Riassunto analitico
In quantum field theory, it is sometimes useful to analytically continue the number of dimensions of spacetime or the number of components of a field and its spin from an integer to a real value. From a group-theoretical point of view, this corresponds to considering groups with real, rather than integer, ranks, and is a rather obscure operation, which can, however, be meaningfully formulated in the language of categories. The goal of this master thesis is to make sense of the orthosymplectic supergroup OSp(n|m), which often appears in supersymmetric contexts, when n,m are non-integer. After briefly reviewing category theory, we introduce Deligne categories and their associated Brauer algebras. These allow us to generalize the notion of irreducible representation to the case on non-integers ranks, as we first illustrate for the bosonic groups O(n) and Sp(m). We then consider their supersymmetric extension OSp(n|m), which is the main original result of this work. We also study theories with a categorical symmetry, focusing on the OSp(n|m) invariant σ-model. A lightning review of supersymmetry, superalgebras and their relevance in physics is contained in an appendix.
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