Extended Homotopy Quantum Field Theories and their Orbifoldization

Based on a detailed definition of extended homotopy quantum field theories we develop a field-theoretic orbifold construction for these theories when the target space is the classifying space of a finite group G, i.e. for G-equivariant topological field theories. More precisely, we use a recently developed bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. One main motivation is the 3-2-1-dimensional case where our orbifold construction allows us to describe the orbifoldization of equivariant modular categories by a geometric construction. As an important ingredient of this result, we prove that a 3-2-1-dimensional G-equivariant topological field theory yields a G-multimodular category by evaluation on the circle. The orbifold construction is a special case of a pushforward operation along an arbitrary morphism of finite groups and provides a valuable tool for the construction of extended homotopy quantum field theories.