Canonical decomposition of linear differential operators with selected differential Galois groups

TL;DR

This paper introduces a canonical decomposition method for certain linear differential operators with special Galois groups, revealing their structure through self-adjoint operators and applying it to complex operators related to Calabi-Yau 3-folds.

Contribution

It presents a new decomposition framework for differential operators with selected Galois groups using self-adjoint operators, extending to infinite families and large order operators.

Findings

Decomposition of order-six operator into three order-two self-adjoint operators.

Operators homomorphic to their adjoint have rational solutions depending only on the rightmost operator.

Application to large operators associated with Calabi-Yau 3-folds.

Abstract

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.