Aomoto–Gelfand Hypergeometric Functions

• Aomoto–Gelfand hypergeometric functions are special multivariable functions defined via twisted period integrals that generalize the classical Gauss function to configuration spaces.
• They feature explicit series expansions and recursion relations derived from combinatorial lattice structures, facilitating analytic continuation and asymptotic analysis.
• These functions bridge representation theory, integrable systems, Feynman integrals, and arithmetic geometry, providing a unifying framework for holonomic PDEs.

Aomoto–Gelfand hypergeometric functions are a class of multivariable special functions defined as solutions to holonomic systems of partial differential equations associated with combinatorial and geometric data, most classically formulated by Gelfand, Kapranov, and Zelevinsky (GKZ). On one hand, these functions generalize the classical Gauss hypergeometric function to the context of configuration spaces and Grassmannians; on the other, they provide a unifying language for the paper of period integrals, monodromy, representation theory, and integrable systems. Central to the Aomoto–Gelfand framework is the realization of these functions as (multi-valued) twisted period integrals on the complement of a hyperplane arrangement, with powerful links to twisted de Rham cohomology and homology. This approach supports rich algebraic structures—such as contiguity relations, monodromy representations, and deep asymptotics—while interfacing with the arithmetic theory (finite fields, l-adic sheaves), representation theory (Gelfand-Tsetlin bases, Lie algebra modules), integrable hierarchies (Toda equations), and Feynman diagrammatics through explicit analytic formulas.

1. Foundational Structures: Definition and Integral Representation

Aomoto–Gelfand hypergeometric functions are defined in the context of a matrix $A \in \mathbb{Z}^{d \times n}$, viewed as encoding a hyperplane arrangement (for instance, the configuration of linear forms $L_j(t) = \text{linear in } t$). The prototypical integral representation is: $$F(z) = \int_{\gamma} \prod_{j=1}^{n} L_j(t;z)^{\alpha_j} \omega,$$ where $z$ encodes the data of the arrangement, $\gamma$ is a twisted cycle in a suitable relative homology group, and $\omega$ is a form representing a class in twisted de Rham cohomology (Abe, 2015, Abe, 2018).

This construction organizes solutions of the A-hypergeometric (GKZ) system associated to $A$ and parameter vector $\beta$: $$\begin{cases} \left( \sum_{j=1}^n a_{ij} x_j \frac{\partial }{\partial x_j} - \beta_i \right) f(x) = 0, & i=1,\ldots,d, \ \Box_u f(x) = 0, & u \in \ker_\mathbb{Z}(A), \end{cases}$$ where the "box" operators $\Box_u$ encode binomial difference conditions among monomials (Abe, 2018, Abe, 2015, Gerasimov et al., 2022).

This system guarantees that the solution space is holonomic and of rank equal to the normalized volume of the associated polytope defined by $A$, with regular singularities along the hyperplane arrangement.

Twisted cohomology and homology play a critical role: the period integral $\int_\gamma \Phi^\alpha \omega$ is interpreted as the pairing

$$H_k(X, \mathcal{L}^\vee) \times H^k(X, \mathcal{L}) \to \mathbb{C},$$

where $X$ is the complement of the arrangement in projective space, $\mathcal{L}$ is the associated rank-one local system, and the basis of solutions is naturally indexed by the twisted homology (Abe, 2015, Abe, 2018, Matsubara-Heo, 2019).

2. Series Expansions, Recursion, and Explicit Formulas

Aomoto–Gelfand functions admit explicit series expansions generalizing the classical ${}_2F_1$, Appell, and Lauricella series. For instance, the GKZ (or $\mathcal{A}$–hypergeometric) $\Gamma$-series associated to a shifted lattice $\gamma+B$ is

$$\mathcal{F}_\gamma(z) = \sum_{t \in \mathbb{Z}_{\geq 0}^k} \frac{z^{\gamma + Bt}}{(\gamma + Bt)!},$$

where $B$ is a basis of relations among the columns of $A$ and $t$ runs over nonnegative multi-indices subject to combinatorial constraints (Artamonov, 2022, Artamonov, 2020, Artamonov, 2021).

Special cases such as the Lauricella $F_D$, $F_C$, Appell $F_4$, and their multidimensional generalizations are admitted as explicit members of the theory (Luo et al., 2011, Duhr et al., 2023, Watanabe et al., 2013).

Integral representations, such as those derived from Euler–type or Euler–Laplace interpolants, produce combinatorial and analytic formulas for special values, expansions, and reductions (Matsubara-Heo, 2019, Zakrzewski et al., 14 Jan 2025).

Recursion relations, often generated via the action of contiguity operators, connect functions with shifted parameters (such as up/down steps in exponent vectors), which are crucial for analytic continuation, reduction formulas, and computational schemes for practical evaluation (Okuyama et al., 2022, Nishiyama, 2010, Artamonov, 2022).

3. Relation to Representation Theory and Lie Algebras

A fundamental bridge links Aomoto–Gelfand functions to representation theory. Notably, basis vectors in the Gelfand–Tsetlin scheme or in certain Zhelobenko-type constructions can be explicitly realized as A–hypergeometric $\Gamma$-series (polynomial or rational) (Artamonov, 2020, Artamonov, 2022, Artamonov, 2021). In these settings, the combinatorial structure of GT diagrams maps directly to shift vectors and summation indices in the lattice of relations underlying the series expansion.

The action of Lie algebra generators translates to explicit recursions or differential operators acting on Aomoto–Gelfand functions. Antisymmetrized versions of the GKZ system arise in the representation theory context, and the so-called hypergeometric constants (values of generalized Horn or $\Gamma$-series at unity) encode transformation coefficients between different bases (Artamonov, 2022).

Matrix element representations, in particular those associated with oscillator Lie algebras $\mathfrak{L}_N$, reveal the solution space as spans of Whittaker-type functions or matrix elements between vacuum states, unifying the analytic and algebraic approaches (Gerasimov et al., 2022).

4. Applications: Integrable Systems, Feynman Integrals, and Arithmetic

Aomoto–Gelfand hypergeometric functions emerge in nonlinear integrable systems, most prominently as tau-function solutions of the two-dimensional Toda–Hirota equation (2dTHE) and its variants (Kimura, 5 Jun 2025, Kimura, 13 Jun 2025). The construction proceeds by identifying solutions to Euler–Poisson–Darboux–type systems arising from the GKZ differential equations, applying Laplace and Darboux transformations, and systematically exploiting contiguity relations generated by root vector shifts. The resulting solutions can be interpreted within the tau-function hierarchy framework, linking the algebraic structures underpinning the Gelfand hypergeometric system and the dynamical symmetries of integrable hierarchies.

In quantum field theory and mathematical physics, Feynman integrals in two or more dimensions, especially those with massless propagators, can be expressed in terms of single-valued analogues of Aomoto–Gelfand hypergeometric functions. The key construction is the pairing of holomorphic and anti-holomorphic (twisted) periods via intersection numbers in twisted homology, producing explicit analytic expressions for amplitudes in terms of generalized Lauricella $F_D^{(r)}$ and generalized hypergeometric ${}_{L+1}F_L$ functions (Duhr et al., 2023).

In arithmetic geometry, the $\ell$-adic GKZ hypergeometric sheaf provides an arithmetic avatar of Aomoto–Gelfand theory over finite fields. Here, the cohomology sheaf is perverse and pure (under non-resonance), and traces of Frobenius yield hypergeometric exponential sums. The weight filtration of these sheaves controls sharp estimates for sums of arithmetic significance (e.g., zeta functions of K3 surfaces) (Fu, 2012, Otsubo, 2021).

5. Algebraic and Analytical Structures: Contiguity, Monodromy, and Stokes Phenomena

A salient feature of Aomoto–Gelfand systems is the existence of a full system of contiguity relations—recurrence formulas that shift parameters in either the exponents of the forms or the deformation variables. For example, operators $S(B; –a_{1k})$ act as down-shifts in exponent vectors, producing parameter-shifted solutions,

$$S(B; -a_{1k}) \cdot \Phi(\gamma; x) = a_k \cdot \Phi(\gamma - a_{1k}; x).$$

These operators are representation-theoretically generated by the action of root vectors associated to $\mathfrak{gl}(N)$ and correspond to explicit first-order differential or difference operators (Nishiyama, 2010, Kimura, 13 Jun 2025, Artamonov, 2022).

Intersection pairings in the twisted (co)homology space, in tandem with the period matrix, govern the monodromy representation of solution spaces and, in the analytic context, the Stokes matrices and connection formulas in their asymptotic expansion. The expansion of quadratic relations in terms of combinatorial data, such as bipartite graphs, further enriches the algebraic structure (Matsubara-Heo, 2019).

Asymptotics, especially WKB expansions, are accessible via stationary phase or saddle point analysis of the underlying multidimensional integrals. The behavior under perturbation, confluence, and parameter variation is governed by this analytic infrastructure, as seen in the context of Airy or confluently degenerate equations (Zakrzewski et al., 14 Jan 2025).

6. Extensions: Incomplete, Logarithmic, and Non-Gaussian Systems

Aomoto–Gelfand theory accommodates inhomogeneities and parameter deformations. Incomplete hypergeometric systems—in which the underlying integral has non-fully spanning integration domains—yield inhomogeneous versions of the GKZ system, described by boundary-term evaluations. These naturally generalize incomplete classical integrals, such as the incomplete Gauss or elliptic integrals, to the multivariable setting (Nishiyama, 2010).

Logarithmic solutions correspond to degenerate exponents and require the development of perturbation techniques to construct explicit bases of solutions, as in the logarithmic A-hypergeometric series method. Here, the combinatorial and algebraic structure of the lattice of relations among exponents (and associated monomials) determines both the exponents and the function space of logarithmic series (Okuyama et al., 2022).

A further generalization replaces the usual Euler or Laplace integrands with arbitrary (non-Gaussian) exponential-polynomial objects. The theory of non-Gaussian integrals provides a broader analytic setting for Aomoto–Gelfand hypergeometric functions, encompassing and simplifying the GKZ framework by enabling new forms of differential operators and hierarchical expansions (Stoyanovsky, 2020).

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The Aomoto–Gelfand hypergeometric functions unify and extend classic special function theory, representation theory, arithmetic geometry, and integrable systems under a comprehensive analytic and algebraic umbrella. Their paper elucidates deep features of monodromy, period maps, and solution space geometry, and supports a host of explicit analytic constructions valuable across pure and applied mathematics.