Supersymmetry had a long history in physics since 1970s. Among them, the 3d N = 2 theory (three-dimensional theory with two supersymmetries) by Aharony et al. is of special interest. In 2011, Dimofte-Gaiotto-Gukov proposed a topological interpretation of such theory and after that, Terashima-Yamazaki made a further step. They utilized Fomin-Shapiro-Thurston's theory of cluster surface to connect Dimofte-Gaiotto-Gukov's interpretation. In this thesis, we are going to provide a mathematical formalization of Terashima-Yamazaki's formulation over operator algebraic aspects and discuss the possibility of generalizing their formulation from mapping cylinders to a larger class of 3-manifolds. We expect that a three dimensional analogy of Fomin-Shapiro-Thurston's surface theory will be helpful....[ Read more ]

Supersymmetry had a long history in physics since 1970s. Among them, the 3d N = 2 theory (three-dimensional theory with two supersymmetries) by Aharony et al. is of special interest. In 2011, Dimofte-Gaiotto-Gukov proposed a topological interpretation of such theory and after that, Terashima-Yamazaki made a further step. They utilized Fomin-Shapiro-Thurston's theory of cluster surface to connect Dimofte-Gaiotto-Gukov's interpretation. In this thesis, we are going to provide a mathematical formalization of Terashima-Yamazaki's formulation over operator algebraic aspects and discuss the possibility of generalizing their formulation from mapping cylinders to a larger class of 3-manifolds. We expect that a three dimensional analogy of Fomin-Shapiro-Thurston's surface theory will be helpful.