Beyond the polynomial ansatz: regular $AdS_3\times M_8$ solutions of M-theory

Abstract: We study regular compact solutions of the cohomogeneity-one $AdS_3\times M_8$ system of Gauntlett, Mac Conamhna, Mateos and Waldram, where $M_8$ is an $S<sup>2$ bundle over three constant-curvature Riemann surfaces. We find that an endpoint argument rules out every non-flat curvature-sign sector without a positive-curvature factor, while an analytic deformation gives regular families with four radial shape parameters in all remaining sectors, with the base geometry held fixed. The known polynomial solutions form a proper analytic locus in these families. Non-polynomiality is therefore generic in parameter space, although the deviation from an endpoint-matching polynomial can be very small. Apart from the completely flat exception, the curvature-sign criterion is necessary and sufficient within this ansatz. Four magnetic flux ratios give local coordinates near the reference solution and its small-curvature deformations, and a rational inverse reconstructs candidate endpoint data directly from finite flux ratios. In the fully positive sector we prove that one explicit four-dimensional finite-ratio neighbourhood lies inside the physical flux image. The magnetic periods and leading holographic central charge reduce to endpoint data, and, once the common Brown--Henneaux normalization is fixed, the nontrivial flux dependence of the equivariant localization formula agrees with the endpoint result as an exact algebraic identity. We also quantify the non-polynomial corrections and analyse finite degenerations under explicit convergence hypotheses.