Editing Moduli (physics) (section)

====N=2 Theories====

In extended 4-dimensional theories with N=2 supersymmetry, corresponding to a single [[Dirac spinor]] supercharge, the conditions are stronger.  The N=2 supersymmetry algebra contains two [[representation theory|representation]]s with scalars, the [[vector superfield|vector multiplet]] which contains a complex scalar and the [[hypermultiplet]] which contains two complex scalars.  The moduli space of the vector multiplets is called the [[Coulomb branch]] while that of the hypermultiplets is called the [[Higgs branch]].  The total moduli space is locally a product of these two branches, as [[supersymmetry nonrenormalization theorems|nonrenormalization theorems]] imply that the metric of each is independent of the fields of the other multiplet.(See for example Argyres, [http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/fgilec.pdf Non-Perturbative Dynamics Of Four-Dimensional Supersymmetric Field Theories], pp.&nbsp;6–7, for further discussion of the local product structure.)

In the case of global N=2 supersymmetry, in other words in the absence of gravity, the Coulomb branch of the moduli space is a [[special Kähler manifold]].  The first example of this restriction appeared in the 1984 article [https://inspirehep.net/record/202378/  Potentials and Symmetries of General Gauged N=2 Supergravity: Yang-Mills Models] by [[Bernard de Wit]] and [[Antoine Van Proeyen]], while a general geometric description of the underlying geometry, called [[special geometry]], was presented by [[Andrew Strominger]] in his 1990 paper [http://inspirehep.net/record/26953  Special Geometry].

The Higgs branch is a [[hyperkähler manifold]] as was shown by [[Luis Alvarez-Gaume]] and [[Daniel Z. Freedman|Daniel Freedman]] in their 1981 paper [https://inspirehep.net/record/10231/  Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model].  Including gravity the supersymmetry becomes local.  Then one needs to add the same Hodge condition to the special Kahler Coulomb branch as in the N=1 case.  [[Jonathan Bagger]] and [[Edward Witten]] demonstrated in their 1982 paper [http://inspirehep.net/record/13231/  Matter Couplings in N=2 Supergravity] that in this case, the Higgs branch must be a [[quaternionic Kähler manifold]].