58 pages, 2 figures, Latex2e, to appear in the proceedings of the Summer School Algebraic Geometry and Hypergeometric Functions in Istanbul in June 2005
This text is based on lectures by the author in the Summer School `Algebraic Geometry and Hypergeometric Functions' in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the `geography' of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.
We study a pair of conjectures on better behaved GKZ hypergeometric systems of PDEs inspired by Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities. We prove the conjectures in the case of dimension two.
We consider a version of the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinski (GKZ) suited for the case when the underlying lattice is replaced by a finitely generated abelian group. In contrast to the usual GKZ hypergeometric system, the rank of the better behaved GKZ hypergeometric system is always the expected one. We give largely self-contained proofs of many properties of this system. The discussion is intimately related to the study of the variations of Hodge structures of hypersurfaces in algebraic tori.
In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in \cite{GM}, \cite{MH}, \cite{MT} and \cite{MT2}. We also discuss the relation between intersection theory and evaluation of an integral of a product of powers of absolute values of polynomials.
We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter $β\in N \otimes {\mathbb C}.$ As an application, we construct a topological mirror symmetry map that produces a complete system of $Γ$--series solutions to the better behaved version of the GKZ hypergeometric system for $β\in N \otimes {\mathbb C}.$
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $\ell$-adic counterpart of the GKZ hypergeometric system, which we call the $\ell$-adic GKZ hypergeometric sheaf. It is an object in the derived category of $\ell$-adic sheaves on the affine space over a finite field. Traces of Frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by Gel…
We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in arXiv:1902.01536.
Given a family of Laurent polynomials, we will construct a morphism between its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The kernel and cokernel of this morphism are very simple and consist of free O-modules. The result above enables us to put a mixed Hodge module structure on certain classes of GKZ systems and shows that they have quasi-unipotent monodromy.
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $\mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points …
We study the mixed twistor D-modules associated to meromorphic functions. In particular, we describe their push-forward and specialization under some situations. We apply the results to study the twistor property of a type of GKZ-hypergeometric systems, and to study their specializations. As a result, we obtain some isomorphisms of mixed TEP-structures in the local mirror symmetry.
We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.
We consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.
Let $A$ be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N}A]$ is normal. Fix a face $F$ of the cone of $A$. We show that the projection and restriction of an $A$-hypergeometric system to the coordinate subspace corresponding to $F$ are essentially $F$-hypergeometric; moreover, at most one of them is nonzero. We also show that, if $A$ is in addition homogeneous, the holonomic dual of an $A$-hypergeometric system is itself $A$-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.
We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on the existence of non-commutative Hodge structures on the reduced quantum D-module of a nef complete intersection inside a toric variety.
We show in several important cases that the $A$-hypergeometric system attached to a Feynman diagram in Lee--Pomeransky form, obtained by viewing the momenta and the nonzero masses as indeterminates, has a normal underlying semigroup. This continues a quest initiated by Klausen, and studied by Helmer and Tellander. In the process we identify several relevant matroids related to the situation and explore their relationships.
We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.
Diagrammatic construction of GKZ systems for polygonal functions
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute（P.N. 列别杰夫物理研究所 I.E. 塔姆理论物理系）；Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科罗蒙诺索夫国立大学理论与数学物理研究所）
The connection between Feynman integrals and GKZ $A$-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities; in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for example, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator p…
If $β\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. We then produce a result, based on a Fourier-Sato transfo…
We give an exposition of Dwork's construction of Frobenius structures associated to generalized hypergeometric equations via the interpretation of the latter due to Gelfand-Kapranov-Zelevinsky in the language of A-hypergeometric systems. As a consequence, we extract some explicit formulas for the degeneration at 0 in terms of the Morita p-adic gamma function.
We study Betti structures in the solution complexes of confluent hypergeometric equations. We use the framework of enhanced ind-sheaves and the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. The main result is a group theoretic criterion that ensures that enhanced solutions of such systems are defined over certain subfields of the complex numbers. The proof uses a description of the hypergeometric systems as exponentially twisted Gauss-Manin systems of certain Laurent polynomials.
Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-α_j)-z\prod_{j=1}^h(D-β_j), \qquad D=z\frac{d}{dz}. \] If the numbers $α_i$ and $β_j$ are real and for all $i$ and $j$ the number $α_i-β_j$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Corti and Golyshev. We also use non-abelian Hodg…
We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.
We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ by using twisted cohomology groups and the intersection form on them. We apply our results to algebraic statistics; numerical evaluation of the normalizing constants of two way contingency tables with fixed marginal sums.
Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $α_i$ and $β_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge str…
For the hypergeometric spectral curve and its confluent degenerations (spectral curves of "hypergeometric type"), we obtain a simple formula expressing the topological recursion free energies as a sum over BPS states (degenerate spectral networks) for a corresponding quadratic differential $φ$. In doing so, we generalize Gaiotto-Moore-Neitzke's construction of BPS structures to include the case where $φ$ has simple poles or supports a degenerate ring domain. For the nine spectral curves of hypergeometric type, we provide a complete description of the corresponding BPS structures over a generic…
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the fami…
