Title: Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version),
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.