Modularity of trace functions in orbifold theory for Z-graded vertex   operator superalgebras
  We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the h\sigma-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where \sigma is the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,Z). 
Chongying Dong and Zhongping Zhao