On Twisted Virasoro Operators and Number Theory

Abstract

We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,χ)$ and $L(-1,χ)$ for $χ$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We introduce a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(-1, χ)$'s for appropriately chosen $χ$'s, and can be understood physically as a kind of 'vacuum Casimir engergy' in our settings. We also note a relation between field theory and our twisted operators. Lastly we give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field.

Some concepts in section 5 are clarified. Some grammatical errors are corrected. Acknowledgements added in Introduction section