  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.
