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arXiv · 2002.10285

Poisson-geometric analogues of Kitaev models

Alexander Spies
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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 $\Gamma$.

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Alexander Spies. 2020-02-24. Poisson-geometric analogues of Kitaev models. https://doi.org/10.1007/s00220-021-03992-5

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