Non-Abelian T-duality in Superspace

In this thesis we take a superspace perspective on T-duality, focusing on sigma models defined on background geometries that are constructed in terms of Lie supergroups. We briefly review Abelian bosonic and fermionic T-duality and the derivation of Buscher’s rules, moving then to the dualisation of principal chiral models on group manifolds. Extension of the latter to the case of supergroup manifolds represents the starting point of our analysis, which features an extended discussion about the explicit dualisation of the supergroup OSp(1|2). While the initial model represents an appropriate three-dimensional supergravity background, the T-dual one hints in the opposite direction, as the ansatz adopted to construct the dual veilbeine fails to satisfy the supegravity torsion constraints. Such result, together with the complexity of the ansatz-based approach, suggests that a more abstract and general point of view should be taken on the dualisation procedure. This represents the next step of our analysis and allows a simpler dualisation of principal chiral models and a clearer argument that the above T-dual model falls outside the class of three-dimensional supergravity backgrounds. Extension of the dualisation procedure to symmetric and semi-symmetric coset space sigma models based on Lie supergroups G/H is also favored by the more abstract perspective, which allows to recover the well-known exchange of equations of motion and Maurer- Cartan equations typically observed in purely bosonic settings, hence leading to the construction of a dual Lax connection and ensuring preservation of classical integrability. While dualisation of principal chiral models can be performed in full generality, for coset models the procedure might be affected by impediments appearing in the process of integrating out the gauge fields in favor of the dual variables, and thus requires a case by case analysis. We proceed by solving those gauge fields equations of motion that allow for a general solution, thus confining the potential obstruction to a single equation, whose solvability depends on the invertibility of two linear operators. We conclude by discussing two explicit examples in which dualisation goes through, the first based on the symmetric space SO(4)/SO(3), well-known for its dualisability, the second on the semi-symmetric space OSp(1|2)/SO(1,1) already approached in the literature from the point of view of holography and representing a Green-Schwarz-like sigma model satisfying the supergravity torsion constraints.