Editing Moduli (physics) (section)

====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>