Comments on the Emergence of 4D Topological Amplitudes in M-Theory

The M-theoretic Emergence Proposal claims that all of the terms in the low-energy action arise from quantum effects. After reviewing the current status of this proposal, we focus on four-dimensional compactifications with $N=2$ supersymmetry, where kinetic terms are encoded in topological string amplitudes, such as the prepotential $F_0$. Evidence for the emergence of such terms was provided recently, where in particular it was shown that the classical cubic term in $F_0$ can be obtained by integrating out the light towers of states in the M-theory limit, using a novel regularization of the infinite sum over Gopakumar/Vafa invariants. We address two issues that were left open. First, we show that the regularization can be equivalently performed in complex structure moduli space and in K\"ahler moduli space. Second, we extend the proposed regularization to the linear terms in the one-loop prepotential $F_1$.