Middle Convolution Techniques

• Middle convolution is a functorial transformation on linear differential equations, local systems, and D‑modules that uses convolution with a rank‑one Kummer object followed by a middle extension.
• Its matrix realization for Fuchsian systems employs explicit block constructions and quotienting by invariant subspaces to compute rank changes and control rigidity.
• Analytically, the operation connects Euler-type integral transforms with q‑difference, irregular, arithmetic, and Hodge‑theoretic settings, offering broad applicability.

Middle convolution is a functorial transformation on linear differential equations, local systems, perverse sheaves, and holonomic $\mathcal{D}$-modules that originated in Katz’s study of rigid local systems and was later reformulated in explicit algebraic forms for Fuchsian systems, extended to Hodge-theoretic and arithmetic settings, and adapted to $q$-difference, logarithmic, irregular, braid-theoretic, Lie-algebraic, and Pfaffian contexts. In the differential setting it is built from convolution with a rank-one Kummer object followed by a middle, or intermediate, extension; in matrix form it is realized by explicit block constructions and quotienting by canonical invariant subspaces. Across these realizations, middle convolution is repeatedly used to control rigidity, transform local data, construct new systems from old ones, and relate analytic integral transforms to algebraic and categorical operations (Dettweiler et al., 2012, Bibilo et al., 2015, Sakai et al., 2014).

1. Katzian formulation and basic definitions

In the $\mathscr{D}$-module framework, middle convolution with a Kummer module is written

$$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$

where $\mathcal{K}_{\lambda_0}$ is the Kummer module

$$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$

and “middle” means the image of the natural map

$M\star_!L_\chi \to M\star_*L_\chi$

in the corresponding convolution formalism (Martin, 2018, Dettweiler et al., 2012, Dettweiler, 2015). This realization places middle convolution simultaneously in the theory of regular holonomic $\mathscr{D}$-modules, perverse sheaves, and local systems.

Katz’s original viewpoint treats middle convolution as one of the basic operations, together with tensoring by rank-one local systems, in the construction of rigid local systems on punctured projective lines. Several later works emphasize that cohomologically rigid local systems can be built from rank-one data by iterating such operations, and that this procedure is compatible with motivic realizations in the cases under consideration (Patrikis, 2014, Dettweiler et al., 2012). In this sense, middle convolution is both a structural operation and an algorithmic device.

A persistent feature across formulations is the use of a rank-one Kummer factor. In additive settings the parameter is usually written $\lambda$ or $\mu$; in multiplicative settings it appears as $q$0 or $q$1; in the $q$2-difference setting it is encoded by $q$3 (Bibilo et al., 2015, Sakai et al., 2014). Although the language varies, the formal role is analogous: middle convolution alters local monodromy or local exponents through a controlled convolution-and-quotient procedure.

2. Matrix realizations for Fuchsian systems

For Fuchsian differential equations, middle convolution admits an explicit linear-algebraic realization due to the Dettweiler–Reiter approach. Starting from a Fuchsian system

$q$4

one forms block “convolution matrices” $q$5 or $q$6, then quotients by canonical invariant subspaces built from kernels of the residue matrices and their global relations (Bibilo et al., 2015, Bibilo et al., 2015). In the additive version, middle convolution $q$7 acts on tuples of residue matrices; in the multiplicative version, $q$8 acts on tuples of monodromy matrices (Bibilo et al., 2015).

A standard feature of this realization is reduction by subspaces usually denoted $q$9 and $\mathscr{D}$0, or variants thereof. In the resonant Fuchsian setting, for instance, one defines $\mathscr{D}$1 from $\mathscr{D}$2, takes $\mathscr{D}$3, and sets $\mathscr{D}$4 before passing to the quotient $\mathscr{D}$5 (Bibilo et al., 2015). This quotient is what distinguishes middle convolution from the unreduced convolution construction.

The multiplicative version carries an explicit dimension formula. If $\mathscr{D}$6, then after $\mathscr{D}$7 the size $\mathscr{D}$8 of the new tuple is

$\mathscr{D}$9

(Bibilo et al., 2015). This formula makes the rank change computable from the original monodromy tuple.

Rigidity is a central invariant in this framework. The literature summarized here repeatedly states that rigidity index is preserved by middle convolution, a fact used both in the classical Fuchsian theory and in the $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$0-difference analogue (Bibilo et al., 2015, Sakai et al., 2014). At the same time, the deformation-theoretic behavior depends on resonance. For non-resonant Fuchsian systems, middle convolution preserves Schlesinger’s deformation equations; for resonant systems, Bolibruch-type non-Schlesinger deformations are not preserved in general (Bibilo et al., 2015). This distinction is one of the main caveats in extending the regular theory beyond the generic non-resonant case.

3. Integral-transform interpretation and analytic constructions

Middle convolution is not only an algebraic operation on tuples of matrices; it is also realized analytically by Euler-type integral transforms. In the Fuchsian setting, if $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$1 is a solution of the original system, then the transformed solution is obtained from contour integrals whose kernels contain powers of $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$2, often over Pochhammer-type contours (Takemura, 2008). This integral-transform interpretation is the differential analogue of the categorical convolution with a Kummer object.

For rank-two Fuchsian systems with four singularities, middle convolution is directly related to integral transformations of Heun’s equation. One explicit form is

$$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$3

which sends a solution of one Fuchsian equation to a solution of another with transformed parameters; under specialization to the Heun locus in the space of initial conditions of Painlevé VI, this becomes an integral transformation of Heun’s equation itself (Takemura, 2008). When the convolution parameter is an integer, the corresponding transform degenerates to a differential-operator realization, identified with generalized Darboux transformations in the cited work (Takemura, 2008).

The same analytic/algebraic duality underlies constructive approaches to the Riemann–Hilbert problem. A general scheme proceeds by applying multiplicative middle convolution to the prescribed monodromy tuple, solving the simpler transformed Riemann–Hilbert problem by explicit methods, and then reconstructing the original system by inverse additive middle convolution. In diagrammatic form,

$$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$4

yielding constructive solutions for classes of Fuchsian systems, including examples with four singular points (Bibilo et al., 2015). In this role, middle convolution functions as a reduction step that moves a monodromy problem into a more tractable region.

4. The $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$5-difference analogue

A $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$6-analogue of middle convolution was introduced for linear $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$7-difference equations with rational coefficients. In the basic form,

$$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$8

with $$\mathrm{MC}_{\lambda_0}(M)=M*_{\mathrm{mid}}\mathcal{K}_{\lambda_0},$$9, one constructs $\mathcal{K}_{\lambda_0}$0-convolution matrices $\mathcal{K}_{\lambda_0}$1 from the tuple $\mathcal{K}_{\lambda_0}$2, defines subspaces

$\mathcal{K}_{\lambda_0}$3

and sets

$\mathcal{K}_{\lambda_0}$4

(Sakai et al., 2014). This is the $\mathcal{K}_{\lambda_0}$5-middle convolution in the Sakai–Yamaguchi theory.

The solution-level realization is a $\mathcal{K}_{\lambda_0}$6-Euler, or Jackson-integral, transform:

$\mathcal{K}_{\lambda_0}$7

with Jackson integral

$\mathcal{K}_{\lambda_0}$8

This establishes the same duality between matrix convolution and integral transformation that is familiar in the differential case (Sakai et al., 2014). A later reformulation uses

$\mathcal{K}_{\lambda_0}$9

introducing a base point $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$0 to sharpen convergence analysis (Arai et al., 2022).

The foundational properties established for the $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$1-analogue closely parallel the classical case. If $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$2 is a Fuchsian type $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$3-difference equation, then $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$4 is again Fuchsian type; under technical conditions $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$5, irreducibility is preserved; and under the same hypotheses the rigidity index is preserved (Sakai et al., 2014). Later reformulations emphasize additivity under composition,

$$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$6

a property singled out as a merit of the reformulated theory (Arai et al., 14 Mar 2025).

Applications in the cited literature include $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$7-hypergeometric equations, variants of $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$8-hypergeometric equations of higher order, the linear problem associated with $$\mathcal{K}_{\lambda_0}=\mathscr{D}_{\mathbb{A}^1}/\mathscr{D}_{\mathbb{A}^1}\cdot (t\partial_t-\gamma_0), \qquad \lambda_0=\exp(-2i\pi\gamma_0),\ \gamma_0\in(0,1),$$9-Painlevé VI, and integral transformations for the $M\star_!L_\chi \to M\star_*L_\chi$0-Heun equation (Arai et al., 2022, Sasaki et al., 2022, Arai et al., 14 Mar 2025). In the $M\star_!L_\chi \to M\star_*L_\chi$1-Painlevé VI setting, the transformation is further interpreted in terms of affine Weyl group symmetry (Sasaki et al., 2022).

5. Hodge-theoretic and arithmetic behavior

Middle convolution has a developed Hodge theory for irreducible variations of polarized complex Hodge structure on punctured affine lines. In this setting, one tracks nearby cycles, vanishing cycles, Hodge numbers, and degrees of Hodge bundles under convolution with Kummer objects. For an irreducible variation $M\star_!L_\chi \to M\star_*L_\chi$2, one formula established in the literature is

$M\star_!L_\chi \to M\star_*L_\chi$3

together with explicit transformation laws for local Hodge data and bundle degrees (Dettweiler et al., 2012). These results upgrade Katz’s algorithm by making the Hodge-theoretic effect of each middle convolution step explicit.

Subsequent work computes the behavior of Hodge data under additive middle convolution in greater generality. One paper gives the nearby-cycle local Hodge numerical data at infinity without assuming scalar monodromy at infinity, thereby generalizing earlier results of Dettweiler and Sabbah (Martin, 2018). Another computes explicit formulas for global Hodge numbers, degrees, local vanishing cycles, and local monodromy at infinity under additive middle convolution for irreducible variations of polarized complex Hodge structures (Dettweiler et al., 2018). In particular, for a generic Kummer module $M\star_!L_\chi \to M\star_*L_\chi$4, one has

$M\star_!L_\chi \to M\star_*L_\chi$5

in the formulation of that paper (Dettweiler et al., 2018).

Arithmetic aspects appear in several forms. In the $M\star_!L_\chi \to M\star_*L_\chi$6-adic setting, middle convolution is related to the $M\star_!L_\chi \to M\star_*L_\chi$7-adic Fourier transform, and the local monodromy and determinant of $M\star_!L_\chi \to M\star_*L_\chi$8 are described explicitly in the tame case using Laumon’s local Fourier theory and local $M\star_!L_\chi \to M\star_*L_\chi$9-constants (Dettweiler, 2015). One consequence stated there is that an at most quadratic determinant is often preserved under $\mathscr{D}$0 if $\mathscr{D}$1 is quadratic (Dettweiler, 2015).

For holonomic $\mathscr{D}$2-modules over the field of algebraic numbers, middle convolution is shown to preserve the categories of $\mathscr{D}$3-connections, globally nilpotent connections, and almost everywhere nilpotent connections, within a six-functor formalism on derived categories of arithmetic type (Wakabayashi, 2023). That stability is then used to prove equivalences between these arithmetic properties for rigid Fuchsian systems, giving an affirmative answer, in the cited rigid case, to the conjecture described by André and Baldassarri (Wakabayashi, 2023).

A different arithmetic line studies degenerating principal variations of Hodge structure arising from Katz’s middle convolution. In dimensions $\mathscr{D}$4 through $\mathscr{D}$5, the cited work constructs principal VHS by iterated convolution and quadratic twists, computes limiting periods explicitly, and exhibits extension classes that are rational multiples of zeta values; in the $\mathscr{D}$6 case, the Mumford–Tate group is of type $\mathscr{D}$7 (Jr. et al., 2014). Related work on generalized Kuga–Satake theory uses middle convolution to construct rigid local systems with motivic significance beyond the Tannakian category generated by abelian varieties (Patrikis, 2014).

6. Irregular, braid-theoretic, and Lie-algebraic extensions

Middle convolution has been extended beyond regular singular Fuchsian systems. For differential equations with irregular singularities, a tentative definition of the index of rigidity was introduced together with a middle convolution operation on enlarged modules of the form $\mathscr{D}$8, reduced by explicit subspaces $\mathscr{D}$9:

$\lambda$0

Under assumptions including semisimplicity and $\lambda$1, the index is preserved, systems of positive index can be reduced to rank one by iterated middle convolution and addition, and index-zero systems reduce to a finite list of terminal patterns (Takemura, 2010).

For linear Pfaffian systems with irregular singularities, a 2025 development introduces the middle Laplace transform as an algebraic counterpart of the Laplace transform in Katz theory and defines middle convolution as

$\lambda$2

The resulting theory generalizes Haraoka’s construction from logarithmic singularities to systems with irregular singularities and establishes invertibility, irreducibility, and additivity

$\lambda$3

(Adachi, 3 Feb 2025).

Another line of development identifies middle convolution with generalized Long–Moody constructions. The Katz–Long–Moody functor

$\lambda$4

extends middle convolution from $\lambda$5-local systems to broader semidirect-product settings such as $\lambda$6, including $\lambda$7-bundles, fiber-type hyperplane arrangements, and link complements in the solid torus (Hiroe et al., 2023). In the KZ-type setting, multiplicative middle convolution for the pure braid group is shown to correspond to the Katz–Long–Moody construction, to preserve unitarity relative to a Hermitian matrix, and to admit an algorithm determining the signature of the induced Hermitian form (Negami, 19 Mar 2025).

A 2026 Lie-algebraic analogue pushes the construction to modules over free Lie algebras, Drinfeld–Kohno Lie algebras, and holonomy Lie algebras of complements of hyperplane arrangements. In that framework, middle convolution is defined homologically by a cokernel of maps

$\lambda$8

recovers the Dettweiler–Reiter additive middle convolution as a special case, is compatible with Haraoka’s convolution for logarithmic connections, and is related by a Riemann–Hilbert correspondence to middle convolution for local systems on hyperplane-complement arrangements (Hiroe, 11 May 2026).

Taken together, these developments present middle convolution not as a single construction tied only to Fuchsian systems, but as a family of closely related functors whose common structural features are convolution with a rank-one object, passage to a middle quotient or image, and the preservation or computable transformation of rigidity, local invariants, and representation-theoretic structure across a wide range of analytic, algebraic, arithmetic, and geometric settings.