A Burchnall-Chaundy-Krichever Theory for Fractional Differential Operators

Abstract

Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Sato's theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Sato's Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.