Editing Supermultiplet

{{Short description|A representation of the supersymmetry algebra}}
{{for|the classification scheme for hadrons|Eightfold way (physics)}}

In [[theoretical physics]], a '''supermultiplet''' is a [[group representation|representation]] of a [[supersymmetry algebra]], possibly with [[extended supersymmetry]].

Then a '''superfield''' is a field on [[superspace]] which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply be viewed as a function on superspace. Formally, it is a [[section (fiber bundle)|section]] of an [[associated vector bundle|associated supermultiplet bundle]].

Phenomenologically, superfields are used to describe [[particles]]. It is a feature of supersymmetric field theories that particles form pairs, called [[superpartner]]s where [[bosons]] are paired with [[fermions]].

These supersymmetric fields are used to build supersymmetric [[quantum field theories]], where the fields are promoted to [[Hermitian operator|operator]]s.

==History==
Superfields were introduced by [[Abdus Salam]] and [[J. A. Strathdee]] in a 1974 article.<ref name="salam_strathdee">{{cite book |last1=Salam |first1=Abdus |last2=Strathdee |first2=J. |title=Super-Gauge Transformations |journal=World Scientific Series in 20th Century Physics |date=May 1994 |volume=5 |pages=404–409 |doi=10.1142/9789812795915_0047 |bibcode=1994spas.book..404S |isbn=978-981-02-1662-7 |url=https://www.worldscientific.com/doi/epdf/10.1142/9789812795915_0047 |access-date=3 April 2023}}</ref>  Operations on superfields and a partial classification were presented a few months later by [[Sergio Ferrara]], [[Julius Wess]] and [[Bruno Zumino]].<ref name="fwz">{{cite journal |last1=Ferrara |first1=Sergio |last2=Wess |first2=Julius |last3=Zumino |first3=Bruno |title=Supergauge multiplets and superfields |journal=Phys. Lett. B |date=1974 |volume=51 |issue=3 |pages=239–241 |doi=10.1016/0370-2693(74)90283-4 |bibcode=1974PhLB...51..239F |url=https://dx.doi.org/10.1016/0370-2693%2874%2990283-4 |access-date=3 April 2023|url-access=subscription }}</ref>

==Naming and classification==
The most commonly used supermultiplets are vector multiplets, chiral multiplets (in <math>d = 4,\mathcal{N} = 1</math> supersymmetry for example), hypermultiplets (in <math>d = 4,\mathcal{N} = 2</math> supersymmetry for example), tensor multiplets and gravity multiplets.  The highest component of a vector multiplet is a [[gauge boson]], the highest component of a chiral or hypermultiplet is a [[spinor]], the highest component of a gravity multiplet is a [[graviton]].  The names are defined so as to be invariant under [[dimensional reduction]], although the organization of the fields as representations of the [[Lorentz group]] changes.

The use of these names for the different multiplets can vary in literature. A chiral multiplet (whose highest component is a spinor) may sometimes be referred to as a ''scalar multiplet'', and in <math>d = 4,\mathcal{N} = 2</math> SUSY, a vector multiplet (whose highest component is a vector) can sometimes be referred to as a chiral multiplet.

== Superfields in d = 4, N = 1 supersymmetry ==
Conventions in this section follow the notes by {{harvs|txt|last=Figueroa-O'Farrill|year=2001}}.

A general complex superfield <math>\Phi(x, \theta, \bar \theta)</math> in <math>d = 4, \mathcal{N} = 1</math> supersymmetry can be expanded as

:<math>\Phi(x, \theta, \bar\theta) = \phi(x) + \theta\chi(x) + \bar\theta \bar\chi'(x) + \bar \theta \sigma^\mu \theta V_\mu(x) + \theta^2 F(x) + \bar \theta^2 \bar F'(x) + \bar\theta^2 \theta\xi(x) + \theta^2 \bar\theta \bar \xi' (x) + \theta^2 \bar\theta^2 D(x)</math>,

where <math>\phi, \chi, \bar \chi' , V_\mu, F, \bar F', \xi, \bar \xi', D</math> are different complex fields. This is not an [[irreducible representation|irreducible]] supermultiplet, and so different constraints are needed to isolate irreducible representations.

=== Chiral superfield ===
A (anti-)chiral superfield is a supermultiplet of <math>d=4, \mathcal{N} = 1</math> supersymmetry.

In four dimensions, the minimal <math>\mathcal{N}=1</math> supersymmetry may be written using the notion of [[superspace]]. Superspace contains the usual space-time coordinates <math>x^{\mu}</math>, <math>\mu=0,\ldots,3</math>, and four extra fermionic coordinates <math>\theta_\alpha,\bar\theta^\dot\alpha</math> with <math>\alpha, \dot\alpha = 1,2</math>, transforming as a two-component (Weyl) [[spinor]] and its conjugate.

In <math>d = 4,\mathcal{N} = 1</math> [[supersymmetry]], a '''chiral superfield''' is a function over '''chiral superspace'''. There exists a projection from the (full) superspace to chiral superspace. So, a function over chiral
superspace can be [[Differential geometry|pulled back]] to the full superspace. Such a function <math>\Phi(x, \theta, \bar\theta)</math> satisfies the covariant constraint <math>\overline{D}\Phi=0</math>, where <math>\bar D</math> is the covariant derivative, given in index notation as
:<math>\bar D_\dot\alpha = -\bar\partial_\dot\alpha - i\theta^\alpha \sigma^\mu_{\alpha\dot\alpha}\partial_\mu.</math>

A chiral superfield <math>\Phi(x, \theta, \bar\theta)</math> can then be expanded as

:<math> \Phi (y , \theta ) = \phi(y) + \sqrt{2} \theta \psi (y) + \theta^2 F(y),</math>
where <math> y^\mu = x^\mu + i \theta \sigma^\mu \bar{\theta} </math>. The superfield is independent of the 'conjugate spin coordinates' <math>\bar\theta</math> in the sense that it depends on <math>\bar\theta</math> only through <math>y^\mu</math>. It can be checked that <math>\bar D_\dot\alpha y^\mu = 0.</math>

The expansion has the interpretation that <math>\phi</math> is a complex scalar field, <math>\psi</math> is a Weyl spinor. There is also the auxiliary complex scalar field <math>F</math>, named <math>F</math> by convention: this is the [[F-term]] which plays an important role in some theories.

The field can then be expressed in terms of the original coordinates <math>(x,\theta, \bar \theta)</math> by substituting the expression for <math>y</math>:
:<math>\Phi(x, \theta, \bar\theta) = \phi(x) + \sqrt{2} \theta \psi (x) + \theta^2 F(x) + i\theta\sigma^\mu\bar\theta\partial_\mu\phi(x) - \frac{i}{\sqrt{2}}\theta^2\partial_\mu\psi(x)\sigma^\mu\bar\theta - \frac{1}{4}\theta^2\bar\theta^2\square\phi(x).</math>

==== Antichiral superfields ====

Similarly, there is also '''antichiral superspace''', which is the complex conjugate of chiral superspace, and '''antichiral superfields'''.

An antichiral superfield <math>\Phi^\dagger</math> satisfies <math>D \Phi^\dagger = 0,</math> where
:<math>D_\alpha = \partial_\alpha + i\sigma^\mu_{\alpha\dot\alpha}\bar\theta^\dot\alpha\partial_\mu.</math>

An antichiral superfield can be constructed as the complex conjugate of a chiral superfield.

==== Actions from chiral superfields ====
For an action which can be defined from a single chiral superfield, see [[Wess–Zumino model]].

=== Vector superfield ===
The vector superfield is a supermultiplet of <math>\mathcal{N} = 1</math> supersymmetry.

A vector superfield (also known as a real superfield) is a function <math>V(x,\theta,\bar\theta)</math> which satisfies the reality condition <math>V = V^\dagger</math>. Such a field admits the expansion

:<math>V = C + i\theta\chi - i \overline{\theta}\overline{\chi} + \tfrac{i}{2}\theta^2(M+iN)-\tfrac{i}{2}\overline{\theta^2}(M-iN) - \theta \sigma^\mu \overline{\theta} A_\mu +i\theta^2 \overline{\theta} \left( \overline{\lambda} + \tfrac{i}{2}\overline{\sigma}^\mu \partial_\mu \chi \right) -i\overline{\theta}^2 \theta \left(\lambda + \tfrac{i}{2}\sigma^\mu \partial_\mu \overline{\chi} \right) + \tfrac{1}{2}\theta^2 \overline{\theta}^2 \left(D + \tfrac{1}{2}\Box C\right).</math>

The constituent fields are
* Two real scalar fields <math>C</math> and <math>D</math>
* A complex scalar field <math>M + iN</math>
* Two Weyl spinor fields <math>\chi_\alpha</math> and <math>\lambda^\alpha</math>
* A real vector field ([[gauge field]]) <math>A_\mu</math>

Their transformation properties and uses are further discussed in [[supersymmetric gauge theory]].

Using gauge transformations, the fields <math>C, \chi</math> and <math>M + iN</math> can be set to zero. This is known as [[Wess–Zumino gauge]]. In this gauge, the expansion takes on the much simpler form
:<math> V_{\text{WZ}} = \theta\sigma^\mu\bar\theta A_\mu + \theta^2 \bar\theta \bar\lambda + \bar\theta^2 \theta \lambda + \frac{1}{2}\theta^2\bar\theta^2 D. </math>
Then <math>\lambda</math> is the [[superpartner]] of <math>A_\mu</math>, while <math>D</math> is an auxiliary scalar field. It is conventionally called <math>D</math>, and is known as the [[D-term]].

==Scalars==
A scalar is never the highest component of a superfield; whether it appears in a superfield at all depends on the dimension of the spacetime. For example, in a 10-dimensional N=1 theory the vector multiplet contains only a vector and a [[Majorana–Weyl spinor]], while its dimensional reduction on a d-dimensional [[torus]] is a vector multiplet containing d real scalars.  Similarly, in an [[eleven-dimensional supergravity|11-dimensional theory]] there is only one supermultiplet with a finite number of fields, the gravity multiplet, and it contains no scalars.  However again its dimensional reduction on a d-torus to a maximal gravity multiplet does contain scalars.

==Hypermultiplet==

A '''hypermultiplet''' is a type of representation of an extended [[supersymmetry algebra]], in particular the matter multiplet of <math>\mathcal{N} = 2</math> supersymmetry in 4 dimensions, containing two complex [[Scalar field|scalars]] ''A''<sub>''i''</sub>, a Dirac [[Spinor field|spinor]] ψ, and two further [[Auxiliary field|auxiliary]] complex scalars ''F''<sub>''i''</sub>.

The name "hypermultiplet" comes from old term "hypersymmetry" for ''N''=2 supersymmetry used by {{harvtxt|Fayet|1976}}; this term has been abandoned, but the name "hypermultiplet" for some of its representations is still used.

== Extended supersymmetry (N > 1) ==
This section records some commonly used irreducible supermultiplets in extended supersymmetry in the <math>d = 4</math> case. These are constructed by a [[highest-weight representation]] construction in the sense that there is a vacuum vector annihilated by the supercharges <math>Q^A, A = 1, \cdots, \mathcal{N}</math>. The irreps have dimension <math>2^\mathcal{N}</math>. For supermultiplets representing massless particles, on physical grounds the maximum allowed <math>\mathcal{N}</math> is <math>\mathcal{N} = 8</math>, while for [[renormalization|renormalizability]], the maximum allowed <math>\mathcal{N}</math> is <math>\mathcal{N} = 4</math>.<ref name="kqs">{{cite arXiv |last1=Krippendorf |first1=Sven |last2=Quevedo |first2=Fernando |last3=Schlotterer |first3=Oliver |title=Cambridge Lectures on Supersymmetry and Extra Dimensions |date=5 November 2010|class=hep-th |eprint=1011.1491 }}</ref>

=== N = 2 ===
The <math>\mathcal{N} = 2</math> '''vector''' or '''chiral multiplet''' <math>\Psi</math> contains a [[gauge field]] <math>A_\mu</math>, two [[Weyl fermion]]s <math>\lambda, \psi</math>, and a scalar <math>\phi</math> (which also transform in the [[adjoint representation]] of a [[gauge group]]). These can also be organised into a pair of <math>\mathcal{N} = 1</math> multiplets, an <math>\mathcal{N} = 1</math> vector multiplet <math>W = (A_\mu, \lambda)</math> and chiral multiplet <math>\Phi = (\phi, \psi)</math>. Such a multiplet can be used to define [[Seiberg–Witten theory]] concisely.

The <math>\mathcal{N} = 2</math> '''hypermultiplet''' or '''scalar multiplet''' consists of two Weyl fermions and two complex scalars, or two <math>\mathcal{N} = 1</math> chiral multiplets.

=== N = 4 ===
The <math>\mathcal{N} = 4</math> '''vector multiplet''' contains one gauge field, four Weyl fermions, six scalars, and [[CPT symmetry|CPT]] conjugates. This appears in [[N = 4 supersymmetric Yang–Mills theory]].

==See also==

* [[Supersymmetric gauge theory]]
* [[D-term]]
* [[F-term]]

==References==
{{reflist}}
*{{Citation | last1=Fayet | first1=P. | title=Fermi-Bose hypersymmetry | doi=10.1016/0550-3213(76)90458-2 | mr=0416304  | year=1976 | journal=Nuclear Physics B | volume=113 | issue=1 | pages=135–155|bibcode = 1976NuPhB.113..135F }}
* Stephen P. Martin. ''A Supersymmetry Primer'', [[arxiv:hep-ph/9709356|arXiv:hep-ph/9709356]] .
* Yuji Tachikawa. ''N=2 supersymmetric dynamics for pedestrians'', [[arxiv:1312.2684|arXiv:1312.2684]].
* {{cite arXiv |first=J. M. |last=Figueroa-O'Farrill |title=Busstepp Lectures on Supersymmetry |year=2001 |eprint=hep-th/0109172}}

{{Supersymmetry topics}}
[[Category:Supersymmetry]]