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Invitation to $p$-adic vertex algebras

Cameron Franc, Geoffrey Mason

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.

Invitation to $p$-adic vertex algebras

Abstract

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

Paper Structure

This paper contains 28 sections, 21 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Vertex algebras
  3. Fields on $V$
  4. $Y$-maps
  5. Vertex algebras over $k$
  6. Vertex operator algebras
  7. $p$-adic Banach spaces
  8. $p$-adic modular forms
  9. $p$-adic vertex algebras
  10. $p$-adic fields on $V$
  11. $p$-adic $Y$-maps
  12. $p$-adic vertex algebras
  13. The $p$-adic state-field correspondence
  14. $p$-adic vertex operator algebras
  15. Construction of some $p$-adic vertex algebras
  16. ...and 13 more sections

Key Result

Theorem 3

$\mathcal{F}(V)$ is a $p$-adic Banach space when equipped with the sup norm, i.e., $|a(z)|:=\sup_n|a(n)|$.

Theorems & Definitions (34)

  • Remark 1
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Definition 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 24 more