In supersymmetry, 4D N=1 global supersymmetry is the theory of global supersymmetry in four dimensions with a single supercharge. It consists of an arbitrary number of chiral and vector supermultiplets whose possible interactions are strongly constrained by supersymmetry, with the theory primarily fixed by three functions, those being the Kahler potential, the superpotential, and the gauge kinetic matrix. Many common models of supersymmetry are special cases of this general theory, such as the Wess-Zumino model, N=1 super Yang-Mills theory, or the Minimal Supersymmetric Standard Model.
Background
Global supersymmetry has a spacetime symmetry algebra given by the super-Poincaré algebra with a single supercharge. In four dimensions this supercharge can be expressed either as a pair of Weyl spinors or as a single Majorana spinor. The particle content of this theory must belong to representations of the super-Poincare algebra, known as supermultiplets.[1] Without including gravity, there are two types of supermultiplets: a chiral supermultiplet consisting of a complex scalar field and its Majorana spinor superpartner, and a vector supermultiplet consisting of a gauge field along with its Majorana spinor superpartner.
In four dimensions, a general theory has an arbitrary number of chiral multiplets indexed by , along with an arbitrary number of gauge multiplets indexed by . Here are complex scalar fields, are gauge fields, and and are Majorana spinors known as chiralini and gaugini, respectively. Supersymmetry imposes stringent conditions on the way that the supermultiplets can be combined in the theory. In particular, most of the structure is fixed by three arbitrary functions of the complex scalar fields.[2]: 287 The dynamics of the chiral multiplets is fixed by the holomorphic superpotential and the Kahler potential , while the mixing between the chiral and gauge sectors is primarily fixed by the holomorphic gauge kinetic matrix . The gauge group must also be consistent with the properties of the chiral sector.
Scalar manifold geometry
The complex scalar fields in the chiral supermultiplets can be seen as coordinates of a -dimensional manifold, known as the scalar manifold. This manifold can be parametrized using complex coordinates , where the barred index represents the complex conjugate . Supersymmetry ensures that the manifold is necessarily a complex manifold, which is a type of manifolds that locally looks as and whose transition functions are holomorphic.[3]: 80 This is because supersymmetry transformations map into left-handed Weyl spinors, and into right-handed Weyl spinors, so the geometry of the scalar manifold must reflect the fermion spacetime chirality by admitting an appropriate decomposition into complex coordinates.[nb 1]
The scalar manifold also admits a metric compatible with its complex structure, with such a manifold then known as a Hermitian manifold.[4]: 325 The only non-zero components of the metric associated to these manifolds is , with a line element given by
The chirality properties inherited from supersymmetry also imply that any closed loop around the scalar manifold has to maintain the splitting between and .[3]: 80–81 This implies that the manifold has a holonomy group. Such manifolds are known as Kahler manifolds and can alternatively be defined as being manifolds that admit a two-form, known as a Kahler form, defined by
such that .[4]: 330 This also implies that the scalar manifold is a symplectic manifold. These manifolds have the useful property that their metric can be expressed in terms of a function known as a Kahler potential through[5]
where this function is invariant up to the addition of the real part of an arbitrary holomorphic function
Such transformations are known as Kahler transformations and since they do not affect the geometry of the scalar manifold, any supersymmetric action must be invariant under these transformations.
Notes
- ^ To see that the manifold must be a complex manifold, consider a general coordinate redefinition . Since this is a mere redefinition of variables, it should not affect any physical quantities such as chirality. The super-Poincare algebra implies the supersymmetry variation of the scalars and , which differ by the chirality. However, a supersymmetry variation a general coordinate redefinition is given by , where the second term can introduce a right-handed Weyl fermion. To avoid this requires the transition map to be holomorphic , implying a complex manifold.
References
- ^ Weinberg, S. (2005). "25". The Quantum Theory of Fields Volume 3: Supersymmetry. Cambridge University Press. p. 43-53. ISBN 978-0521670555.
- ^ Freedman, D.Z.; Van Proeyen, A. (2012). Supergravity. Cambridge: Cambridge University Press. ISBN 978-0521194013.
- ^ a b Dall'Agata, G.; Zagermann, M. (2021). Supergravity: From First Principles to Modern Applications. Springer. ISBN 978-3662639788.
- ^ a b Nakahara, M. (2003). Geometry, Topology and Physics (2 ed.). CRC Press. ISBN 978-0750306065.
- ^ Tong, D. (2021), "3", Supersymmetric Field Theory (PDF)