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K-theory classification of RR fluxes
In the classical limit of type II string theory, which is type II supergravity, the Ramond?Ramond field strengths are differential forms.
In the quantum theory the well-definedness of the partition functions of D-branes implies that the RR field strengths obey Dirac quantization conditions when spacetime is compact,
or when a spatial slice is compact and one considers only the (magnetic) components of the field strength which lie along the spatial directions.
This led twentieth century physicists to classify RR field strengths using cohomology with integral coefficients.

However some authors have argued that the cohomology of spacetime with integral coefficients is too big. For example, in the presence of Neveu?Schwarz H-flux or non-spin cycles some RR fluxes dictate the presence of D-branes.
In the former case this is a consequence of the supergravity equation of motion which states that the product of a RR flux with the NS 3-form is a D-brane charge density.
Thus the set of topologically distinct RR field strengths that can exist in brane-free configurations is only a subset of the cohomology with integral coefficients.

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The Atiyah?Hirzebruch spectral sequence constructs twisted K-theory, with a twist given by the NS 3-form field strength, as a quotient of a subset of the cohomology with integral coefficients.
In the classical limit, which corresponds to working with rational coefficients, this is precisely the quotient of a subset described above in supergravity.
The quantum corrections come from torsion classes and contain mod 2 torsion corrections due to the Freed-Witten anomaly.

Thus twisted K-theory classifies the subset of RR field strengths that can exist in the absence of D-branes quotiented by large gauge transformations.
Daniel Freed has attempted to extend this classification to include also the RR potentials using differential K-theory.
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