Supersymmetry and Irrelevant Deformations

The $T \overline{T}$ operator provides a universal irrelevant deformation of two-dimensional quantum field theories with remarkable properties, including connections to both string theory and holography beyond $\mathrm{AdS}$ spacetimes. In particular, it appears that a $T \overline{T}$-deformed theory is a kind of new structure, which is neither a local quantum field theory nor a full-fledged string theory, but which is nonetheless under some analytic control. On the other hand, supersymmetry is a beautiful extension of Poincar\'e symmetry which relates bosonic and fermionic degrees of freedom. It is natural to ask what one can learn about irrelevant deformations in supersymmetric quantum field theories. In this thesis, we describe a presentation of the $T \overline{T}$ deformation in manifestly supersymmetric settings. We define a "supercurrent-squared" operator, which is closely related to $T \overline{T}$, in any two-dimensional theory with $(0, 1)$, $(1, 1)$, or $(2, 2)$ supersymmetry. This deformation generates a flow equation for the superspace Lagrangian of the theory, which therefore makes the supersymmetry manifest. In certain examples, the deformed theories produced by supercurrent-squared are related to superstring and brane actions, and some of these theories possess extra non-linearly realized supersymmetries. We then show that $T \overline{T}$ defines a new theory of both abelian and non-abelian gauge fields coupled to charged matter, which includes models compatible with maximal supersymmetry. Most of this thesis is adapted from previous publications. However, the final chapter presents new results on $T \overline{T}$-like deformations of general gauge theories in an arbitrary number of spacetime dimensions; this analysis has not appeared in past work.