Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 515where Ω is the curvature of the connection.The Chern character is useful in part because it facilitates the computationof the Chern class of a tensor product. Specifically, it obeys thefollowing identities:ch(V ⊕ W ) = ch(V ) + ch(W ), ch(V ⊗ W ) = ch(V )ch(W ).Using the Grothendieck Additivity Axiom for Chern classes, the first ofthese identities can be generalized to state that ch is a homomorphism ofAbelian groups from the K−theory K(X) into the rational cohomology ofX. The second identity establishes the fact that this homomorphism alsorespects products in K(X), and so ch is a homomorphism of rings. TheChern character is used in the Hirzebruch–Riemann–Roch Theorem.The so–called twisted K–theory a particular variant of K−theory, inwhich the twist is given by an integral 3D cohomology class. In physics,it has been conjectured to classify D−branes, Ramond–Ramond fieldstrengths and in some cases even spinors in type II string theory.4.5.4 Atiyah’s View on K−TheoryAccording to Michael Atiyah [Atiyah and Anderson (1967); Atiyah (2000)],K–theory may roughly be described as the study of additive (or, Abelian)invariants of large matrices. The key point is that, although matrix multiplicationis not commutative, matrices which act in orthogonal subspacesdo commute. Given ‘enough room’ we can put matrices A and B into theblock form( ) A 0,0 1( ) 1 0,0 Bwhich obviously commute. Examples of Abelian invariants are traces anddeterminants.The prime motivation for the birth of K−theory came from Hirzebruch’sgeneralization of the classical Riemann–Roch Theorem (see [Hirzebruch(1966)]). This concerns a complex projective algebraic manifold X anda holomorphic (or algebraic) vector bundle E over X. Then one has thesheaf cohomology groups H q (X, E), which are finite–dimensional vectorspaces, and the corresponding Euler characteristicsχ(X, E) =n∑(−1) q dim H q (X, E),q=0