Semi-Flatland

TL;DR

This work develops a semiflat framework to engineer non-geometric Type II vacua via $T^3$-fibered compactifications with duality monodromies, linking these to M-theory on $G_2$ manifolds and Joyce manifolds. By embedding the perturbative duality group into an auxiliary $T^4$ and using $SL(4,Z)$, the authors map T-duality actions to a geometric setting, enabling systematic construction and spectrum matching with M-theory in several cases. They build explicit backgrounds with ${D_4}$ and ${E_n}$-type singularities, including almost non-geometric asymmetric orbifolds, and analyze their massless spectra and dualities, illustrating how non-geometric data can correspond to geometric uplifts in M-theory. The study also investigates chiral Scherk-Schwarz reductions and outlines routes to lifting moduli via RR fluxes, aiming to broaden the landscape of ${ m obreak N=1}$ vacua with controlled duality properties and potential links to $G_2$ compactifications.

Abstract

We study perturbative compactifications of Type II string theory that rely on a fibration structure of the extra dimensions a la SYZ. Non-geometric spaces are obtained by using T-dualities as monodromies. These vacua generically preserve N=1 supersymmetry in four dimensions, and are U-dual to M-theory on G2 manifolds. Several examples are discussed, some of which admit an asymmetric orbifold description. The massless spectrum is matched to that of the dual M-theory compactification on a Joyce manifold when a comparison is possible. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber. We also give an explanation from this semiflat viewpoint for the Hanany-Witten brane-creation effect and for the equivalence of the Type IIA orientifold on T5/Z2 and Type IIB on S1 x K3.

Figures (35)

• Figure 1: A possible fundamental domain (gray area) for the action of the $SL(2,\mathbb{Z})$ modular group on the upper half-plane. The upper-half plane parametrizes the possible values of $\tau$ (or $\rho$): the moduli of a two-torus. The gray domain can be folded into an $S^2$ with three special points (the two orbifold points: $\tau_{\mathbb{Z}_6} = e^{2\pi i /6}$ and $\tau_{\mathbb{Z}_4} = i$, and the decompactification point: $\tau\rightarrow i \infty$).
• Figure 2: Base of the $T^4/\mathbb{Z}_2$ orbifold. The $\mathbb{Z}_2$ action inverts the base coordinates and has four fixed points denoted by red stars. They have $180^\circ$ deficit angle. As the arrows show, one has to fold the diagram and this gives an $S^2$.
• Figure 3: Flat $S^2$ base constructed from four triangles: base of $K3$ in the $\mathbb{Z}_2$ orbifold limit.
• Figure 4: Singularities in the base of $T^6 / \mathbb{Z}_2 \times \mathbb{Z}_2$. The big dashed cube is the original $T^3$ base. The orbifold group generates the singular lines as depicted in the figure. The red dots show the intersection points of these edges.
• Figure 5: (i) Rhombic dodecahedron: fundamental domain for the base of $T^6/\mathbb{Z}_2\times\mathbb{Z}_2$. Six pyramids are glued on top of the faces of a cube. Neighboring pyramid triangles give rhombi since the vertices are coplanar ( e.g. $ABCD$). (ii) The $S^3$ base can be constructed by identifying triangles as shown by the arrows. After gluing, the deficit angle around cube edges is $180^\circ$ which is appropriate for a $D_4$ singularity. The dihedral angles of the dashed lines are $120^\circ$ and since three of them are glued together, there is no deficit angle. The tips of the pyramids get identified and the space finally becomes an $S^3$.