Contents

Contents

Idea

The KK-compactification of F-theory on Spin(7)-manifolds to 4d. Differs subtly from F-theory on CY4.

Related by T-duality to 3d M-theory on Spin(7)-manifolds

Properties

(Non-)Supersymmetry and (vanishing) cosmological constant

What has been called Witten's dark fantasy in Heckmann-Lawrie-Lin-Zoccarato 19, Section 8 is an argument, going back to Witten 95a, Witten 95b, Sec. 3, Witten 00, p. 7 for the existence of non-perturbative non-supersymmetric 4d string vacua/string phenomenology with fundamentally vanishing cosmological constant (“dark energy”).

The original idea was formulated in terms of 3d M-theory on 8-manifolds decompactified at strong coupling to 4d via duality between M-theory and type IIA string theory (recall the super 2-brane in 4d).

Based on the observation of Vafa 96, Section 4.3 that the argument should have a natural realization in 4d F-theory on Spin(7)-manifolds (T-dual to the previous perspective), a detailed construction was finally laid out in Bonetti-Grimm-Pugh 13, Heckmann-Lawrie-Lin-Zoccarato1 18, Heckman-Lawrie-Lin-Sakstein-Zoccarato 19.

The key technical point is the claim that a careful analysis of D=4 N=1 supergravity obtained after KK-compactification of F-theory on Spin(7)-manifolds T-dual to M-theory on Spin(7)-manifolds reveals a “1/2 supersymmetry” where

1. the vacuum state is supersymmetric and hence has vanishing cosmological constant;

2. but no finite-energy-excitation of the vacuum appears supersymmetrically,

hence fermions and bosons in the model do not appear in supersymmetric spectra.

Relation to J-twisted Cohomotopy

On a spin-manifold of dimension 8 a choice of topological Spin(7)-structure is equivalently a choice of cocycle in J-twisted Cohomotopy cohomology theory. This follows (FSS 19, 3.4) from

1. the standard coset space-structures on the 7-sphere (see here)

2. the fact that coset spaces $G/H$ are the homotopy fibers of the maps $B H \to B G$ of the corresponding classifying spaces (see here)

F-theory KK-compactified on elliptically fibered complex analytic fiber $\Sigma$

$dim_{\mathbb{C}}(\Sigma)$12345
F-theoryF-theory on CY2F-theory on CY3F-theory on CY4F-theory on CY5

References

General

The concept goes back to