On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups

TL;DR

The paper establishes a concrete link between M-theory $G_4$ flux quantization in F-theory and Freed-Witten anomalies of Type IIB 7-branes, showing a half-integer shift in $G_4$ whenever the dual D7-brane wraps a non-spin submanifold. By constructing nontrivial, non-complete-intersection 4-cycles in resolved elliptically fibered Calabi–Yau fourfolds, the authors derive explicit formulas for $\int_{C_{(4)}} c_2(\tilde Z_4)$ that reflect the spin properties of the brane locus, with distinct treatments for Sp, SU($2N$), SU($2N+1$), and SU(5) examples. The analysis employs two Tate-model Ansätze to define detecting cycles, connects to Sen limits for weak coupling, and extends to $U(1)$-restricted configurations, showing broad validity across unitary and symplectic gauge groups except for $SU(3)$. These results enhance understanding of flux quantization consistency, moduli stabilization implications, and the global structure of F-theory compactifications with realistic gauge sectors. The work provides a practical framework for predicting half-integer shifts from purely geometric data, with potential impact on global model building and anomaly cancellation.”

Abstract

We study the quantization of the M-theory G-flux on elliptically fibered Calabi-Yau fourfolds with singularities giving rise to unitary and symplectic gauge groups. We seek and find its relation to the Freed-Witten quantization of worldvolume fluxes on 7-branes in type IIB orientifold compactifications on Calabi-Yau threefolds. By explicitly constructing the appropriate four-cycles on which to calculate the periods of the second Chern class of the fourfolds, we find that there is a half-integral shift in the quantization of G-flux whenever the corresponding dual 7-brane is wrapped on a non-spin submanifold. This correspondence of quantizations holds for all unitary and symplectic gauge groups, except for SU(3), which behaves mysteriously. We also perform our analysis in the case where, in addition to the aforementioned gauge groups, there is also a 'flavor' U(1)-gauge group.

Figures (4)

• Figure 1: This shows the extended Dynkin diagram of $Sp(2)$ (right) obtained by folding the extended Dynkin diagram of $SU(4)$ (left). The monodromy responsible for the folding acts as a reflection with respect to the vertical dashed line drawn on the left figure.
• Figure 2: This shows the transition from the extended Dynkin diagram of $Sp(2)$ (left) to the extended Dynkin diagram of $SU(5)$ (right) happening along the curve $\{P=Q=0\}$ due to the singularity enhancement. The fifth D-brane of the $SU(5)$ stack is given by the Whitney-type brane. The orange nodes are the fibers of the 4-cycles on which it is possible to detect the Freed-Witten anomaly.
• Figure 3: This shows the transition from the extended Dynkin diagram of $SU(4)$ (left) to the extended Dynkin diagram of $SU(5)$ (right) happening along the curve $\{P=Q=0\}$ due to the singularity enhancement. The fifth D-brane of the $SU(5)$ stack is given by the Whitney-type brane. The orange nodes are the fibers of the 4-cycles on which it is possible to detect the Freed-Witten anomaly.
• Figure 4: This shows the transition from the extended Dynkin diagram of $SU(5)$ (left) to the extended Dynkin diagram of $SO(10)$ (right) happening along the curve $\{P=Q=0\}$ due to the singularity enhancement. Nodes connected by arrows are identified. The orange nodes are the fibers of the 4-cycles on which it is possible to detect the Freed-Witten anomaly.