Invariants and divergences in half-maximal supergravity theories

TL;DR

The paper classifies invariants in half-maximal supergravity in $D=4,5$ up to dimension eight, examining how the $\mathfrak{sl}_2\mathds{R}$ anomaly in $D=4$ and the absence of a one-loop dilaton anomaly in $D=5$ constrain divergences. Using harmonic superspace and the ectoplasm formalism, it shows that certain $R^4$-type invariants can be written either as full-superspace integrals with non-duality-invariant integrands or as twelve- or sixteen-theta harmonic integrals with manifest duality invariance, leading to an F/D borderline for potential counterterms. Under the (strong) assumption of a full $16$-supercharge off-shell formulation that preserves duality, the authors argue that eligible three-loop ($D=4$) and two-loop ($D=5$) divergences must be duality-invariant and thus absent for pure half-maximal theories, though vector multiplets introduce nontrivial anomaly terms that can alter the argument in $D=4$. They also analyze six- and eight-derivative invariants, including a class of protected $R^4$-type terms, and discuss non-renormalisation theorems, descent equations for duality co-forms, and implications for string theory effective actions and perturbative finiteness. The results connect field-theoretic symmetry constraints with string-theoretic expectations, highlighting the intricate interplay between duality, supersymmetry, and higher-derivative invariants in half-maximal supergravity.

Abstract

The invariants in half-maximal supergravity theories in D=4,5 are discussed in detail up to dimension eight (e.g. R^4). In D=4, owing to the anomaly in the rigid SL(2,R) duality symmetry, the restrictions on divergences need careful treatment. In pure N=4 supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L>1 loops would also have to be writable as full-superspace integrals. At the three-loop order such a duality-invariant full-superspace integral candidate counterterm exists, but its duality invariance is marginal in the sense that the full-superspace counter-Lagrangian is not itself duality-invariant. We show that such marginal invariants are not allowable as counterterms in a 16-supercharge off-shell formalism. It is not possible to draw the same conclusion when vector multiplets are present because of the appearance of F^4 terms in the SL(2,R) anomaly. In D=5 there is no one-loop anomaly in the shift invariance of the dilaton, and we argue that this implies finiteness at two loops, again subject to the assumption that 16 supercharges can be preserved off-shell.