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

TL;DR

This paper discusses the emergence of topological string amplitudes in 4D N=2 M-theory compactifications, focusing on regularization methods for deriving kinetic terms from infinite state sums.

Contribution

It demonstrates the equivalence of regularization procedures in different moduli spaces and extends the regularization to linear terms in the one-loop prepotential.

Findings

Regularization can be performed in both complex structure and Kähler moduli spaces.

The regularization method is extended to linear terms in the one-loop prepotential F_1.

Evidence supports the emergence of topological string amplitudes from quantum effects in M-theory.

Abstract

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 $F_1$.