Modularity in orbifold theory for vertex operator algebra

This paper gives an analogue of Ag( V) and modular invariance theories for a vertex operator superalgebra V and an automorphism g of finite order.;Given a vertex operator superalgebra V and a finite automorphism group G of V, for any g ∈ G, the relation between the g-twisted V-modules and Ag( V)-modules is established. And we also show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under C 2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C2-cofinite and g-rational for any g ∈ G..;We also generalized the above theories to Z -graded vertex operator super algebra.