Math 2024, 17: Calabi-Yau, Heterotic Compactifications & Type II Orientifolds

Kaehler Manifolds have holonomy group U(n) and there can be defined a Ricci (1,1)-form, so that the Ricci-class directly defines the first Chern class.

By the Calabi-Yau theorem, A Kaehler n-fold admits a unique Ricci-flat metric for given Kaehler class iff the first Chern class is 0. CYₙ has several topological properties, which can be fixed in a Hodge diamond.

Deformations of the internal CY-metric retaining Ricci-flatness have a corresponding scalar moduli field. Such deformations are 2-fold. One group is the volume moduli, in which the Kaehler moduli measures volumes.

The other is such, that the entries of the Hodge diamond give the massless complex scalars

The traditional way to obtain 4d N=1 compactifications with gauge dynamics starts from the heterotic string in CY₃. Assume gaugino variation on G either SO(32) or E₈ ⨉ E₈

For the spin connection, SU(3) is a gauge field and the structure group of TM. SU(3) then needs to be embedded into G and G breaks to a SU(3)-commutant.

Applying the SUSY condition from the gaugino variation will give hermitian Young-Mills equations as equivalent formulation of the condition. They imply that F is the curvature of a holomorphic vector bundle V, and one of the equations can be solved iff V is stable with respect to the slope.

The vacuum on the target space then is additionally defined by the choice of said slope-stable vector bundle by the second Chern-form. Such a choice could break G again to a commutant of the structure group of V within G. The interactions in 4D follow from 10D SYM interactions by dimensional reductions. The couplings in 4D correspond to the internal wave function overlapping on the target space.

type IIA/B on ℝ(1, 3) ⨉ M⁶, M⁶ = CY₃ → 4D N=2. Spacetime-filling Dp-branes with worldvolume

is subject to stability, supersymmetry at KK scale, and Gauss' law, which is expressed here through the tadpole cancellation condition. These reinforce a Poincare-duality between the k-th homology group and the k-th cohomology group, fix 4d N=1 SUSY at KK-scale, and give the Bianchi-identity for the relevant F-forms

On compact spaces then, the total charge vanishes, since the flux lines can't escape to infinity. To avoid instability, take several branes along Σ, which is incompatible with SUSY at first glance. To re-enable this construction, SUSY objects of negative RR-charge are required, which exist in the type II orientifolds. These objects fix WS-parity Ω and holomorphic involution σ in type IIB SUSY. type IIA orientifolds take the anti-holomorphic involution instead. The construction then is

We also require a gauge group ΠU(N) where the total rank is constrained by the size of the Oₚ-planes. At this point, embedding SU(3) ⨉ SU(2) ⨉ U(1) into the gauge group in configuration with 3 chiral massless generation of standard model matter yields the standard model, along with extra hidden sectors generated by other branes and moduli/axions.