Construction of a Fuchsian equation from a monodromy representation

1990 ◽  
Vol 48 (5) ◽  
pp. 1090-1099 ◽  
Author(s):  
A. A. Bolibrukh
Keyword(s):  
Monodromy Representation ◽  
Fuchsian Equation

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals Hyperbolic three-string vertex

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Atakan Hilmi Fırat
Keyword(s):  
Boundary Value Problem ◽  
Conservation Laws ◽  
Riemann Surfaces ◽  
Higher Order ◽  
Local Coordinates ◽  
Pants Decomposition ◽  
Fuchsian Equation ◽  
String Amplitudes ◽  
Hyperbolic Riemann Surfaces ◽  
Liouville’S Equation

Abstract We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

scholarly journals The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4

2015 ◽  
Vol 217 ◽  
pp. 61-94
Author(s):  
Yoshiaki Goto ◽  
Keiji Matsumoto
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Series ◽  
Integral Representations ◽  
Monodromy Representation ◽  
Homology Groups ◽  
Linearly Independent ◽  
Fundamental Systems

AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

scholarly journals Mixed Hilbert modular forms and families of abelian varieties

1997 ◽  
Vol 39 (2) ◽  
pp. 131-140 ◽  
Author(s):  
Min Ho Lee
Keyword(s):  
Modular Forms ◽  
Fuchsian Group ◽  
Automorphic Forms ◽  
Elliptic Surface ◽  
Hilbert Modular Forms ◽  
Monodromy Representation ◽  
General Elliptic ◽  
Hilbert Modular ◽  
Upper Half Plane ◽  
Image Position

In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the formfor z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the monodromy representation, and w: ℋ→ℋ the lifting of the period map of the elliptic surface.

scholarly journals A linear Fuchsian equation with variable indices

2003 ◽  
Vol 190 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Ali Bentrad ◽  
Satyanad Kichenassamy
Keyword(s):  
Fuchsian Equation

MONODROMY ARISING FROM THE MARKOFF THEORY

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1819-1832
Author(s):  
SERGE PERRINE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Free Group ◽  
Monodromy Group ◽  
Hilbert Problem ◽  
Monodromy Representation ◽  
Poincare Group ◽  
Complete Family ◽  
Computation Theory ◽  
Riemann Hilbert Problem

In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two generators. We also identify a link with some eigenvalues of a Laplacian. The introduction explains the links that we see with information and computation theory (classical or quantum).

The structure of a local system associated with a hypergeometric system of rank 9

2020 ◽  
Vol 31 (03) ◽  
pp. 2050021
Author(s):  
Jyoichi Kaneko ◽  
Keiji Matsumoto ◽  
Katsuyoshi Ohara
Keyword(s):  
Differential Equations ◽  
Fundamental Group ◽  
Fundamental System ◽  
Local System ◽  
Base Space ◽  
Integral Conditions ◽  
Monodromy Representation

We study a local system associated with a system [Formula: see text] of hypergeometric differential equations in two variables of rank [Formula: see text] with seven parameters [Formula: see text] and [Formula: see text]. We modify the fundamental system of solutions to [Formula: see text] given in [A system of hypergeometric differential equations in two variables of rank 9, Internat. J. Math. 28 (2017), 1750015, 34 pp] so that it is valid even in cases where [Formula: see text] satisfy some integral conditions. By using this fundamental system, we show the irreducibility of the monodromy representation of [Formula: see text] under some conditions on the parameters. We characterize the fundamental group of the base space of this local system as the group generated by three loops with four relations among them.

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