{k}{12} E_2 f,$15. For non-degenerate modular $D_k f = f' - \frac{k}{12} E_2 f,$16 equations over $D_k f = f' - \frac{k}{12} E_2 f,$17, the $D_k f = f' - \frac{k}{12} E_2 f,$18-expansion

$D_k f = f' - \frac{k}{12} E_2 f,$19

of the analytic solution coincides, under their hypotheses, with the Fourier expansion of the weight-$D_k f = f' - \frac{k}{12} E_2 f,$20 newform attached to the spectral elliptic curve of the equation, so the coefficients $D_k f = f' - \frac{k}{12} E_2 f,$21 are multiplicative (Golyshev et al., 2012). This identifies a branch of MDE theory in which the differential equation, the modular form, and the elliptic curve determine one another.

6. Noncommutative modular operators and higher-genus generalization

The most recent generalization replaces commutative coefficient rings by Ore algebras over possibly noncommutative differential algebras. For a monic binomially normalized operator

$D_k f = f' - \frac{k}{12} E_2 f,$22

the gauge change $D_k f = f' - \frac{k}{12} E_2 f,$23 acts by conjugation $D_k f = f' - \frac{k}{12} E_2 f,$24, and one introduces the shifted derivation

$D_k f = f' - \frac{k}{12} E_2 f,$25

together with noncommutative Bell polynomials $D_k f = f' - \frac{k}{12} E_2 f,$26 and covariant Bell polynomials $D_k f = f' - \frac{k}{12} E_2 f,$27 (Jafari, 8 Mar 2026).

The first structural result is a closed Miura/oper expansion:

$D_k f = f' - \frac{k}{12} E_2 f,$28

where the $D_k f = f' - \frac{k}{12} E_2 f,$29 are gauge covariants. Theorem 5.5 gives the universal formula

$D_k f = f' - \frac{k}{12} E_2 f,$30

Under reparametrization $D_k f = f' - \frac{k}{12} E_2 f,$31 with central jets, these covariants acquire higher-Schwarzian anomaly terms, and the corrected Wilczynski covariants

$D_k f = f' - \frac{k}{12} E_2 f,$32

transform projectively:

$D_k f = f' - \frac{k}{12} E_2 f,$33

(Jafari, 8 Mar 2026).

The paper globalizes this calculus to Riemann surfaces and modular connections. In genus $D_k f = f' - \frac{k}{12} E_2 f,$34, one defines a modular connection $D_k f = f' - \frac{k}{12} E_2 f,$35 of eccentricity $D_k f = f' - \frac{k}{12} E_2 f,$36 and the covariant derivative

$D_k f = f' - \frac{k}{12} E_2 f,$37

which recovers Serre’s derivative when $D_k f = f' - \frac{k}{12} E_2 f,$38 in the scalar $D_k f = f' - \frac{k}{12} E_2 f,$39 case. Modular linear differential operators are then characterized by the equivariance condition

$D_k f = f' - \frac{k}{12} E_2 f,$40

(Jafari, 8 Mar 2026). The same construction yields noncommutative Rankin–Cohen brackets and Maurer–Cartan realizations in genus $D_k f = f' - \frac{k}{12} E_2 f,$41, and, in higher genus, Siegel modular connections

$D_k f = f' - \frac{k}{12} E_2 f,$42

and $D_k f = f' - \frac{k}{12} E_2 f,$43-equivariant derivations

$D_k f = f' - \frac{k}{12} E_2 f,$44

on Siegel space (Jafari, 8 Mar 2026).

This noncommutative generalization changes the status of MDEs in two ways. First, it shows that gauge covariance, projective covariance, and modular covariance can be organized uniformly through oper-style invariants $D_k f = f' - \frac{k}{12} E_2 f,$45 and Wilczynski covariants $D_k f = f' - \frac{k}{12} E_2 f,$46. Second, it extends the subject from scalar modular forms to matrix-valued, bundle-valued, and higher-genus automorphic objects. A plausible implication is that the classical MLDE, the Jacobi heat equation, and the flavored Jacobi PDE are all low-dimensional manifestations of a broader invariant calculus on differential operators over automorphic differential algebras.