Matter coupling and anomaly cancellation in supersymmetric σ - models

Generalized matter couplings to four-dimensional supersymmetric sigma models on general Kähler manifolds are presented, preserving all holomorphic symmetries. Our generalization allows assignment of arbitrary U(1) charges to additional matter fermions, in all representations of (the holomorphic part of) the isometry group. This can be used to eliminate unwanted γ5 anomalies, in particular for the U(1) symmetry arising from the complex structure of the target space. A consistent gauging of this isometry group, or any of its subgroups, then becomes possible. When gauged in the presence of a chiral scalar multiplet, the U(1) symmetry is broken spontaneously, generating a mass for the U(1) vector multiplet via the supersymmetric Higgs effect. As an example we discuss the case of the homogeneous coset space E6/SO(10) × U(1). Supersymmetric sigma-models in four space-time dimensions are formulated in terms of chiral superfields Φ, the physical components of which are complex scalars z and (left-handed) chiral spinors ψ L. Because of the complex nature of the fields and the restrictions imposed by supersymmetry, the target space of the scalars is a Kähler manifold [1, 2]. For models with rigid supersymmetry in flat space-time, the kinetic part of the lagrangean is given in terms of a real composite vector superfield K(Φ̄,Φ) by the following supersymmetric expression Lσ = K(Φ̄,Φ)|D = −gαα ( ∂z̄∂μz α + ψ̄ α L D/ψ α L ) + 1 8 Rααββ ψ̄ α L γ ψ L ψ̄ β L γμψ β L. (1) The real symmetric scalar function K(z̄, z) is the local Kähler potential from which the complex hermitean metric gαα is derived as its mixed second derivative gαα = K,αα. (2) The corresponding connection and curvature components are Γ α βγ = g gβα,γ, Rααββ = gαγ Γ γ αβ, β. (3) The covariant derivative of the spinor field is formed with the holomorphic pullback of the connection Γ α βγ Dμψ α L = ∂μψ α L − ∂μz Γ α βγ ψ L. (4) The lagrangean Lσ is by construction invariant under a U(1) symmetry multiplying the superfields Φ, hence all its components (z, ψ L), by a universal phase factor e. In geometrical language the symmetry can be represented in terms of a holomorphic Killing vector R θ (z) by the transformations δθz α = θR θ (z) = iθq(α)z , δθψ α L = θR α θ,β ψ β L = iθq(α)ψ α L. (5) Here the q(α) represent the U(1) charges of the fields. Other symmetries may be present, depending on the properties of the Kähler manifold on which the scalars take their values. In particular, there may be a larger set of holomorphic Killing vectors R i (z) defining a Lie-algebra with structure constants f k ij : R i R α j,β − R jR i,β = f k ij R k . (6) Then the lagrangean (1) is invariant under the infinitesimal transformations generated by the derivation δ = θδi: δz = θR i (z), δz̄ α = θR̄ α i (z̄), δψ L = θ R i, β(z)ψ , δψ̄ α L = θ R̄ α i, β(z̄) ψ̄ . (7)