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Schwarzian conditions for linear differential operators with selected differential Galois groups. (English) Zbl 1382.34017

Summary: 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 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.

MSC:

34A30 Linear ordinary differential equations and systems
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34C14 Symmetries, invariants of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms

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