Supersymmetric gauge theory
= SUSY in 4D (with 4 real generators) =
theoretical physics, one often analyzes theories with supersymmetrywhich also haveinternal gauge symmetries. So, it is important to come up with a supersymmetric generalizationof gauge theories.In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates , transforming as a two-component spinorand its conjugate.
Every superfield, i.e. a field that depends on all coordinates of the superspace, may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called
chiral superfields, that only depend on the variables but not their conjugates (more precisely, ). However, a vector superfielddepends on all coordinates. It describes a gauge fieldand its superpartner, namely a Weyl fermionthat obeys a Dirac equation.
V is the vector superfield (prepotential) and is real (). The fields on the right hand side are component fields.
gauge transformations act as:where Λ is any chiral superfield.
It's easy to check that the chiral superfield:is gauge invariant. So is its complex conjugate .
A nonSUSY covariant gauge which is often used is the
Wess-Zumino gauge. Here, C, χ, M and N areall set to zero. The residual gauge symmetries are gauge transformations of the traditional bosonictype.
A chiral superfield X with a charge of q transforms as::The following term is therefore gauge invariant:
is called a bridge since it "bridges" a field which transforms under Λ only with a field which transforms under only.
More generally, if we have a real gauge group G that we wish to supersymmetrize, we first have to
complexifyit to Gc. e-qV then acts a compensator for the complex gauge transformations in effect absorbing them leaving only the real parts. This is what's being done in the Wess-Zumino gauge.
Let's rephrase everything to look more like a conventional Yang-Mill gauge theory. We have a U(1) gauge symmetry acting upon full superspace with a 1-superform gauge connection A. In the analytic basis for the tangent space, the covariant derivative is given by . Integrability conditions for chiral superfields with the chiral constraint leave us with . A similar constraint for antichiral superfields leaves us with . This means that we can either gauge fix or but not both simultaneously. Call the two different gauge fixing schemes I and II respectively. In gauge I, and in gauge II, . Now, the trick is to use two different gauges simultaneously; gauge I for chiral superfields and gauge II for antichiral superfields. In order to bridge between the two different gauges, we need a gauge transformation. Call it e-V (by convention). If we were using one gauge for all fields, would be gauge invariant. However, we need to convert gauge I to gauge II, transforming X to (e-V)qX. So, the gauge invariant quantity is .
In gauge I, we still have the residual gauge where and in gauge II, we have the residual gauge satisfying . Under the residual gauges, the bridge transforms as . Without any additional constraints, the bridge wouldn't give all the information about the gauge field. However, with the additional constraint , there's only one unique gauge field which is compatible with the bridge modulo gauge transformations. Now, the bridge gives exactly the same information content as the gauge field.
Theories with 8 or more SUSY generators
In theories with higher supersymmetry (and perhaps higher dimension), a vector superfield typically describes not only a gauge field and a Weyl fermion but also at least one complex
Wikimedia Foundation. 2010.
Look at other dictionaries:
Seiberg–Witten gauge theory — In theoretical physics, Seiberg Witten gauge theory is a set of calculations that determine the low energy physics mdash; namely the moduli space and the masses of electrically and magnetically charged supersymmetric particles as a function of… … Wikipedia
Theory of everything — A theory of everything (TOE) is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories.… … Wikipedia
String theory — This article is about the branch of theoretical physics. For other uses, see String theory (disambiguation). String theory … Wikipedia
K-theory (physics) — In string theory, the K theory classification refers to a conjectured application of K theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond Ramond field strengths as well as the charges of stable D… … Wikipedia
Wess-Zumino gauge — In particle physics, the Wess Zumino gauge is a particular choice of a gauge transformation in a gauge theory with supersymmetry. In this gauge, the supersymmetrized gauge transformation is chosen in such a way that most components of the vector… … Wikipedia
Mirror symmetry (string theory) — In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi Yau manifolds. It happens, usually for two such six dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they… … Wikipedia
Superstring theory — is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantum gravity.… … Wikipedia
Grand unification theory — articleissues article=y refimprove=June 2006 expert=Physics tone=August 2008 update=March 2008Grand Unification, grand unified theory, or GUT refers to any of several very similar unified field theories or models in physics that predicts that at… … Wikipedia
Topological string theory — In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry.… … Wikipedia
Kaluza–Klein theory — In physics, Kaluza–Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921 and was discovered by the mathematician Theodor Kaluza … Wikipedia