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 of an associated supermultiplet bundle.
Superfields were introduced by Abdus Salam and J. A. Strathdee in a 1974 article. Operations on superfields and a partial classification were presented a few months later by Sergio Ferrara, Julius Wess and Bruno Zumino.
Naming and classification
The most commonly used supermultiplets are vector multiplets, chiral multiplets (in supersymmetry for example), hypermultiplets (in 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 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 Figueroa-O'Farrill (2001).
A general complex superfield in supersymmetry can be expanded as
where are different complex fields. This is not an irreducible supermultiplet, and so different constraints are needed to isolate irreducible representations.
A (anti-)chiral superfield is a supermultiplet of supersymmetry.
In four dimensions, the minimal supersymmetry may be written using the notion of superspace. Superspace contains the usual space-time coordinates , , and four extra fermionic coordinates with , transforming as a two-component (Weyl) spinor and its conjugate.
In 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 pulled back to the full superspace. Such a function satisfies the covariant constraint , where is the covariant derivative, given in index notation as
A chiral superfield can then be expanded as
where . The superfield is independent of the 'conjugate spin coordinates' in the sense that it depends on only through . It can be checked that
The expansion has the interpretation that is a complex scalar field, is a Weyl spinor. There is also the auxiliary complex scalar field , named 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 by substituting the expression for :
Similarly, there is also antichiral superspace, which is the complex conjugate of chiral superspace, and antichiral superfields.
An antichiral superfield satisfies where
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.
The vector superfield is a supermultiplet of supersymmetry.
A vector superfield (also known as a real superfield) is a function which satisfies the reality condition . Such a field admits the expansion
The constituent fields are
- Two real scalar fields and
- A complex scalar field
- Two Weyl spinor fields and
- A real vector field (gauge field)
Their transformation properties and uses are further discussed in supersymmetric gauge theory.
Using gauge transformations, the fields and can be set to zero. This is known as Wess-Zumino gauge. In this gauge, the expansion takes on the much simpler form
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 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.
A hypermultiplet is a type of representation of an extended supersymmetry algebra, in particular the matter multiplet of supersymmetry in 4 dimensions, containing two complex scalars Ai, a Dirac spinor ψ, and two further auxiliary complex scalars Fi.
The name "hypermultiplet" comes from old term "hypersymmetry" for N=2 supersymmetry used by 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 case. These are constructed by a highest-weight representation construction in the sense that there is a vacuum vector annihilated by the supercharges . The irreps have dimension . For supermultiplets representing massless particles, on physical grounds the maximum allowed is , while for renormalizability, the maximum allowed is .
N = 2
The vector or chiral multiplet contains a gauge field , two Weyl fermions , and a scalar (which also transform in the adjoint representation of a gauge group). These can also be organised into a pair of multiplets, an vector multiplet and chiral multiplet . Such a multiplet can be used to define Seiberg–Witten theory concisely.
The hypermultiplet or scalar multiplet consists of two Weyl fermions and two complex scalars, or two chiral multiplets.
N = 4
- Salam, Abdus; Strathdee, J. (May 1994). Super-Gauge Transformations. pp. 404–409. Bibcode:1994spas.book..404S. doi:10.1142/9789812795915_0047. ISBN 978-981-02-1662-7. Retrieved 3 April 2023.
- Ferrara, Sergio; Wess, Julius; Zumino, Bruno (1974). "Supergauge multiplets and superfields". Phys. Lett. B. 51 (3): 239–241. Bibcode:1974PhLB...51..239F. doi:10.1016/0370-2693(74)90283-4. Retrieved 3 April 2023.
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- Fayet, P. (1976), "Fermi-Bose hypersymmetry", Nuclear Physics B, 113 (1): 135–155, Bibcode:1976NuPhB.113..135F, doi:10.1016/0550-3213(76)90458-2, MR 0416304
- Stephen P. Martin. A Supersymmetry Primer, arXiv:hep-ph/9709356 .
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