Harmonicity in N=4 supersymmetry and its quantum anomaly

TL;DR

The work generalizes the holomorphicity of ${ m N}=2$ vector-couplings to ${ m N}=4$ 1/2-BPS operators using harmonic superspace, showing that their moduli-dependent couplings ${ m F}_g$ satisfy first- and second-order harmonicity constraints with anomalies. These constraints arise from worldsheet boundary contributions in string theory, yielding recursion relations for the non-analytic parts of the couplings, and are checked against explicit one-loop heterotic amplitudes and their six-dimensional decompactifications. The authors extend the framework to ${ m N}=4$ conformal supergravity, derive covariantized expressions, and demonstrate that the same analyticity structure persists under decompactification to six dimensions. The results provide a unified description of analyticity and anomalies for 1/2-BPS operators across dimensions, with implications for topological amplitudes and moduli-space dynamics in extended supersymmetry.

Abstract

The holomorphicity property of N=1 superpotentials or of N=2 F-terms involving vector multiplets is generalized to the case of N=4 1/2-BPS effective operators defined in harmonic superspace. The resulting harmonicity equations are shown to control the moduli dependence of the couplings of higher dimensional operators involving powers of the N=4 Weyl superfield, computed by N=4 topological amplitudes. These equations can also be derived on the string side, exhibiting an anomaly from world-sheet boundary contributions that leads to recursion relations for the non-analytic part of the couplings.