Invitation to $p$-adic vertex algebras

TL;DR

This paper explores the development of $p$-adic vertex algebras, focusing on their construction, properties, and connections to $p$-adic modular forms, expanding the mathematical framework of vertex operator algebras in the $p$-adic setting.

Contribution

It introduces the concept of $p$-adic vertex algebras and discusses their completion, with specific focus on the $p$-adic Heisenberg VOA and its relation to $p$-adic modular forms.

Findings

Construction of $p$-adic vertex algebras from $p$-adically normed spaces

Establishment of the $p$-adic Heisenberg VOA framework

Connections between $p$-adic VOAs and $p$-adic modular forms

Abstract

An overview of the authors' ideas about the process of completing a $p$-adically normed space in the setting of vertex operator algebras. We focus in particular on the $p$-adic Heisenberg VOA and its connections with $p$-adic modular forms.