  The duality between $E_8\times E_8$ heteritic string on manifold $K3\times
T^2$ and Type IIA string compactified on a Calabi-Yau manifold induces a
correspondence between vector bundles on $K3\times T^2$ and Calabi-Yau
manifolds. Vector bundles over compact base space $K3\times T^2$ form the set
of isomorphism classes, which is a semi-ring under the operation of Whitney sum
and tensor product. The construction of semi-ring $Vect\ X$ of isomorphism
classes of complex vector bundles over X leads to the ring $KX=K(Vect\ X)$,
called Grothendieck group. As K3 has no isometries and no non-trivial
one-cycles, so vector bundle winding modes arise from the $T^2$
compactification. Since we have focused on supergravity in d=11, there exist
solutions in d=10 for which space-time is Minkowski space and extra dimensions
are $K3\times T^2$. The complete set of soliton solutions of supergravity
theory is characterized by RR charges, identified by K-theory. Toric
presentation of Calabi-Yau through Batyrev's toric approximation enables us to
connect transitions between Calabi-Yau manifolds, classified by enhanced
symmetry group, with K-theory classification.
