Fixing D7 Brane Positions by F-Theory Fluxes

TL;DR

The paper addresses fixing D7-brane positions in F-theory via fluxes, using the M-theory dual on $K3\times K3$ to derive an explicit, $SO(3)\times SO(3)$-covariant flux potential and map moduli to D7 data. By analyzing Minkowski minima and the F-theory limit, it shows how to choose flux quanta to stabilize specific brane configurations (e.g., $SO(8)^4$) and even move branes by turning on particular fluxes, while detailing when and how gauge fields become massive. The work provides concrete constructions demonstrating the stabilization of gauge groups, partial moduli fixing, and the potential for supersymmetric vacua (e.g., $\mathcal{N}=2$ in 4d) within controlled F-theory settings, and it outlines paths to extend to more general Calabi–Yau fourfolds. Overall, it offers a calculable framework for engineering brane configurations and gauge content via fluxes in F-theory, with implications for realistic model building and GUT scenarios.

Abstract

To do realistic model building in type IIB supergravity, it is important to understand how to fix D7-brane positions by the choice of fluxes. More generally, F-theory model building requires the understanding of how fluxes determine the singularity structure (and hence gauge group and matter content) of the compactification. We analyse this problem in the simple setting of M-theory on K3xK3. Given a certain flux which is consistent with the F-theory limit, we can explicitly derive the positions at which D7 branes or stacks of D7 branes are stabilised. The analysis is based on a parameterization of the moduli space of type IIB string theory on T^2/Z_2 (including D7-brane positions) in terms of the periods of integral cycles of M-theory on K3. This allows us, in particular, to select a specific desired gauge group by the choice of flux numbers.

Figures (2)

• Figure 1: For an $SO(8)^4$ configuration, the two degrees of freedom of the O-planes and the dilaton are encoded in cycles that surround two of the four blocks in the $\mathbb{C}\mathbb{P}^1$ base and wrap an arbitrary direction in the fibre torus. The four $SO(8)$ blocks are denoted by A,B,C and D. The four cycles displayed here form a basis that is dual to the four forms $e_i, \alpha$ and $\beta$. Note that we also have indicated the fibre part of each cycle.
• Figure 2: (a) The assignment between the geometrically constructed cycles between D-branes and the cycles that are given in the table above. This assignment is the same for each of the four $SO(8)$ blocks. The cross marks the position of the O plane, grey dots denote the D7 branes. Due to the fibre involution in the O-plane monodromy, cycles 3 and 4 do not intersect. (b) The corresponding gauge enhancement: The cycles become the nodes of the Dykin diagram, lines are drawn for intersections.