On the Coordinate Transformation of a Vertex Operator Algebra

Abstract

We provide first a purely VOA-theoretic guide to the theory of coordinate transformations for a VOA in direct accordance with its first appearance in a paper of Zhu. Among these results, we are able to obtain new closed-form expressions for the square-bracket Heisenberg modes. We then elaborate on the connection to p-adic modular forms which arise as characters of states in p-adic VOAs. In particular, we show that the image of the p-adic character map for the p-adic Heisenberg VOA contains infinitely-many p-adic modular forms of level one which are not quasi-modular. Finally, we introduce a new VOA structure obtained from the Artin-Hasse exponential, and serving as the p-adic analogue of the square-bracket formalism.