(Pseudo-)trace functions and modular invariance of vertex operator algebra
Abstract:
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 C_{2}-finiteness
condition. As it is well observed in many finite models, the space of characters
of modules is SL_{2}(Z)-invariant. Zhu showed that
if S_{W}(t) is a character of
module W then S_{W}(-1/t)
is a linear sum of characters under the above two conditions.
In this talk, we will show that C_{2}-finiteness is sufficient to get SL_{2}(Z)-invariance in a sense. In this case, we have to consider not only characters (linear sums of powers of q = e^{2pit}), but also extended characters (including logarithmic forms), which are defined by pseudo-trace functions induced from symmetric linear functions of extended Zhu-algebra.
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