GKZ Hypergeometric System Research Articles

Symbol alphabets from the Landau singular locus

We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal A-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal A-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal A-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel’fand, Graev, Kapranov and Zelevinskiĭ theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal A-determinant.

Perverse schobers and GKZ systems

Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such “quasi-symmetric” GKZ hypergeometric systems.

On duality of certain GKZ hypergeometric systems

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.

GKZ-hypergeometric systems for Feynman integrals

Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller's transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.

ℓ-adic GKZ hypergeometric sheaves and exponential sums

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 ℓ-adic counterpart of the GKZ hypergeometric system, which we call the ℓ-adic GKZ hypergeometric sheaf. It is an object in the derived category of ℓ-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 Gelfand and Graev. We prove that the ℓ-adic GKZ hypergeometric sheaf is perverse, calculate its rank, and prove that it is irreducible under the non-resonance condition. We also study the weight filtration of the GKZ hypergeometric sheaf, determine its lisse locus, and apply our result to the study of weights of twisted exponential sums.

Mirror symmetry, D-brane superpotentials and Ooguri–Vafa invariants of Calabi–Yau manifolds**Supported by Y4JT01VJ01 and NSFC(11475178)

The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi–Yau manifolds. By using the generalized GKZ hypergeometric system, we will calculate the D-brane superpotentials of two non-Fermat type compact Calabi–Yau hypersurfaces in toric varieties, respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri–Vafa invariants for the mirror Calabi–Yau manifolds.

Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules

AbstractWe endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.

On stringy cohomology spaces

We modify the definition of the families of $A$ and $B$ stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces have the same dimension as the ones defined in the joint paper with Mavlyutov \cite{BM}, but they admit natural flat connections with respect to the appropriate parameters. This solves a longstanding question of relating GKZ hypergeometric system to stringy cohomology. We construct products on these spaces by vertex algebra techniques. In the process, we fix a minor gap in \cite{BM} and prove a statement on intersection cohomology of dual cones that may be of independent interest.

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Convex and Algebraic Geometry

The workshop Convex and Algebraic Geometry was organized by Klaus Altmann (Berlin), Victor Batyrev (Tübingen), and Bernard Teissier (Paris). Both title subjects meet primarily in the theory of toric varieties. These constitute the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Thus, toric geometry and its several generalizations provide a kind of section from polyhedral into algebraic geometry. While this reflects only a thin slice of algebraic geometry, it is general enough to display many important phenomena, techniques, and methods. It serves as a wonderful testing ground for general theories such as the celebrated mirror symmetry in its different flavours. In particular, much of the popularity of toric geometry originates in mathematical physics. The meeting was attended by almost 50 participants from many European countries, Canada, the USA, and Japan. The program consisted of talks by 23 speakers, among them many young researchers. Most subjects fit more or less into the following main areas: One of the major discussions during the meeting concerned the existence of strongly exceptional sequences on toric varieties which consist of line bundles. A full exceptional sequence provides a kind of “basis” for the derived category. While Hille and Perling presented an example that does not carry such a sequence of full length, Bondal suggested a method to link this question to sheaves on the dual real torus that are constructible with respect to a certain stratification. In general, one expects to gain exceptional sequences from the universal bundles on moduli spaces. Using this method, Craw constructs those sequences on smooth toric Fano threefolds. In this context, Maclagan and Ueda consider the case of three-dimensional abelian quotient singularities. Ueda investigates the Fukaya category of the corresponding potential on the dual torus explicitly. Using mirror symmetry, Horja establishes a connection between the orbifold K -theory of toric Deligne-Mumford stacks and solutions to GKZ-hypergeometric D -modules. Gross and Siebert have developed a program to understand mirror symmetry as the duality of certain degeneration data. The special fibers split into toric components, and the degeneration is encoded in a topological manifold B with an affine and a polytopal structure. Duality is now inherited from discrete geometry, and the topology of B reflects the topology of the general fiber. In particular, if B is a ( \mathbb Q -homology) \mathbb P^n_{\mathbb C} , then this construction might lead to (compact) Hyperkähler varieties. Considering, in a special case, a certain contraction of the total space of these families leads to a description of torus actions on algebraic varieties via divisors on their Chow quotients. These divisors carry polytopes or even polyhedral complexes as their coefficients, compare the talks of Hausen, Süss, and Vollmert. In a similar setting, but with an explicit manipulation of Pfaffians, Brown and Reid construct smoothings of certain non-isolated singularities giving rise to four-dimensional flips. The most rigorous degeneration of a variety is the tropical one. Here, everything takes place over the so-called tropical semiring, and one ends up with piecewise linear spaces. In fact, Siebert's degeneration data mentioned above correspond to these objects. Itenberg and Shustin use this approach to calculate the Welschinger invariants, which are a kind of real version of Gromov–Witten invariants. Along the lines of the method of Gathmann and Markwig, there is a recursive formula for theses invariants. In the case of del Pezzo surfaces, it turns out that both invariants are (log-) asymptotically equivalent. A generalization of toric varieties in a different direction from the torus actions mentioned above is given by the notion of spherical varieties. Pasquier considers horospherical Fano varieties and comes up with an adapted notion of (generalized, coloured) reflexive polytopes. Bruns, Haase, and Hering deal with ordinary polytopes and their relations to syzygies of toric varieties. For an integral matrix A one obtains a semigroup algebra \mathbb C[\mathbb N A] (leading to the usual affine toric variety) and a GKZ-hypergeometric system of differential equations. The latter depends on a parameter \beta , and Miller has reported on a result that relates the set of \beta where the rank of the system jumps to the set of those multidegrees where the semigroup algebra \mathbb C[\mathbb N A] carries local cohomology. In particular, the Cohen–Macaulay property is equivalent to the constant rank condition, answering an old question of Sturmfels. One of the nighttime discussions gave rise to the suggestion to not include normality in the definition of a toric variety, thus overcoming the cumbersome term of a “not necessarily normal toric variety”. The workshop was closed on Friday night by an informal piano recital by Benjamin Nill and Milena Hering featuring Strawinsky, Liszt, and Chopin.

GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces

We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Grobner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up toh 1,1=3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.