Editing Moduli (physics) (section)

==Moduli spaces of supersymmetric gauge theories==
In general quantum field theories, even if the classical potential energy is minimized over a large set of possible expectation values, once quantum corrections are included it is generically the case that nearly all of these configurations cease to minimize the energy.  The result is that the set of vacua of the [[Quantum mechanics|quantum theory]] is generally much smaller than that of the [[classical theory]].  A notable exception occurs when the various vacua in question are related by a [[symmetry]] which guarantees that their energy levels remain exactly degenerate.

The situation is very different in [[supersymmetry|supersymmetric]] quantum field theories.  In general, these possess large moduli spaces of vacua which are not related by any symmetry, for example, the masses of the various excitations may differ at various points on the moduli space.  The moduli spaces of supersymmetric gauge theories are in general easier to calculate than those of nonsupersymmetric theories because supersymmetry restricts the allowed geometries of the moduli space even when quantum corrections are included.

===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]].