Yang-mills Theories and Whitham Integrable Hierarchies

We review recent work on the study of N = 2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy, mainly focusing on three main results: (i) We develop a new recursive method to compute the whole instanton expansion of the low-energy effective prepotential; (ii) We interpret the slow times of the hierarchy as additional couplings and promote them to spurion superfields that softly break N = 2 supersymmetry down to N = 0 through deformations associated to higher Casimir operators of the gauge group; (iii) We show that the Seiberg–Witten-Whitham equations provide a set of non-trivial constraints on the form of the strong coupling expansion in the vicinity of the maximal singularities. We use them to check a proposal that we make for the value of the off-diagonal couplings at those points of the moduli space.