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p-adic vertex operator algebras.

Cameron Franc1, Geoffrey Mason2

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PubMed
Summary

This study introduces axioms for nonarchimedean 2D conformal field theory using p-adic Banach spaces. It constructs p-adic versions of key algebraic structures, revealing connections to p-adic modular forms.

Keywords:
Serre p-adic modular formsp-adic Banach spacesp-adic vertex operator algebras

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Area of Science:

  • Theoretical Physics
  • Algebraic Quantum Field Theory
  • Number Theory

Background:

  • Traditional conformal field theories (CFTs) are built upon Hilbert spaces.
  • Nonarchimedean analysis offers alternative mathematical frameworks.
  • Vertex operator algebras (VOAs) are central to CFT and string theory.

Purpose of the Study:

  • To postulate axioms for a chiral half of a 2D bosonic nonarchimedean CFT.
  • To replace the traditional Hilbert space with a p-adic Banach space.
  • To explore the consequences of these axioms and construct new mathematical objects.

Main Methods:

  • Postulation of novel axioms for nonarchimedean vertex operator algebras.
  • Utilizing p-adic Banach spaces as the underlying vector space.
  • Investigating the algebraic structures arising from these axioms.

Main Results:

  • Construction of p-adic commutative Banach rings.
  • Development of p-adic analogues of the Virasoro, Heisenberg, and Moonshine module VOAs.
  • Emergence of Serre p-adic modular forms as limits of classical 1-point functions.

Conclusions:

  • The proposed axioms provide a foundation for nonarchimedean CFT.
  • The study demonstrates the existence of rich mathematical structures in this new framework.
  • Connections between p-adic CFT and p-adic modular forms are established.