U(n) Spectral Covers from Decomposition

TL;DR

The paper develops decomposed spectral covers for heterotic bundles with $S(U(1)\times U(4))$, $S(U(2)\times U(3))$, and $S(U(1)\times U(1)\times U(3))$, showing that realizing a low-energy $U(1)$ requires tuning both the $SU(5)$ spectral data and the Calabi–Yau complex structure. By embedding $SU(5)$ covers into $Sp(5)$ language and analyzing global sections, it constructs explicit parameterizations (cases I and II) that isolate $U(1)$ factors within each decomposed structure, while enforcing holomorphy constraints tied to the base geometry. In F-theory, the dual analysis in the stable degeneration limit reveals that the monodromy locus for two-cycles factorizes with squared factors, signaling reduced monodromy and the presence of $U(1)$ symmetries; this is illustrated concretely in $E_6$ gauge theory with a $(1+2)$ decomposition. The work provides a systematic framework for engineering $U(1)$ symmetries in F-theory via heterotic-inspired spectral data and modular tuning, with potential implications for proton decay suppression and flavor structure in realistic models.

Abstract

We construct decomposed spectral covers for bundles on elliptically fibered Calabi-Yau threefolds whose structure groups are S(U(1) x U(4)), S(U(2) x U(3)) and S(U(1) x U(1) x U(3)) in heterotic string compactifications. The decomposition requires not only the tuning of the SU(5) spectral covers but also the tuning of the complex structure moduli of the Calabi-Yau threefolds. This configuration is translated to geometric data on F-theory side. We find that the monodromy locus for two-cycles in K3 fibered Calabi-Yau fourfolds in a stable degeneration limit is globally factorized with squared factors under the decomposition conditions. This signals that the monodromy group is reduced and there is a U(1) symmetry in a low energy effective field theory. To support that, we explicitly check the reduction of a monodromy group in an appreciable region of the moduli space for an $E_6$ gauge theory with (1+2) decomposition. This may provide a systematic way for constructing F-theory models with U(1) symmetries.

Figures (9)

• Figure 1: The configuration of the twelve 7-branes and two-cycles in the rational elliptic surface $W_1 = \pi_{Y_4}^{-1}(b)$.
• Figure 2: The extended Dynkin diagram of the $E_8$ Lie algebra. Each node corresponds to a two-cycle in Figure \ref{['fig:2-cycles']}.
• Figure 3: The left figure (a) shows the five branch points in the $a_0$-plane. The cross mark represents the base point $b$. The right figure (b) shows the loops corresponding to the five branch points in the $a_0$-plane.
• Figure 4: The configuration of 7-branes at the base point $b$.
• Figure 5: The configuration of 7-branes at the base point $b_1$.