Supersymmetry, Supergravity and Non--Perturbative Dynamics of Gauge Theories

Abstract

We present a review of supersymmetry, supergravity, and the non-perturbative dynamics of gauge theories, tracing a path from the supersymmetry algebra to moduli stabilisation and de~Sitter vacua in string theory. Representations of the supersymmetry algebra, the superspace formalism, and basic models including the Wess--Zumino model and $\mathcal{N}=1$ supersymmetric Yang--Mills theory are discussed. The non-perturbative dynamics of $\mathcal{N}=2$ gauge theories is analysed through the Seiberg--Witten solution: the curve, prepotential, Picard--Fuchs system, BPS spectrum, and confinement via monopole condensation. The transition to $\mathcal{N}=1$ supergravity is carried out in three steps, showing how the K\"{a}hler potential $K$ and superpotential $W$ determine all five Lagrangian sectors and how the scalar potential acquires its exponential prefactor and gravitationally induced negative contribution. String theory applications include D-brane gauge theories, the AdS/CFT correspondence, geometric engineering of the Seiberg--Witten solution, and reduction of $\mathcal{N}=4$ to $\mathcal{N}=1$ supersymmetry. The KKLT moduli stabilisation mechanism is analysed in detail, including $\alpha'^3$ corrections to the K\"{a}hler potential. Three regimes of the scalar potential are identified -- classical KKLT, corrected KKLT with a shifted AdS minimum, and a runaway regime -- and the critical parameter $\hat{\xi}_c$ separating controlled de~Sitter vacua from decompactification is determined. The tension with the de~Sitter swampland conjecture is discussed.