Tautological systems and free divisors

TL;DR

This paper introduces tautological systems linked to prehomogeneous group actions, showing their connection to mixed Hodge modules and deriving one-dimensional differential systems that generalize quantum differential equations.

Contribution

It defines tautological systems for prehomogeneous actions and relates them to mixed Hodge modules, providing a new perspective and generalization of quantum differential equations.

Findings

Tautological systems are associated with prehomogeneous group actions.

These systems relate to mixed Hodge modules under certain conditions.

Derived one-dimensional differential systems generalize quantum differential equations.

Abstract

We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.