# Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)

## TL;DR

This paper generalizes Schwarzian differential equations from hypergeometric functions to higher-order linear differential operators with specific Galois groups, revealing conditions for modular solutions and applications in physics and combinatorics.

## Contribution

It introduces a framework for Schwarzian conditions on higher-order operators with selected Galois groups, extending previous work and analyzing modular correspondences and Calabi-Yau operator equivalences.

## Key findings

- Pullback invariance reduces to symmetric powers of order-two operators for orders three and four.
- Conditions for modular correspondence solutions of Schwarzian equations are established.
- Analysis of hypergeometric examples shows invariance up to operator homomorphisms.

## Abstract

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.