The simple supergravity in a 6-dimensional spacetime has four complex supercharges which assemble into a Weyl spinor of SO(1, 5). There are also four different supermultiplet as follows:

i) Gravity multiplet consisting of the 6-bein e_μ ^a , the gravitino ψ_μ and a 2-form potential B_μν⁺ with a self-dual field strength,

ii) Yang-Mills multiplet consisting of a vector field A_μ and a Weyl spinor λ,

iii) Tensor multiplet consisting of the dilaton σ in 2-form potential B_μν⁻ and a Weyl spinor χ,

iv) Hypermatter multiplet consisting of 4 real scalars ϕ ^a and two Weyl spinors ψ ^a .

The SO(1,5) chiralities of ψ_μ and λ or identical and opposite to those of χ and ψ ^a .

An N = 1 supergravity in D = 6 in general will contain one gravity multiplet, n _V gauge multiplets, k tensor multiplets and _H hypermatter multiplets . The vectors and spinors in the vector multiplet transform in the adjoint representation of a gauge group G, i.e. n _V = dim G. G can be a product of several factors. The tensor multiplet is always a singlet of G, while the scalar multiplet can transform non-trivially under G.

For k ≠ 1 there is no covariant action describing the general model, although covariant field equations can be written down. When k = 1 a covariant action can be constructed [1].

For arbitrary values of k, n _V and n _H the modal has chiral anomalies. Unlike D = 4 in D = 6 in addition to pure gauge anomalies there also exist pure gravity and mixed-gauge-gravity anomalies. These anomalies can be cancelled using the Green-Schwarz anomaly cancellation mechanism. To apply this mechanism one needs to construct an 8-form from the Riemann curvature 2-form R and the Yang-Mills 2-form F. In order for the GS mechanism to be applicable, two conditions are necessary [2]:

i) n _V − n _H − s = 29k − 273, where s is the number of hypermatter multiplets which are singlets of the gauge group .

ii) It is also necessary that the quartic invariants in the gauge field stenght, tr F⁴, in the representations of interest factorize into sums of products of trF² in some “fundamental” representation. If these two conditions are satisfied all the chiral anomalies can be eliminated by appropriately transforming the B-fields under local gauge symmetries .

The anomaly 8-forms for various fields are as follows [2]

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where P₁ is the contribution of a chiral 2-form potential. We have also assumed that the gravitino is charged under a U(1) component of the gauge group as in the model of [2].