Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

TL;DR

This paper explores the symmetries and covariance properties of linear differential equations related to hypergeometric functions, revealing connections to modular forms, Schwarzian equations, and algebraic transformations, with implications for physics and elliptic curves.

Contribution

It introduces new examples of rational transformations preserving differential equations, extends covariance results to more general functions, and links these to modular forms and Schwarzian conditions in a novel way.

Findings

Examples of rational transformations leaving differential equations covariant.

New covariance results for Heun and higher genus hypergeometric functions.

Connections between Schwarzian equations, modular forms, and algebraic transformations.

Abstract

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that emerged in a paper by Casale. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to $_3F_2$, hypergeometric functions, and show that one just reduces to the previous $_2F_1$ cases through a Clausen identity. In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or $_2F_1$ hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.