Integral invariants in N = 4 SYM and the effective action for coincident D - branes

The construction of supersymmetric invariant integrals is discussed in a superspace setting. The formalism is applied to D = 4, N = 4 SYM and used to construct the F 2 , F 4 and (F 5 + ∂ 2 F 4) terms in the effective action of coincident D-branes. The results are in agreement with those obtained by other methods. A simple derivation of the abelian ∂ 4 F 4 invariant is given and generalised to the non-abelian case. We also find some double-trace invariants. The invariants are interpreted in terms of superconformal multiplets: the F 2 and F 4 terms are given by one-half BPS multiplets, the (F 5 + ∂ 2 F 4) arises as a full superspace integral of the Konishi multiplet K and the abelian ∂ 4 F 4 term comes from integrating the fourth power of the field strength superfield. Counterparts of the abelian invariants are exhibited for the D = 6, (2, 0) tensor multiplet and the D = 3, N = 8 scalar multiplet. The method is also applied to D = 4, N = 8 supergravity. All invariants in the linearised theory (with SU (8) symmetry) which arise from partial superspace integrals are constructed.