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

TL;DR

The paper develops a superspace method to construct integral invariants for theories with sixteen supersymmetries, applying harmonic superspace to D=4, N=4 SYM to derive the $F^2$, $F^4$, and $(F^5 + \partial^2 F^4)$ terms in the coincident D-brane effective action. It shows that these invariants correspond to specific superconformal multiplets, notably one-half BPS and Konishi descendants, and derives an abelian $\partial^4 F^4$ invariant with non-abelian generalisations, including double-trace structures. The framework is extended to D=6,(2,0) and D=3,N=8 multiplets and to D=4,N=8 supergravity in its linearised form, yielding a finite set of sub-superspace invariants that align with known higher-derivative corrections and potential loop-counterterm structures. Overall, the approach reproduces stringy tree-level terms up to $\alpha'^4$, clarifies the role of BPS versus long multiplets in higher-derivative corrections, and suggests a path toward understanding non-abelian Born-Infeld-type actions within a manifestly supersymmetric, group-theoretic framework.

Abstract

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 + \del^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 \del^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+\del^2F^4) arises as a full superspace integral of the Konishi multiplet K and the abelian \del^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.