Fusion algebras and cohomology of toroidal orbifolds

Abstract

In this thesis we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the group G = (Z/2)3 via the pairing θg(φ)R(G) ⊗θh(φ) R(G) →θgh(φ) R(G) where θ : H4(G,Z)→ H3(G,Z) is the inverse transgression map and φ is a carefully chosen cocycle class. We find the rank of the fusion algebra X(G) = ∑ g∈G θg(φ)R(G) as well as the relation between its basis elements. We also give some applications to topological gauge theories. We next show that the twisted fusion algebra of the group (Z/p)3 is isomorphic to the non-twisted fusion algebra of the extraspecial p-group of order p3 and exponent p. The final point of my thesis is to explicitly compute the cohomology groups H∗(X/G;Z) where X/G is a toroidal orbifold and G = Z/p for a prime number p. We compute the particular case where X is induced by the ZG-module (IG)n, where IG is the augmentation ideal.