Emergence of CY Triple Intersection Numbers in M-theory

TL;DR

This paper demonstrates that classical Yukawa couplings in type IIA Calabi-Yau compactifications can be derived from quantum one-loop effects involving BPS states, supporting the M-theoretic Emergence Proposal.

Contribution

It introduces a novel regularization method for infinite sums of Gopakumar-Vafa invariants, linking classical intersection numbers to quantum effects in string compactifications.

Findings

Classical Yukawa couplings derived from quantum one-loop integrals.

Regularization scheme involving Calabi-Yau degeneration limits.

Validation through explicit period calculations for specific threefolds.

Abstract

To give more credence to the M-theoretic Emergence Proposal it is important to show that also classical kinetic terms in a low energy effective action arise as a quantum effect from integrating out light towers of states. We show that for compactifications of type IIA on Calabi-Yau manifolds, the classical weak coupling Yukawa couplings, which are the triple intersection numbers of the Calabi-Yau threefold, can be obtained from the 1/2-BPS protected one-loop Schwinger integral over $D2$-$D0$ bound states, after employing a novel regularization for the final infinite sum of Gopakumar-Vafa invariants. Approaching the problem in a consecutive manner from 6D decompactification over emergent string to the ultimate M-theory limits, we arrive at a mathematically concrete regularization that involves finite distance degeneration limits of Calabi-Yau threefolds in an intriguing way. We test and challenge this proposal by the concrete determination of the periods around such degeneration points for threefolds with one K\"ahler modulus and the two examples $\mathbb P_{1,1,1,6,9}[18]$ and $\mathbb P_{1,1,2,2,6}[12]$.