Related papers: A note on generalized hypergeometric functions, KZ…

We study Feynman integrals in the framework of Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. The latter defines a class of functions wherein Feynman integrals arise as special cases, for any number of loops and kinematic…

Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann…

We review various generalizations of supersymmetry and discuss their relationship. In particular, we show how supersymmetry, parasupersymmetry, fractional supersymmetry, orthosupersymmetry, and the Z_n-graded topological symmetries are…

Embedding Feynman integrals in Grassmannians, we can write Feynman integrals as some finite linear combinations of generalized hypergeometric functions. In this paper we present a general method to obtain Gauss relations among those…

Both Feynman integrals and holographic Witten diagrams can be represented as multivariable hypergeometric functions of a class studied by Gel'fand, Kapranov & Zelevinsky known as GKZ or $\mathcal{A}$-hypergeometric functions. Among other…

We consider the $sl(2)$ quantized Knizhnik-Zamolodchikov equation (qKZ), defined in terms of rational R-matrices. The properties of the equation change when the step of the equation takes a resonance value. In this case the discrete…

A conjectural formula for the $k$-point generating function of Gromov--Witten invariants of the Riemann sphere for all genera and all degrees was proposed in \cite{DY2}. In this paper, we give a proof of this formula together with an…

In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue…

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten…

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…

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology…

Let $X$ be a prehomogeneous vector space under a connected reductive group $G$ over $\mathbb{R}$. Assume that the open $G$-orbit $X^+$ admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz…

In this paper, we obtain new results related to Minkowski fractional integral inequality using generalized k-fractional integral operator which is in terms of the Gauss hypergeometric function.

We develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of…

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational…

We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that…

We discuss the hypergeometric solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero and show that they give all solutions of the qKZ equation. We completely describe linear relations between the hypergeometric…

In this thesis we will study Feynman integrals from the perspective of A-hypergeometric functions, a generalization of hypergeometric functions which goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This point of…

We shall describe a simple generalization of commutative rings. The category GR of such "rings", contains the ordinary commutative rings (fully faithfully), but also the "integers" and "residue field" at a real or complex place of a field ;…