nLab F/M-theory on elliptically fibered Calabi-Yau 4-folds

Idea

General

The KK flux compactification of M-theory/F-theory on an elliptically fibered Calabi-Yau variety of (complex) dimension 4 (hence real dimension 8, see also at M-theory on 8-manifolds) yields type IIB superstring theory (with some S-duality-related non-perturbative effects included) KK-compactified on Calabi-Yau 3-folds.

This is subtly different from F-theory on Spin(7)-manifolds, see at Witten's Dark Fantasy.

Realization of moduli stabilization of those flux compactifications is by considering higher dimensional gauge fields in the compact space, the curvature of those fields parametrizes the mass term.

Motivation from string model building/phenomenology

By the discussion at supersymmetry and Calabi-Yau manifolds such compactifications of type IIB superstring theory yield the UV-completion of an effective field theory in 4-dimensions which exhibits $N =1$ supersymmetry.

KK-compactifications of higher dimensional supergravity with minimal ($N=1$) supersymmetry:

perspective	KK-compactification with $N=1$ supersymmetry
M-theory	M-theory on G₂-manifolds
F-theory	F-theory on CY4-manifolds
heterotic string theory	heterotic string theory on CY3-manifolds

For this reason F/M-theory compactifications on elliptically fibered Calabi-Yau 4-folds are a major contender in string phenomenology model building. See at string phenomenology – Models in type II / F-theory for more on this.

Motivation from mathematics

At the same time these types of F-theory KK-compactifications are of interest also for their pure mathematics and as such received a fair amount of attention notably in algebraic geometry.

For instance the duality between F-theory and heterotic string theory translates into a class of rich equivalences of moduli spaces of bundles on different algebraic varieties which is rather non-obvious without the string theory-perspective.

From the abstract of (Donagi 98).

The heterotic string compactified on an $(n-1)$-dimensional elliptically fibered Calabi-Yau $Z \to B$ is conjectured to be dual to F-theory compactified on an $n$-dimensional Calabi-Yau $X \to B$, fibered over the same base with elliptic K3 fibers. In particular, the moduli of the two theories should be isomorphic. The cases most relevant to the physics are $n=2$, $3$, $4$, i.e. the compactification is to dimensions $d=8$, $6$ or $4$ respectively. Mathematically, the richest picture seems to emerge for $n=3$, where the moduli space involves an analytically integrable system whose fibers admit rather different descriptions in the two theories.

Moreover, the description of the supergravity C-field (in the physics literature: the “4-form flux”, see at flux compactification), being (up to various twists and subtleties, see there) a cocycle in degree-4 ordinary differential cohomology on the Calabi-Yau variety involves the moduli spaces of (higher) line bundles on the $CY_4$ which under decomposition of the C-field into various cup-product factors associated with the various divisor of the $CY_4$ involves intermediate Jacobians and Artin-Mazur formal groups in degrees 0,1,2 with rich theory.

Related concepts

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

$dim_{\mathbb{C}}(\Sigma)$	1	2	3	4	5
	F-theory	F-theory on CY2	F-theory on CY3	F-theory on CY4	F-theory on CY5

• F-theory on Spin(7)-manifolds

KK-compactification of M-theory

• M-theory on a circle

• M-theory on 8-manifolds/on Spin(7)-manifolds

• M-theory on G₂-manifolds

  • G₂-MSSM

• M-theory on Sp(1)-manifolds

• M-theory on hyperbolic manifolds

• heterotic M-theory on ADE-orbifolds

• M-theory on Calabi-Yau manifolds

  • on CY2-folds

  • on CY3-folds

  • on CY4-folds

  • on CY5-folds

References

Reviews of the general setup include

• Thomas Grimm, Raffaele Savelli, sections 3.1, 3.2 of Gravitational Instantons and Fluxes from M/F-theory on Calabi-Yau fourfolds (arXiv:1109.3191)

Discussion of the duality between F-theory and heterotic string theory for compactifications on elliptically fibered Calabi-Yau manifolds includes

• Ron Donagi, ICMP lecture on heterotic/F-theory duality (arXiv:hep-th/9802093)

• Ron Donagi, Eyal Markman, Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles (arXiv:alg-geom/9507017)

Discussion of relation to M-theory on G₂-manifolds/M-theory on Spin(8)-manifolds includes

• Mariana Graña, C. S. Shahbazi, Marco Zambon, $Spin(7)$-manifolds in compactifications to four dimensions, JHEP11(2014)046 (arXiv:1405.3698)

For more on this see at M-theory on Spin(7)-manifolds