K-theory and phase transitions at high energies

The duality between E_8xE_8 heteritic string on manifold K3xT^2 and Type IIA string compactified on a Calabi-Yau manifold induces a correspondence between vector bundles on K3xT^2 and Calabi-Yau manifolds. Vector bundles over compact base space K3xT^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 compactihcation. 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 K3xT^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.