Poisson-geometric analogues of Kitaev models

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 $G^{\ast }$) 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 $\Gamma$.

arXiv comment49 pages. v2: Revised Lemma 3.17 (i). v3: Revised introduction; fixed oversight in proof of Theorem 3.29; improved Remark 3.11; fixed typos and improved wording in several places

References in corpus (16)

• Fault-tolerant quantum computation by anyons
• String-net condensation: A physical mechanism for topological phases
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Symplectic structures associated to Lie-Poisson groups
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Two dimensional lattice gauge theory based on a quantum group
• A hierarchy of topological tensor network states
• Turaev-Viro invariants as an extended TQFT
• Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix
• Kitaev lattice models as a Hopf algebra gauge theory
• Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
• Kitaev's Lattice Model and Turaev-Viro TQFTs
• Hopf algebra gauge theory on a ribbon graph