Monodromy of A-hypergeometric functions

Abstract

We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.

Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation

“…In contrast to our previous considerations [71,[80][81][82], the case where some additional differential equations are generated simultaneously was analysed in the present paper. Our technique is based on the methods developed for the analysis of the monodromy of GKZ hypergeometric functions [33,[49][50][51][52][53][54][55][56][57][58].…”

“…(2.6) with undetermined coefficients. To fix these coefficients, it is necessary to evaluate the Mellin-Barnes integral as a power series solution [33]. In particular, under the condition that the monodromy is irreducible (see eq.…”

“…(2.5) are equal to zero for arbitrary values of z i [44][45][46]. The same recipe works in its application to Mellin-Barnes integrals [33], which can be treated as a particular case of the Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric system [47,48]. However, the inverse differential operators have a very complicated structure (see, for example, ref.…”

“…[43]), which gives rise to technical problems in the analysis of the number of independent PDEs. Fortunately, there is a simpler way [33,[49][50][51][52][53][54][55][56][57][58]) to find the conditions of reducibility and to define the dimension of the (ir)reducible subspace of solutions. In our case, the 5 Another example of such a cancellation was presented in ref.…”

Feynman Integrals, Toric Geometry and Mirror Symmetry

“…The relation between hypergeometric functions and the GKZ differential system can be simply understood as follows (see [48,52,53]).…”