Poisson-geometric Analogues of Kitaev Models

Alexander Spies

doi:10.1007/s00220-021-03992-5

Abstract

We define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph Gamma by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double {mathcal {H}}(G). Each vertex (face) of Gamma defines a Poisson action of G (of G^*) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph Gamma and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from Gamma .

Highlights

Kitaev’s lattice models were initially introduced to model a quantum memory on a surface that is governed by its topological properties and allows for intrinsically fault-tolerant quantum computation [23]
Meusburger related Kitaev models to the combinatorial quantization of Chern-Simons theories [27], that was obtained by Alekseev, Grosse and Schomerus [1,2,3] and Buffenoir and Roche [14,15], and axiomatized as a Hopf algebra gauge theory
We show that the actions for a pair of a vertex v and adjacent face f can be combined to obtain a Poisson action of the classical double D(G) that corresponds to the module algebra structure of H(H )⊗E over the Drinfeld double D(H )

Summary

Introduction

The natural symplectic structure on the moduli space of flat D(G)-bundles on a compact oriented surface with boundary can be obtained from Fock and Rosly’s Poisson manifold D(G)×E for an embedded graph by taking the quotient with respect to the D(G)-action for every vertex. We use this fact to relate the Poisson algebra of functions on the moduli space with Poisson-Kitaev models.

Background

Poisson Analogues of Kitaev Models

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