Spin(7)-Manifolds as Generalized Connected Sums and 3d N=1 Theories

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four-fold and a $G_2$-holonomy manifold times a circle, respectively, which both asymptote to a Calabi-Yau three-fold times a cylinder. The generalized connected sum construction is first exemplified for Joyce orbifolds, and is then used to construct examples of new compact manifolds with $\mathrm{Spin}(7)$-holonomy. In instances when there is a K3-fibration of the $\mathrm{Spin}(7)$-manifold, we test the spectra using duality to heterotic on a $T^3$-fibered $G_2$-holonomy manifold, which are shown to be precisely the recently discovered twisted-connected sum constructions.Comment: 49 pages, 4 figures; v2: added reference

The Noether-Lefschetz Problem and Gauge-Group-Resolved Landscapes: F-Theory on K3 x K3 as a Test Case

Four-form flux in F-theory compactifications not only stabilizes moduli, but gives rise to ensembles of string vacua, providing a scientific basis for a stringy notion of naturalness. Of particular interest in this context is the ability to keep track of algebraic information (such as the gauge group) associated with individual vacua while dealing with statistics. In the present work, we aim to clarify conceptual issues and sharpen methods for this purpose, using compactification on ${\rm K3} \times {\rm K3}$ as a test case. Our first approach exploits the connection between the stabilization of complex structure moduli and the Noether-Lefschetz problem. Compactification data for F-theory, however, involve not only a four-fold (with a given complex structure) $Y_4$ and a flux on it, but also an elliptic fibration morphism $Y_4 \longrightarrow B_3$, which makes this problem complicated. The heterotic-F-theory duality indicates that elliptic fibration morphisms should be identified modulo isomorphism. Based on this principle, we explain how to count F-theory vacua on ${\rm K3} \times {\rm K3}$ while keeping the gauge group information. Mathematical results reviewed/developed in our companion paper are exploited heavily. With applications to more general four-folds in mind, we also clarify how to use Ashok-Denef-Douglas' theory of the distribution of flux vacua in order to deal with statistics of sub-ensembles tagged by a given set of algebraic/topological information. As a side remark, we extend the heterotic/F-theory duality dictionary on flux quanta and elaborate on its connection to the semistable degeneration of a K3 surface.Comment: 81 pages, 5 figure

Restrictions on infinite sequences of type IIB vacua

Ashok and Douglas have shown that infinite sequences of type IIB flux vacua with imaginary self-dual flux can only occur in so-called D-limits, corresponding to singular points in complex structure moduli space. In this work we refine this no-go result by demonstrating that there are no infinite sequences accumulating to the large complex structure point of a certain class of one-parameter Calabi-Yau manifolds. We perform a similar analysis for conifold points and for the decoupling limit, obtaining identical results. Furthermore, we establish the absence of infinite sequences in a D-limit corresponding to the large complex structure limit of a two-parameter Calabi-Yau. In particular, our results demonstrate analytically that the series of vacua recently discovered by Ahlqvist et al., seemingly accumulating to the large complex structure point, are finite. We perform a numerical study of these series close to the large complex structure point using appropriate approximations for the period functions. This analysis reveals that the series bounce out from the large complex structure point, and that the flux eventually ceases to be imaginary self-dual. Finally, we study D-limits for F-theory compactifications on K3\times K3 for which the finiteness of supersymmetric vacua is already established. We do find infinite sequences of flux vacua which are, however, identified by automorphisms of K3.Comment: 35 pages. v2. Typos corrected, ref. added. Matches published versio

PALP - a User Manual

This article provides a complete user's guide to version 2.1 of the toric geometry package PALP by Maximilian Kreuzer and others. In particular, previously undocumented applications such as the program nef.x are discussed in detail. New features of PALP 2.1 include an extension of the program mori.x which can now compute Mori cones and intersection rings of arbitrary dimension and can also take specific triangulations of reflexive polytopes as input. Furthermore, the program nef.x is enhanced by an option that allows the user to enter reflexive Gorenstein cones as input. The present documentation is complemented by a Wiki which is available online.Comment: 71 pages, to appear in "Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer". PALP Wiki available at http://palp.itp.tuwien.ac.at/wiki/index.php/Main_Pag

On Mirror Maps for Manifolds of Exceptional Holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities