On Twisted Virasoro Operators and Number Theory

TL;DR

This paper connects divergent series, conformal field theory, and number theory by introducing twisted Virasoro operators using Dirichlet characters, providing new methods to compute special L-values and interpret them physically.

Contribution

It introduces a novel twist of the Virasoro algebra with Dirichlet characters, linking divergent series, number theory, and physics, and offers new computational and interpretative frameworks.

Findings

New method to compute $L(0, ext{chi})$ and $L(-1, ext{chi})$

Explicit expressions for fractional powers in modular products

Physical interpretation of divergent series values as vacuum energy

Abstract

We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,\chi)$ and $L(-1,\chi)$ for $\chi$ 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 $L(-1, \chi)$'s for appropriately chosen $\chi$'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.