Anomaly-Free Spectra, Unimodular Lattices and 6D R-Symmetry Gauged Supergravity

Abstract

We study the classification problem for anomaly-free 6D $\mathcal{N}=(1,0)$ supergravities with a gauged abelian R-symmetry and one tensor multiplet. We present eleven new models with gauge group $G_{\mathrm{non}-\mathrm{Abelian}}\times U(1{)}_R$ that satisfy the local Green--Schwarz factorization condition, together with several recently proposed global consistency conditions. In particular, the low-rank models we found are precisely where some of the recent enumeration literature is least directly applicable. These examples suggest that the landscape of anomaly-free gauged $U(1{)}_R$ supergravities may be richer than previously recognized while still remaining highly constrained. We analyze the arithmetic structure of the anomaly coefficients, including their integral pairings, embeddability into rank-two unimodular charge lattices, the characteristic-vector condition and ghost-free gauge-field conditions. We show that $n_V\equiv 8(\mathrm{mod}12)$ is necessary and sufficient for the unimodular embeddability in the rank-two case, when the gauge group does not contain $SU(2)$, $SU(3)$ and $G_2$. For the characteristic-vector condition we verify sufficiency for the branches realized by our examples and identify a remaining branch requiring additional exclusion. We also present a detailed discussion of the contribution to the anomaly polynomial when the $D_4$ Lie algebra is present. These results sharpen the boundary between anomaly-free 6D spectra, global-consistency constraints, and possible UV realization in string theory or F-theory.