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On Integration and the Volumes of Supermanifolds
[Thesis]. Manchester, UK: The University of Manchester; 2021.
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Abstract
In this thesis we investigate the volumes of certain supermanifolds. The volumes of supermanifolds have been studied before in particular in [1]. This thesis builds on that work. We develop the necessary tools to study mainly the volume of the complex Grassmannian supermanifolds. In the first two chapters we review the problem and how it has been solved for ordinary Grassmannian manifolds. We contrast that with the super case and then introduce briefly what a supermanifold is and give an exposition on what integration entails in the super case. In the third chapter we develop the tools we need to calculate the volume of the Grassmannian supermanifolds as Hermitian supermanifolds. We develop Hermitian forms in the super case and we conclude that the natural Hermitian form on the space of matrices isn't positive definite. We then develop the Kronecker product and vectorisation in the super case. With these developed we show the relation between the Berezinian, or superdeterminant, and the Kronecker product. In the fourth chapter we investigate what the volume element of the Grassmannian supermanifolds coming from a natural Hermitian form is and apply the results of the previous chapter so that we can calculate it. In the fifth and last chapter of the main part of the thesis we calculate the volume of the Grassmannian supermanifolds for different values of the relevant parameters. In [1] there is a conjectured formula for the volume of the Grassmannian supermanifolds and we contrast our results with that. We have provided an appendix on superalgebra to provide a guide on the conven- tions and notations used in the main text.