Research about supersymmetric higher dimensional non-commutative geometry and topologial quantum many-body states

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We established supersymmetric higher dimensional non-commutative geomery and applied its mathematics to the topological matters. In 2011, we explored non-commutative geometry for supersymmetric higher dimensional spheres and hyperboloids. Besides, we applied the mathematics to the supersymmetric quantum spin systems and non-Hermitian quantum mechanics and exploited their topological structures such as edge states and hidden order. In 2012, we further investigated the supersymmetric quantum spin systems especially about the recently proposed quantum entanglement. In 2013, we applied the non-commutative geometry to the topological insulators (in particular A-class nad AIII-class) and analyzed their physical properties.