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https://en.wikipedia.org/wiki/K-theory_%28physics%29
K-theory (physics)
In string theory, the K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond?Ramond field strengths as well as the charges of stable D-branes.
In condensed matter physics K-theory has also found important applications, specially in the topological classification of topological insulators, superconductors and stable Fermi surfaces (Kitaev (2009), Horava (2005)).

History
This conjecture, applied to D-brane charges, was first proposed by Minasian & Moore (1997).
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).

Applications
The K-theory classification of D-branes has had numerous applications.
For example, Hanany & Kol (2000) used it to argue that there are eight species of orientifold one-plane. Uranga (2001) applied the K-theory classification to derive new consistency conditions for flux compactifications.
K-theory has also been used to conjecture a formula for the topologies of T-dual manifolds by Bouwknegt, Evslin & Varghese (2004). Recently K-theory has been conjectured to classify the spinors in compactifications on generalized complex manifolds.
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