F-theory and Dark Energy

TL;DR

This work develops a framework for 4D cosmology from F-theory on warped Spin(7) backgrounds, using a warped $\mathbb{R}_{\text{time}}\times S^3\times Y_8$ geometry with an $S^3$ WZW factor at level $N$. Through a dual M-theory description and a detailed Killing-spinor analysis, it shows that two real supercharges can be preserved in a $(2,2)$ setup, while finite-energy excitations break supersymmetry in Lorentzian signature, leading to a 1D quantum mechanics for the scale factor and a TeV-scale mass splitting $\,\Delta m_{4D}\sim \sqrt{M_{IR}M_{UV}}$ between superpartners. The study combines a local 7D Donaldson–Witten twisted gauge theory for intersecting 7-branes with a non-holomorphic Spin(7) fibration, producing a 4D theory with an effectively broken $\mathcal{N}=1/2$ structure and a dark-energy–driven cosmology, including runaway moduli and a scale-factor dynamics amenable to quantum cosmology. A novel Spin(7) spectral-cover approach ties localized matter to a local GUT-like setup, and the large-$N$ WZW/M-theory correspondence provides a UV completion via untwisted sectors while signaling the necessity of including twisted sectors for full unitarity. Collectively, the results offer a mechanism to address the cosmological constant problem, connect dark energy to a TeV-scale split spectrum, and motivate further exploration of Spin(7) compactifications in global models and their phenomenological consequences.

Abstract

Motivated by its potential use as a starting point for solving various cosmological constant problems, we study F-theory compactified on the warped product $\mathbb{R}_{\text{time}} \times S^3 \times Y_{8}$ where $Y_{8}$ is a $Spin(7)$ manifold, and the $S^3$ factor is the target space of an $SU(2)$ Wess--Zumino--Witten (WZW) model at level $N$. Reduction to M-theory exploits the abelian duality of this WZW model to an $S^3 / \mathbb{Z}_N$ orbifold. In the large $N$ limit, the untwisted sector is captured by 11D supergravity. The local dynamics of intersecting 7-branes in the $Spin(7)$ geometry is controlled by a Donaldson--Witten twisted gauge theory coupled to defects. At late times, the system is governed by a 1D quantum mechanics system with a ground state annihilated by two real supercharges, which in four dimensions would appear as "$\mathcal{N} = 1/2$ supersymmetry" on a curved background. This leads to a cancellation of zero point energies in the 4D field theory but a split mass spectrum for superpartners of order $Δm_\text{4D} \sim \sqrt{M_\text{IR} M_\text{UV}}$ specified by the IR and UV cutoffs of the model. This is suggestively close to the TeV scale in some scenarios. The classical 4D geometry has an intrinsic instability which can produce either a collapsing or expanding Universe, the latter providing a promising starting point for a number of cosmological scenarios. The resulting 1D quantum mechanics in the time direction also provides an appealing starting point for a more detailed study of quantum cosmology.

Figures (5)

• Figure 1: The WZW-interpretation of F-theory on $S^3 \times Y_{8}$ allows for a T-dual description in terms of the Hopf fiber of the $S^3$ with $N$ units of flux (left and middle). The large $N$ limit projects onto the untwisted sector which corresponds to M-theory on $S^2 \times Y_{8}$ (right).
• Figure 2: Depiction of the spectral cover description obtained from 7D super-Yang--Mills theory wrapped over a four-manifold $X_{\text{GUT}}$, as governed by the Donaldson--Witten twist. The total space is a non-compact $G_2$ manifold given by the bundle of self-dual two-forms over $X_{\text{GUT}}$. Pairwise intersections lead to matter localized on Riemann surfaces and triple intersections produce Yukawa couplings.
• Figure 3: Depiction of the local model generated by gauge theory on a $6$-brane wrapped on a four-manifold $X_{\text{GUT}} = S_\text{GUT}$ in the limit of 3D $\mathcal{N} = 2$ supersymmetry (left) and its generalization to 3D $\mathcal{N} = 1$ supersymmetry (right). In the case of enhanced supersymmetry, there is a local Calabi--Yau fourfold, and the moduli of the $Spin(7)$ model correspond to a real slice through these parameters. In the corresponding spectral cover construction, this real slice amounts to a $\mathbb{Z}_2$ identification of two spectral covers with a fixed $5$-cycle "at infinity." In the limit where the $6$-brane sits on top of the $\mathbb{Z}_2$ invariant locus one obtains a different twisted gauge theory on the four-manifold. The local geometry experienced by a $6$-brane wrapped on such a four-manifold is an $\mathbb{RP}^2$ bundle over $X_{\text{GUT}}$.
• Figure 4: Depiction of the proposed physical construction of $Spin(7)$ manifolds using elliptically fibered Calabi--Yau fourfold building blocks, glued along a $\text{CY}_3^\text{Het}$ in an asymptotically cylindrical region. In the Type IIB picture, we wrap NS5-branes on a local 5-cycle $S_\text{GUT} \times S^1$, which is stable against perturbations because of a running dilaton profile. After taking into account backreaction, the NS5-branes dissolve into flux, leaving behind a $Spin(7)$ geometry with a fluxed $S^3$ in the 4D spacetime.
• Figure 5: Depiction of the instability of the effective Newtonian potential for the scale factor $a$ in our model. Much as in the case of the Einstein static Universe, this can lead to either a collapsing or expanding Universe.