Quon language: surface algebras and Fourier duality

TL;DR

This paper introduces surface algebras within Quon language, a 3D pictorial framework, to connect quantum information, modular tensor categories, and Fourier duality, revealing new algebraic identities and extending planar algebra concepts to higher genus surfaces.

Contribution

It extends planar algebras to surface algebras on higher genus surfaces and establishes a unique one-parameter extension, linking pictorial Fourier duality with algebraic duality in modular tensor categories.

Findings

Established a unique one-parameter extension of surface algebras.

Connected pictorial Fourier duality with algebraic duality via the S matrix.

Derived new algebraic identities, including generalizations of the Verlinde formula.

Abstract

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the $S$ matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.