(Pseudo-)trace functions and modular invariance of vertex operator algebra
Vertex Operator Algebra (shortly VOA) is a mathematical object to study a chiral algebra of conformal field theory from a view point of axioms. So we consider it algebraically. In order to consider a finite type, there are two conditions: completely reducibility and C2-finiteness condition. As it is well observed in many finite models, the space of characters of modules is SL2(Z)-invariant. Zhu showed that if SW(t) is a character of module W then SW(-1/t) is a linear sum of characters under the above two conditions.
In this talk, we will show that C2-finiteness is sufficient to get SL2(Z)-invariance in a sense. In this case, we have to consider not only characters (linear sums of powers of q = e2pit), but also extended characters (including logarithmic forms), which are defined by pseudo-trace functions induced from symmetric linear functions of extended Zhu-algebra.