Localizing the M-Theory Path Integral

Abstract

We propose a supersymmetric localization formula for the local perturbative contribution to the M-theory path integral, reducing it to an integral over equivariant flux data. Relative equivariant extensions of the eleven-dimensional Chern--Simons couplings determine the localized action, while for M2-brane backgrounds a tangent-cotangent character computes the one-loop determinant. This gives the first closed form expression for the complete perturbative ABJM partition function at general squashing and chemical potentials, and new analytic results for the ABJM topologically twisted index describing black holes. Applied to the non-toric $V^{5,2}$ theory, the same prescription yields a closed analytic Airy constant previously known only numerically. Our formulas reproduce all available independent field theory results, and provide analytic predictions at general parameters where no field theory answer is currently known. We derive an all-genus result for the Type IIA expansion which is an equivariant form of the usual topological string constant map result, and also extract integrated correlators.