It was popularized by Witten (1998) who demonstrated that in type IIB string theory arises naturally from Ashoke Sen's realization of arbitrary D-brane configurations as stacks of D9 and anti-D9-branes after tachyon condensation.
Such stacks of branes are inconsistent in a non-torsion Neveu–Schwarz (NS) 3-form background, which, as was highlighted by Kapustin (2000), complicates the extension of the K-theory classification to such cases.
Bouwknegt & Varghese (2000) suggested a solution to this problem: D-branes are in general classified by a twisted K-theory, that had earlier been defined by Rosenberg (1989).
Diaconescu, Moore & Witten (2003) argued that the K-theory classification is incompatible with S-duality in IIB string theory.
One needs to choose a half of the fluxes to quantize, or a polarization in the geometric quantization-inspired language of Diaconescu, Moore, and Witten and later of Varghese & Sati (2004).
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.
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.
Thus to obtain the space of inequivalent field strengths from the forementioned subset of integral cohomology we must quotient by these large gauge transformations.
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.
This corresponds to a process in which the inserted branes decay via a Dp-brane that forms, wraps the forementioned cycle and then disappears.
Diaconescu, Moore, and Witten have pointed out that the twisted K-theory classification is not compatible with the S-duality covariance of type IIB string theory.
However the MMS prescription for calculating twisted K-theory is easily S-covariantized, as the Freed-Witten anomalies respect S-duality.
Bouwknegt et al. (2006) use this approach to prove Diaconescu, Moore, and Witten's conjectured constraint on the 3-fluxes, and they show that there is an additional term equal to the D3-brane charge.
For example, Kriz & Sati (2005) propose that instead of twisted K-theory, II string theory configurations should be classified by elliptic cohomology.
Prominent researchers in this area include Edward Witten, Peter Bouwknegt, Angel Uranga, Emanuel Diaconescu, Gregory Moore, Anton Kapustin, Jonathan Rosenberg, Ruben Minasian, Amihay Hanany, Hisham Sati, Nathan Seiberg, Juan Maldacena, Alexei Kitaev, Daniel Freed, and Igor Kriz.
A very comprehensible introduction to the twisted K-theory classification of conserved D-brane charges on a 9-dimensional timeslice in the presence of Neveu–Schwarz flux is Maldacena, Moore & Seiberg (2001).