Emergence of CY Triple Intersection Numbers in M-theory

TL;DR

This work investigates whether CY triple intersection numbers, which govern the cubic prepotential in type IIA compactifications, can emerge as quantum effects from integrating out light towers of D2–D0 states in the isotropic M-theory limit. It develops a regularization scheme based on modular forms and Harvey–Moore identities to assign finite values to divergent sums over Gopakumar–Vafa invariants and tests it in CYs with h_{11}=1 and h_{11}=2, including elliptic and K3 fibrations. The results show that, in carefully chosen degeneration points and paths, the zero-point Yukawa couplings can reproduce the classical TINs, supporting a nuanced, working realization of the Emergence Proposal, though full universality and all degeneration channels remain open. The findings connect GV growth, CY degeneration geometry, and M-theory infrared physics, suggesting a deep link between quantum gravity constraints and Calabi–Yau moduli-space structure, with GLSM perspectives clarifying where quantum-corrected singularities sit.

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 Kähler modulus and the two examples $\mathbb P_{1,1,1,6,9}[18]$ and $\mathbb P_{1,1,2,2,6}[12]$.

Figures (6)

• Figure 1: Schematic view of the intersections of degeneration loci $D_1=\{\Delta_1=0\}$, $D_2=\{\Delta_2=0\}$, $D_0$ and their resolution.
• Figure 2: Parametric plot of $({\rm Im}( t_1-t_1^{(0)}),{\rm Im}(t_2-t_2^{(0)}))$ for varying $(x_1,x_2)$ with ${\rm Im}(x_i)=0$ and $0\le {\rm Re}(x_1)\le 0.1$, $0.01\le {\rm Re}(x_2)\le 2.0$.
• Figure 3: Schematic view of the intersections of degeneration loci $D_1=\{\Delta_1=0\}$, $D_2=\{\Delta_2=0\}$, $D_0$, $D_\infty$ and their resolution.
• Figure 4: Parametric plot of $({\rm Im}( t_1-t_1^{(0)}),{\rm Im}(t_2))$ for varying $(x_1,x_2)$ with ${\rm Im}(x_i)=0$ and $0\le {\rm Re}(x_1)\le 0.1$, $0.1\le {\rm Re}(x_2)\le 1$.
• Figure 5: Parametric plot of $({\rm Im}( t_1-t_1^{(0)}),{\rm Im}(t_2))$, evaluated in the patches around $\{P_{23},P_{13},P_{03}\}$. We vary the respective coordinates $(x_1,x_2)$ with ${\rm Im}(x_1)=0$, ${\rm Im}(x_2)=0$. For $P_{23}$ we have $0.1<{\rm Re}(x_1)<1$ and $0<{\rm Re}(x_2)<0.1$, for $P_{13}$ it's $0.01<{\rm Re}(x_1)<1$ and $0<{\rm Re}(x_2)<0.05$ and for $P_{03}$ we choose $0.1<{\rm Re}(x_1)<0.5$ and $0<{\rm Re}(x_2)<0.1$.