More on Dimension-4 Proton Decay Problem in F-theory -- Spectral Surface, Discriminant Locus and Monodromy

TL;DR

This work examines whether a factorized spectral surface in F-theory can robustly forbid dimension-4 proton decay operators. By moving beyond the 7+1D gauge-theory description to a geometric analysis of monodromy on 2-cycles in a Calabi–Yau 4-fold (via a K3-fibered model and string junctions, with Heterotic duals), the authors show that in simple factorization limits the monodromy is reduced, but the full geometry induces monodromies that mix E8 cycles and destroy any unbroken U(1). Consequently, proton-decay operators are likely generated in the factorized scenario, though several loopholes—such as special topologies, alternative factorization schemes, or duality-based cancellations—could still salvage proton stability. The results underscore the limits of gauge-theory approximations and highlight monodromy as a key diagnostic for persistent U(1) symmetries in F-theory compactifications. Overall, the paper provides a geometrical critique of the factorized spectral-surface approach and outlines potential pathways to reestablish proton stability through more refined constructions or dual descriptions.

Abstract

Factorized spectral surface scenario has been considered as one of solutions to the dimension-4 proton decay problem in supersymmetric compactifications of F-theory. It has been formulated in language of gauge theory on 7+1 dimensions, but the gauge theories descriptions can capture physics of geometry of F-theory compactification only approximately at best. Given the severe constraint on the renormalizable couplings that lead to proton decay, it is worth studying without an approximation whether or not the proton decay operators are removed completely in this scenario. We clarify how the behavior of spectral surface and discriminant locus are related, study monodromy of 2-cycles in a Calabi--Yau 4-fold geometry, and find that the proton decay operators are likely to be generated in a simple factorization limit of the spectral surface. A list of loopholes in this study, and hence a list of chances to save the factorized spectral surface scenario, is also presented.

Figures (10)

• Figure 1: The independent 2-cycles of a K3 surface. Dotted lines represent the branch cuts of 7-branes. Black blobs represent $A$-branes, open circles are B-branes, open boxes correspond to C-branes and open triangles are "D"-branes whose $[p,q]$ charge is [3, 1]. $A8^{\prime} \sim D^{\prime}$ 7-branes have the same $[p,q]$ charges as those without a $'$, and are ordered from left to right as in the same way as $A8 \sim D$ 7-branes. 7-branes from $A7$ to $C2$ and those from $A7'$ to $C2'$ constitute separate sets of $[A^7BC^2]$ 7-brane configuration of $E_8$.
• Figure 2: 2-cycles inside $E_8$ and their intersection. Note that $C_{-\theta}$ is not the same as $C_{A87}$; see footnote \ref{['fn:K3-RES']} for more.
• Figure 3: The monodromy locus $\tilde{\Delta}_{\mathrm{lower-quadr}} = 0$ in (\ref{['eq:tilde_discriminant_lqp']}) on the $a_2$-plane and independent loops around these points. $a_2 = a_{2-A}$ is a triple point and correspond to the first factor $a_2 = 0$ of $\tilde{\Delta}_{\mathrm{lower-quadr}}$. $a_2 = a_{2-1,2,3}$ are single points, correspond to the second factor of $\tilde{\Delta}_{\mathrm{lower-quadr}}$. The base point is denoted by a cross mark.
• Figure 4: The monodromy locus $\tilde{\Delta}_{\mathrm{lower-quad}} = 0$ on the $c_2$-plane and independent loops around these points. The cross mark is the base point $c_2 = c_{2,0}$.
• Figure 5: The $B=0$ component of the monodromy locus consists of four points on the $a_0$-plane containing the base point (cross mark). Among them, only $a_{0-out}$ lies in the 8D gauge theory region $|\epsilon_K^2 \epsilon_\eta| \ll |a_0|$, and all others $a_{0-4,5,6} \sim {\cal O}(\epsilon_K^2 \epsilon_\eta)$ are located outside of the 8D gauge theory region (in a shaded region in (a)).