Oriented Closed Polyhedral Maps and the Kitaev Model

TL;DR

This paper introduces arrow presentations to encode oriented closed polyhedral complexes for the Kitaev model, enabling a combinatorial approach to prove ribbon operator identities and topological invariance.

Contribution

It develops a new combinatorial framework using arrow presentations to analyze the Kitaev model on polyhedral maps, linking dual structures and ribbon operators.

Findings

Proves ribbon operator identities for finite-dimensional C*-Hopf algebras.

Establishes a combinatorial notion of homotopy for ribbon curves.

Formulates topological invariance of states created by ribbon operators.

Abstract

A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate ''topological invariance'' of states created by ribbon operators.