GKZ Systems in Algebra and Geometry

• Gelfand–Kapranov–Zelevinsky systems are holonomic PDEs defined by an integer matrix and a parameter vector that unify hypergeometric functions in several variables.
• They connect algebraic geometry, toric varieties, and combinatorial structures like Newton polytopes to characterize solution spaces and singularities.
• Applications of GKZ systems span mirror symmetry, Feynman integrals, and arithmetic geometry, utilizing D-module theory and cohomological techniques.

Gelfand–Kapranov–Zelevinsky (GKZ) systems describe a class of holonomic partial differential equations associated with torus actions on affine spaces. These systems unify the theory of hypergeometric functions in several complex variables, connecting algebraic geometry, representation theory, D-module theory, combinatorics of Newton polytopes, and arithmetic geometry. The structural underpinnings of GKZ systems are determined by monomial configurations, resulting in deep links to toric varieties, discriminants, cohomological interpretations, and applications in mathematical physics, notably in mirror symmetry and Feynman integral computations.

1. Definition and Algebraic Structure

GKZ systems, or $A$-hypergeometric systems, are defined by an integer matrix $A = [a_{ij}]$ specifying the exponents of a set of monomials and by a parameter vector $\beta \in \mathbb{C}^d$. Given $n$ variables $x_1,\ldots,x_n$ and $A\in \mathbb{Z}^{d\times n}$ (columns $a_j$), the system is described by:

• Euler operators:

$$E_i = \sum_{j=1}^n a_{ij} x_j \partial_{x_j} - \beta_i, \quad 1\leq i \leq d$$

• Toric (box) operators for $u\in\ker_{\mathbb{Z}}A$:

$$\Box_u = \prod_{u_j > 0} \partial_{x_j}^{u_j} - \prod_{u_j < 0} \partial_{x_j}^{-u_j}$$

The GKZ $D$-module is $M_A(\beta) = D / H_A(\beta)$, where $D$ is the Weyl algebra and $H_A(\beta)$ is the left ideal generated by all Euler and box operators. The system is holonomic, and for generic $\beta$ the module's (holonomic) rank equals the normalized volume of the convex hull of $A$. For specific "resonant" or rank-jumping $\beta$, the solution space dimension may increase, governed by local cohomology of the associated semigroup ring (Reichelt et al., 2020).

Generalizations to coefficient modules over arbitrary finitely generated abelian grading groups introduce $A$-twisted-toric modules, widening the applicability and leading to "better behaved" GKZ systems with constant rank under particular coefficient choices (Reichelt et al., 1 Feb 2024).

2. Geometric Interpretation and Cohomological Realizations

GKZ systems are deeply intertwined with the geometry of toric varieties and period integrals. The key geometric object is the Newton polytope, the convex hull of the monomial exponents.

A fundamental result is the identification of GKZ solutions with relative cohomology groups of pairs arising from toric varieties and their divisors. For a matrix $A$ associated to a toric variety $X$, and a parameter $x$ specifying a complete intersection $Y_x \subset X$, the solution space is isomorphic to a relative cohomology group:

$$\operatorname{Sol}^\circ(M_{A,\beta_0}, \mathcal{O}_{V, x}) \cong H_n(U_x, U_x \cap D)$$

where $U_x = X \setminus Y_x$ and $D$ is the union of torus-invariant divisors (Lee et al., 2019). This identification frames periods and mirror maps in Calabi–Yau mirror symmetry as GKZ solutions and allows characterization of "rank one points," crucial for applications to moduli and degeneration phenomena.

3. Combinatorics: Newton Polytopes, Discriminants, and Matroid Theory

The behavior of solutions, singular loci, and discriminants of GKZ systems is governed by the geometry and combinatorics of Newton polytopes. The mixed volume of the polytopes determines the generic number of solutions (Bernstein's theorem). More refined invariants, such as the $A$-discriminant and principal $A$-determinant, describe the global singular structure of the associated D-module (Klausen, 2021).

Central to comprehension are the concepts of Minkowski linear independence and the associated Minkowski matroid. Bases of this matroid correspond to maximally independent collections of Newton polytopes, circuits to minimal dependencies (defect $-1$), and cyclics to essential tuples appearing in the structure of discriminants. The irreducibility and number of components of discriminants can be deduced from the matroid structure and the classification of circuits and cyclics (Pokidkin, 3 Sep 2025):

Combinatorial Structure	Defect	Significance
Basis (BK-tuple)	$\geq 0$	Maximal linearly independent set
Circuit	$-1$	Minimal dependency, factors discriminant
Cyclic/Essen. tuple	varies	Controls singularity degenerations

Explicit factorization results for the discriminant, such as

$$E_A(f) = \pm \prod_{\tau \subset \operatorname{Newt}(f)} \Delta_{A\cap\tau}(f_\tau)^{\mu(A,\tau)}$$

highlight how faces and subtuples of Newton polytopes control the singularities and reducibility of GKZ discriminants.

4. Representation Theory and Dualities

GKZ functions admit realization as matrix elements in representations of both reductive and non-reductive Lie algebras. Specifically, they can be represented as matrix coefficients in oscillator Lie algebras, establishing a duality with traditional constructions using reductive algebras (e.g., for Whittaker functions of $GL_{\ell+1}(\mathbb{R})$) (Gerasimov et al., 2022). Such duality extends to the categorical mirror symmetry setting, where better-behaved GKZ systems (bbGKZ) feature explicit correspondences with $K$-theory of toric Deligne–Mumford stacks and analytic Poincaré dualities of solution spaces (Borisov et al., 2019).

In exceptional Lie algebra representation theory, certain GKZ-type systems serve as functional models for irreducible representations, with the solutions forming bases corresponding to Gelfand–Tsetlin-type diagrams under suitable chain reductions of subalgebras (e.g., $\mathfrak{g}_2 \supset \mathfrak{sl}_3$) (Artamonov, 13 Oct 2025).

5. Arithmetic and Cohomological Aspects

The $\ell$-adic and $p$-adic analogues of GKZ systems translate fundamental structures into arithmetic geometry. Over finite fields, the $\ell$-adic GKZ hypergeometric sheaf is a perverse, pure, and, under a nonresonance condition, irreducible object whose Frobenius traces recover hypergeometric exponential sums, including and generalizing Gauss and Kloosterman sums (Fu, 2012). The rank and weight filtration are controlled by the combinatorics of the associated convex polytope. For $p$-adic settings, the GKZ hypergeometric complex is constructed as an overconvergent de Rham complex, yielding an overconvergent $F$-isocrystal on a nondegenerate locus and providing trace formulas for $L$-functions of exponential sums (Fu et al., 2018).

Frobenius structures on GKZ systems are further analyzed via Dwork's theory, with explicit descriptions of degeneration given in terms of the Morita $p$-adic gamma function, illuminating congruence properties and local monodromy of arithmetic differential equations (Kedlaya, 2019).

6. Applications to Mathematical Physics and Mirror Symmetry

GKZ systems provide a structural and computational backbone in several domains of mathematical physics:

• Quantum Field Theory and Feynman Integrals: Feynman parameter integrals in the Lee–Pomeransky or Mellin–Barnes representation can be identified as solutions to GKZ systems. Horn-type hypergeometric series representations, derived via triangulations of Newton polytopes, allow for efficient computation and analytic continuation, with the convergence domains, singular loci (Landau varieties), and branch structure expertly controlled by the $A$-hypergeometric machinery (Klausen, 2019, Feng et al., 2019, Klausen, 2021, Weinzierl, 2022).
• Computation of Master Integral Differential Equations: D-module and Macaulay matrix techniques enable systematic derivation of the Pfaffian systems satisfied by master integrals, reducing the calculation of high-order perturbative contributions to linear algebra in the rational Weyl algebra (Chestnov et al., 2022, Munch, 2022).
• Cosmological Correlators: Reduction algorithms based on the resonance of GKZ parameters (resonant sets and reduction operators) enable partitioning large GKZ systems into manageable subsystems that directly reflect central physical properties such as locality (Grimm et al., 20 Sep 2024).
• Toric Mirror Symmetry: GKZ systems capture period integrals of Landau–Ginzburg models mirror to toric varieties and govern Dubrovin-type quantum connections, with Hodge-theoretic refinements tracking the deep connection between the GKZ D-module and quantum cohomology (Reichelt et al., 2020).

7. Extensions and Generalizations

Recent work extends the notion of GKZ systems to actions of more general commutative linear algebraic groups $G = TU$, incorporating both tori and unipotent parts via Fourier–orbit constructions. Generic holonomicity is established via symbolic moment ideals, and new families of systems exhibiting Airy-type irregularities are produced in the presence of unipotents. Lower bounds and explicit holonomic ranks are determined using symbolic tools, with the classical volume formula recovered in pure torus (classical GKZ) cases (Okuyama, 4 Sep 2025).

Extensions to GKZ systems with grading by arbitrary finitely generated abelian groups require the introduction of twisted-toric modules and adaptation of the Euler-Koszul complex, leading to the existence of regular holonomic systems and robust duality properties even for groups with torsion (Reichelt et al., 1 Feb 2024).

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Through this synthesis, GKZ systems emerge as a unifying structure in algebraic analysis, arithmetic geometry, combinatorics, representation theory, and mathematical physics, providing both the conceptual framework and computational techniques necessary for diverse applications.