Supermultiplet

In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra. It consists of a collection of particles, called superpartners, corresponding to operators in a quantum field theory which in superspace are represented by superfields.

Superfields were introduced by Abdus Salam and J. A. Strathdee in their 1974 article Supergauge Transformations. Operations on superfields and a partial classification were presented a few months later by Sergio Ferrara, Julius Wess and Bruno Zumino in Supergauge Multiplets and Superfields.

The most commonly used supermultiplets are vector multiplets, chiral multiplets (in 4d N=1 supersymmetry for example), hypermultiplets (in 4d N=2 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.

Note, however, that the use of these names for the different multiplets can vary in literature. Sometimes a chiral multiplet (whose highest component is a spinor) can be referred to as a scalar multiplet. Also, in N=2 SUSY, a vector multiplet (whose highest component is a vector) can sometimes be referred to as a chiral multiplet.

Especially in theories with extended supersymmetry, supermultiplets can be divided to short supermultiplets and long supermultiplets, essentially according to the dimensionality. The short supermultiplets coincide with the BPS states.

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.

Chiral Superfield

In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace. Superspace contains the usual space-time coordinates ${\displaystyle x^{\mu }}$, ${\displaystyle \mu =0,\ldots ,3}$, and four extra fermionic coordinates ${\displaystyle \theta ^{1},\theta ^{2},{\bar {\theta }}^{1},{\bar {\theta }}^{2}}$, transforming as a two-component (Weyl) spinor and its conjugate.

In N=1 supersymmetry in 3+1D, 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 ${\displaystyle {\overline {D}}f=0}$. Similarly, we also have an antichiral superspace, which is the complex conjugate of chiral superspace, and antichiral superfields.

${\displaystyle \Phi (y,\theta )=A(y)+{\sqrt {2}}\theta \psi (y)+\theta \theta F(y)}$

${\displaystyle y^{\mu }=x^{\mu }+i\theta \sigma ^{\mu }{\bar {\theta }}}$

Vector Multiplet

A vector superfield depends on all coordinates. It describes a gauge field and its superpartner, namely a Weyl fermion that obeys a Dirac equation.

${\displaystyle V=C+i\theta \chi -i{\overline {\theta }}{\overline {\chi }}+{\tfrac {i}{2}}\theta ^{2}(M+iN)-{\tfrac {i}{2}}{\overline {\theta ^{2}}}(M-iN)-\theta \sigma ^{\mu }{\overline {\theta }}v_{\mu }+i\theta ^{2}{\overline {\theta }}\left({\overline {\lambda }}+{\tfrac {i}{2}}{\overline {\sigma }}^{\mu }\partial _{\mu }\chi \right)-i{\overline {\theta }}^{2}\theta \left(\lambda +{\tfrac {i}{2}}\sigma ^{\mu }\partial _{\mu }{\overline {\chi }}\right)+{\tfrac {1}{2}}\theta ^{2}{\overline {\theta }}^{2}\left(D+{\tfrac {1}{2}}\Box C\right)}$

V is the vector superfield (prepotential) and is real (V = V). The fields on the right hand side are component fields.

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

Hypermultiplet

A hypermultiplet is a type of representation of an extended supersymmetry algebra, in particular the matter multiplet of N=2 supersymmetry in 4 dimensions, containing two complex scalars A_i, a Dirac spinor ψ, and two further auxiliary complex scalars F_i.

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.

See also

• Supersymmetric gauge theory
• D-term
• F-term

References

• 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 .
• Yuji Tachikawa. N=2 supersymmetric dynamics for pedestrians, arXiv:1312.2684.