Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if \({{\mathcal {P}}}\) is any of its spectral projections, the Booleanization of the fundamental group \(\pi _1(M)\) can be embedded inside the group of invertible elements of the corner algebra \({{\mathcal {P}}}\, \mathrm{CAR} \, {{\mathcal {P}}}\). As a consequence, \({{\mathcal {P}}}\) decomposes in \(4^g\) lower projections. Furthermore, a projective representation of \({{\mathbb {Z}}}_2^{4g}\) is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group \((2^X,\Delta )\), where X is the set of fermion sites and \(\Delta \) is the symmetric difference of its sub-sets.

考虑属g的曲面M的有限三角剖分，并假设无自旋费米子遍布三角剖分的边缘。此类粒子的量子动力学发生在规范反换向关系（CAR）的代数内部。根据Kitaev在复曲面模型上的工作，我们确定了由与三角剖分的三角形和顶点关联的元素生成的CAR的子代数。我们表明，从该子代数得出的任何哈密顿量都显示出拓扑光谱退化。更准确地说，如果\（{{\ mathcal {P}}} \）是其任何光谱投影，则基本组\（\ pi _1（M）\）的布尔化可以嵌入到的可逆元素组中角代数\（{{\ mathcal {P}}} \，\ mathrm {CAR} \，{{\ mathcal {P}}} \）。结果，\（{{\ mathcal {P}}} \\）在\（4 ^ g \）个较低的投影中分解。此外，\（{{\ mathbb {Z}}} _ 2 ^ {4g} \）的投影表示也明确地构造在该角代数内。所有这些的关键是将CAR作为布尔乘积\（（（2 ^ X，\ Delta）\）的叉积来表示，其中X是费米子位点的集合，而\（\ Delta \）是它的子集。