Editing Moduli (physics) (section)

===Allowed moduli spaces of 4-dimensional theories===
The more supersymmetry there is, the stronger the restriction on the vacuum manifold. Therefore, if a restriction appears below for a given number N of spinors of supercharges, then it also holds for all greater values of N.

====N=1 Theories====

The first restriction on the geometry of a moduli space was found in 1979 by [[Bruno Zumino]] and published in the article "Supersymmetry and Kähler Manifolds".<ref>{{Cite journal |last=Zumino |first=B. |date=Nov 1979 |title=Supersymmetry and Kähler manifolds |url=https://inspirehep.net/record/142186/?ln=en |journal=Physics Letters B |language=en |volume=87 |issue=3 |pages=203–206 |doi=10.1016/0370-2693(79)90964-X}}</ref>  He considered an [[4D N = 1 global supersymmetry|N=1 theory in 4-dimensions]] with global supersymmetry. N=1 means that the fermionic components of the supersymmetry algebra can be assembled into a single [[Majorana spinor|Majorana]] [[supercharge]]. The only scalars in such a theory are the complex scalars of the [[chiral superfield]]s.  He found that the vacuum manifold of allowed vacuum expectation values for these scalars is not only complex but also a [[Kähler manifold]].

If [[gravity]] is included in the theory, so that there is local supersymmetry, then the resulting theory is called a [[supergravity]] theory and the restriction on the geometry of the moduli space becomes stronger.  The moduli space must not only be Kähler, but also the Kähler form must lift to integral [[cohomology]].  Such manifolds are called [[Hodge manifold]]s.  The first example appeared in the 1979 article "Spontaneous Symmetry Breaking and Higgs Effect in Supergravity Without Cosmological Constant"<ref>{{Cite journal |last1=Cremmer |first1=E. |last2=Julia |first2=B. |last3=Scherk |first3=J. |last4=Ferrara |first4=S. |last5=Girardello |first5=L. |last6=van Nieuwenhuizen |first6=P. |date=Jan 1979 |title=Spontaneous symmetry breaking and Higgs effect in supergravity without cosmological constant |url=http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B147,105 |journal=Nuclear Physics B |language=en |volume=147 |issue=1–2 |pages=105–131 |doi=10.1016/0550-3213(79)90417-6 |bibcode=1979NuPhB.147..105C |url-status=dead |archive-url=https://archive.today/20121210095718/http://inspirehep.net/search?p=find+j+nupha,b147,105 |archive-date= 10 Dec 2012 }}</ref> and the general statement appeared 3 years later in "Quantization of Newton's Constant in Certain Supergravity Theories".<ref>{{Cite journal |last1=Witten |first1=Edward |last2=Bagger |first2=Jonathan |date=Sep 1982 |title=Quantization of Newton's constant in certain supergravity theories |url=https://inspirehep.net/record/11988/ |journal=Physics Letters B |language=en |volume=115 |issue=3 |pages=202–206 |doi=10.1016/0370-2693(82)90644-X|bibcode=1982PhLB..115..202W |url-access=subscription }}</ref>

====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]].

====N>2 Supersymmetry====

In extended supergravities with N>2 the moduli space must always be a [[symmetric space]].