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Wikipedia / Supermultiplet

In theoretical physics, a supermultiplet is a 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 of an associated supermultiplet bundle.

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

These supersymmetric fields are used to build supersymmetric quantum field theories, where the fields are promoted to operatorss.

History

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

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 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 .

supersymmetry can be expanded as

  1. ,

are different complex fields. This is not an irreducible supermultiplet, and so different constraints are needed to isolate irreducible representations.

Chiral superfield

supersymmetry.

supersymmetry may be written using the notion of superspace, transforming as a two-component (Weyl) spinor and its conjugate.

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 is the covariant derivative, given in index notation as

can then be expanded as

by convention: this is the F-term which plays an important role in some theories.

:

Antichiral superfields

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

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.

Vector superfield

supersymmetry.

. Such a field admits the expansion

The constituent fields are

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

can be set to zero. This is known as Wess–Zumino gauge. In this gauge, the expansion takes on the much simpler form

is the superpartner, 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 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 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 ; this term has been abandoned, but the name "hypermultiplet" for some of its representations is still used.

Extended supersymmetry (N > 1)

case. These are constructed by a highest-weight representation, while for renormalizability.

N = 2

vector or chiral multiplet contains a gauge field, two Weyl fermions (which also transform in the adjoint representation of a gauge group. Such a multiplet can be used to define Seiberg–Witten theory concisely.

hypermultiplet or scalar multiplet chiral multiplets.

N = 4

vector multiplet contains one gauge field, four Weyl fermions, six scalars, and CPT conjugates. This appears in N = 4 supersymmetric Yang–Mills theory.

See also

References