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arXiv · hep-th/9511155

A Generalization of the Bargmann-Fock Representation to Supersymmetry by Holomorphic Differential Geometry

Hans-Peter Thienel
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Abstract

In the Bargmann-Fock representation the coordinates $z^i$ act as bosonic creation operators while the partial derivatives $\partial_{z^j}$ act as annihilation operators on holomorphic $0$-forms as states of a $D$-dimensional bosonic oscillator. Considering also $p$-forms and further geometrical objects as the exterior derivative and Lie derivatives on a holomorphic ${\bf C}^D$, we end up with an analogous representation for the $D$-dimensional supersymmetric oscillator. In particular, the supersymmetry multiplet structure of the Hilbert space corresponds to the cohomology of the exterior derivative. In addition, a 1-complex parameter group emerges naturally and contains both time evolution and a homotopy related to cohomology. Emphasis is on calculus.

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Hans-Peter Thienel. 1995-11-22. A Generalization of the Bargmann-Fock Representation to Supersymmetry by Holomorphic Differential Geometry. https://doi.org/10.1088/0305-4470%2F29%2F21%2F028

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