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Characters in p-adic Heisenberg and Lattice Vertex Operator Algebras

Daniel Barake, Cameron Franc

Abstract

We study characters of states in $p$-adic vertex operator algebras. In particular, we show that the image of the character map for both the $p$-adic Heisenberg and $p$-adic lattice vertex operator algebras contain infinitely-many non-classical $p$-adic modular forms which are not contained in the image of the algebraic character map.

Characters in p-adic Heisenberg and Lattice Vertex Operator Algebras

Abstract

We study characters of states in -adic vertex operator algebras. In particular, we show that the image of the character map for both the -adic Heisenberg and -adic lattice vertex operator algebras contain infinitely-many non-classical -adic modular forms which are not contained in the image of the algebraic character map.
Paper Structure (12 sections, 11 theorems, 87 equations)

This paper contains 12 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Vertex Operator Algebras
  4. p-adic Vertex Operator Algebras
  5. p-adic modular forms
  6. p-adic characters in the Heisenberg VOA
  7. The Heisenberg VOA and the square-bracket formalism
  8. Some square-bracket states
  9. Proof of Theorem 1.1
  10. P-adic characters in lattice VOAs
  11. Lattice VOAs
  12. Proof of Theorem 1.2

Key Result

Theorem 1.1

The image of the character map on the $p$-adic Heisenberg VOA $S_{1}$ contains infinitely-many non-classical $p$-adic modular forms. In particular, the image contains which is not quasi-modular of level one, and where $t,l \geq 1$ are odd and $l \not\equiv -1 \mod p-1$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Remark
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark
  • ...and 11 more