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TL;DR: It is the wedge product $\wedge$ and the exterior derivative/differential $d$ (which squares to zero $d^2=0$) that give rise to Grassmann-odd elements and supersymmetry. More concretely, Ref. 1 first writes down a (non-relativistic) SUSY algebra ${\cal A}$ $$\tag{10} \{Q_1,Q_1\}_+~=~2H~=~\{Q_2,Q_2\}_+, \qquad \{Q_1,Q_2\}_+~=~0, $$ spanned by two ...


This is a standard particle-vortex duality transformation. The idea is there exists a "dual" description of a superfluid, in which the bosons (to be precise, the Goldstone mode) is described by a 2-form(tensor) gauge field, and the vortices are now string-like "matter" charged under the gauge field. The effective action for phase fluctuations (i.e. Goldstone ...

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