From Cracked Polytopes to Fano Threefolds

We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.Comment: New introduction and section on the connection with the Gross-Siebert program. 46 page

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

Geometric constraints in dual F-theory and heterotic string compactifications

We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. Specifically, we give a complete classification of models where the heterotic theory is compactified on a smooth Calabi-Yau threefold that is elliptically fibered with a single section and carries smooth irreducible vector bundles, and the dual F-theory model has a corresponding threefold base that has the form of a P^1 bundle. We formulate simple conditions for the geometry on the F-theory side to support an elliptically fibered Calabi-Yau fourfold. We match these conditions with conditions for the existence of stable vector bundles on the heterotic side, and show that F-theory gives new insight into the conditions under which such bundles can be constructed. In particular, we find that many allowed F-theory models correspond to vector bundles on the heterotic side with exceptional structure groups, and determine a topological condition that is only satisfied for bundles of this type. We show that in many cases the F-theory geometry imposes a constraint on the extent to which the gauge group can be enhanced, corresponding to limits on the way in which the heterotic bundle can decompose. We explicitly construct all (4962) F-theory threefold bases for dual F-theory/heterotic constructions in the subset of models where the common twofold base surface is toric, and give both toric and non-toric examples of the general results.Comment: 81 pages, 2 figures; v2, v3: references added, minor corrections; v4: minor errors, Table 5 correcte

Interactions between Algebraic Geometry and Noncommutative Algebra

The workshop discussed the interactions between algebraic geometry and various areas of noncommutative algbebra including finite dimensional algebras, representation theory of algebras and noncommutative algebraic geometry. More than 45 mathematicians participated with a notable number of young mathematicians present

Period, Central Charge and Effective Action on Ricci-Flat Manifolds with Special Holonomy

Motivated by the Gamma conjecture, we introduce a notion of numerical vectors, such as Chern character and Mukai vector, and a notion of numerical t-stabilities and numerical slope functions on triangulated categories. The study of the derived categories of Calabi-Yau manifolds leads us to a conjecture which gives a relation between numerical t-stability and Bridgelands stability on smooth varieties. And when there exists generalized twisted Mukai vectors, we also obtain the results regarding the cohomological Fourier-Mukai (FM) transforms associated to the FM ones on the level of derived categories. In some cases, these cohomological FM transforms agree with the ones on the derived categories of twisted sheaves. In the second part, we discuss geometric and topological properties of G_2 manifolds due to the Kovalev's twisted connected sum constrction. In the Kovalev limit the Ricci-flat metrics on X_{L/R} approximate the Ricci-flat G_2-metrics and we identify the universal modulus, called the Kovalevton, that parametrizes this limit. Moreover, the low energy effective theory exhibits gauge theory sectors with extended supersymmetry in this limit. The universal (semi-classical) Kähler potential of the effective N=1 supergravity action is a function of the Kovalevton and the volume modulus of the G_2-manifold. We describe geometric degenerations in X_{L/R}, which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of G_2-manifolds

Global topology of the Hitchin system

Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.Comment: 41 pages, small changes and clarifications, to appear in Handbook of Modul