Hidden Selection Rules, M5-instantons and Fluxes in F-theory

TL;DR

The paper develops a refined framework for M5-instanton effects in F-theory by solving the Bianchi identity through perturbative and non-perturbative homology, connecting the self-dual worldvolume flux $T_3$, bulk flux $G_4$, and M2 insertions wrapping fibral curves. It identifies three distinct selection-rule regimes depending on $G_4|_D$ and $T_3$, including cases where charged/neutral F-terms arise via M5–M2 bound states and cases where Yukawa hierarchies receive non-perturbative corrections. By leveraging stable degeneration and algebraic cycles (rational equivalence in CH$^2$), the work clarifies how massive $U(1)$ fluxes constrain instanton contributions and how local $U(1)$ enhancements along instantons modify selection rules in multi-section fibrations. The results bridge M/F-theory descriptions with weakly coupled Type IIB insights, predict novel non-perturbative Yukawa corrections, and point to rich phenomenological implications for moduli stabilisation and flavor structure. The analysis also outlines future directions for explicitly constructing the necessary 3-chains and flux configurations beyond the stable degeneration limit.

Abstract

We introduce a new approach to investigate the selection rules governing the contributions of fluxed M5-instantons to the F-theory four-dimensional effective action, with emphasis on the generation of charged matter F-terms. The structure of such couplings is unraveled by exploiting the perturbative and non-perturbative homological relations, introduced in our companion paper arXiv:1506.06764, which encode the interplay between the self-dual 3-form flux on the M5-brane, the background 4-form flux and certain fibral curves. The latter are wrapped by time-like M2-branes representing matter insertions in the instanton path integral. In particular, we clarify how fluxed M5-instantons detect the presence of geometrically massive $U(1)$s which are responsible for `hidden' selection rules. We discuss how for non-generic embeddings the M5-instanton can probe `locally massless' $U(1)$ symmetries if the rank of its Mordell-Weil group is enhanced compared to that of the bulk. As a phenomenological off-spring we propose a new type of non-perturbative corrections to Yukawa couplings which may change the rank of the Yukawa matrix. Along the way, we also gain new insights into the structure of massive $U(1)$ gauge fluxes in the stable degeneration limit.

Figures (1)

• Figure 1: Fibre topology over the curve $C_{\bf 10}$, corresponding to triangulation $T_{11}$ of Krause:2011xj. The fibral curves arise from the fibral curves $\mathbb P^1_i$ corresponding to the simple roots over $C_{\bf 10}$ via the splittings $\mathbb P^1_1 \rightarrow \mathbb P^1_{14}$, $\mathbb P^1_2 \rightarrow \mathbb P^1_{2B} +\mathbb P^1_{24}$, $\mathbb P^1_3 \rightarrow \mathbb P^1_{3C}$, $\mathbb P^1_4 \rightarrow \mathbb P^1_{4D} +\mathbb P^1_{14} + \mathbb P^1_{24}$.