Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged linear sigma models (GLSMs) with non-Abelian gauge groups, where the dual geometries arise as phases of the same GLSM. Using the modular bootstrap approach of topological string theory, one can compute all-genus Gopakumar-Vafa invariants of these Calabi-Yaus. We observe that homological projective duality acts as an element of Γ_0(5) on the topological string partition function and the partition functions of dual geometries transform into each other. Moreover, we study the geometries from an M-/F-theory perspective. We compute the F-theory spectrum and show how the genus one-fibered Calabi-Yaus are connected to certain Calabi-Yaus in toric varieties via a series of Higgs transitions. Based on the F-theory physics, we conjecture that dual geometries are elements of the same Tate-Shafarevich group. Our analysis also leads to a classification of 5-section geometries, as well as the construction of F-theory models with charge 5 hypermultiplets.

We consider the fourteen families W of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in P. Mirror symmetry identifies the masses of even–dimensional D– branes of the mirror Calabi-Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H3(W,Z). It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ0(25), whose Hecke eigenvalues are determined by the Hasse-Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse-Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga-Sato variety. We also comment on simpler applications to local CalabiYau threefolds. ∗kilian@mpim-bonn.mpg.de, ⋄aklemm@th.physik.uni-bonn.de, †esche@bicmr.pku.edu.cn, ‡dbz@mpimbonn.mpg.de

Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series J(Q, q, t) that satisfies a system of linear differential equations with respect to t and q-difference equations with respect to Q. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small J-function J(Q, q, 0) which, in the case of Fano manifolds, is a vector-valued q-hypergeometric function. On the other hand, for the quintic 3-fold we formulate an explicit conjecture for the small J-function and its small linear q-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the case of Fano manifolds, the coefficients of the small linear q-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small J-function agrees with a proposal of Jockers-Mayr.

We propose a formula for the exact central charge of a B-type D-brane that is expected to hold in all regions of the Kahler moduli space of a Calabi-Yau. For Landau-Ginzburg orbifolds we propose explicit expressions for the mathematical objects that enter into the central charge formula. We show that our results are consistent with results in FJRW theory and the hemisphere partition function of the gauged linear sigma model.

We show that the hemisphere partition function for certain U(1) gauged linear sigma models (GLSMs) with D-branes is related to a particular set of Mellin-Barnes integrals which can be used for analytic continuation to the singular point in the K\"ahler moduli space of an $h^{1,1}=1$ Calabi-Yau (CY) projective hypersurface. We directly compute the analytic continuation of the full quantum corrected central charge of a basis of geometric D-branes from the large volume to the singular point. In the mirror language this amounts to compute the analytic continuation of a basis of periods on the mirror CY to the conifold point. However, all calculations are done in the GLSM and we do not have to refer to the mirror CY. We apply our methods explicitly to the cubic, quartic and quintic CY hypersurfaces.

We describe a Lie Algebra on the moduli space of non-rigid compact Calabi–Yau threefolds enhanced with differential forms and its relation to the Bershadsky–Cecotti–Ooguri–Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions \({{\bf F}_{g}^{\rm alg}, g \geq 1}\), which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck’s algebraic de Rham cohomology and on the algebraic Gauss–Manin connection. In this way, we recover a result of Yamaguchi–Yau and Alim–Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi–Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.

We perform the analytic continuation of solutions to the hypergeometric differential equation of order n to the third regular singularity, usually denoted z=1, with the help of recurrences of their Mellin–Barnes integral representations. In the resonant case, there are necessarily logarithmic solutions. We apply the result to Picard-Fuchs equations of certain one–parameter families of Calabi–Yau manifolds, known as the mirror quartic and the mirror quintic.

We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local P^2 and local del Pezzo geometries with E_5,E_6,E_7 and E_8 type singularities. We provide the analogous special polynomial ring for the quintic.

We study topological string theory on elliptically fibered Calabi-Yau threefolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the moduli space. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at the boundary in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which governs this finer structure and which can be related to the anomaly equations for correlation functions.

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.

Gauge Field Theory, Anomalies, and Supersymmetry: BRST Symmetry and Cohomology (N Dragon and F Brandt) Aspects of Supersymmetric BRST Cohomology (F Brandt) Character Expansion for HOMFLY Polynomials: Integrability and Difference Equations (A Mironov, A Morozov and A Morozov) Bicategories in Field Theories - An Invitation (T Nikolaus and C Schweigert) The Compactification of IIB Supergravity on S5 Revisited (P van Nieuwenhuizen) String Theory and Algebraic Geometry: Max Kreuzer's Contributions to the Study of Calabi - Yau Manifolds (P Candelas) Calabi - Yau Three-Folds: Poincare Polynomials and Fractals (A Ashmore and Y-H He) Conifold Degenerations of Fano 3-Folds as Hypersurfaces in Toric Varieties (V Batyrev and M Kreuzer) Nonassociativity in String Theory (R Blumenhagen) Counting Points and Hilbert Series in String Theory (V Braun) Standard Models and Calabi - Yaus (R Donagi) The String Landscape and Low Energy Supersymmetry (M R Douglas) The Cardy - Cartan Modular Invariant (J Fuchs, C Schweigert and C Stigner) A Projection to the Pure Spinor Space (S Guttenberg) Mathieu Moonshine and Symmetries of K3 Sigma Models (S Hohenegger) Toric Deligne - Mumford Stacks and the Better Behaved Version of the GKZ Hypergeometric System (R P Horja) Fano Polytopes (A M Kasprzyk and B Nill) Dual Purpose Landscaping Tools: Small Extra Dimensions in AdS/CFT (J Polchinski and E Silverstein) Notes on the Relation Between Strings, Integrable Models and Gauge Theories (R C Rashkov) E11, Generalised Space-Time and IIA String Theory: The R ⊗ R Sector (A Rocen and P West) The Kreuzer Bi-Homomorphism (A N Schellekens) Emergent Spacetime and Black Hole Probes from Automorphic Forms (R Schimmrigk) How to Classify Reflexive Gorenstein Cones (H Skarke) PALP - A Package for Analyzing Lattice Polytopes: PALP - A User Manual (A P Braun, J Knapp, E Scheidegger, H Skarke and N-O Walliser).

We discuss B-type tensor product branes in mirrors of two-parameter Calabi-Yau hypersurfaces, using the language of matrix factorizations. We determine the open string moduli of the branes at the Gepner point. By turning on both bulk and boundary moduli we then deform the brane away from the Gepner point. Using the deformation theory of matrix factorizations we compute Massey products. These contain the information about higher order deformations and obstructions. The obstructions are encoded in the F-term equations, which we obtain from the Massey product algorithm. We show that the F-terms can be integrated to an effective superpotential. Our results provide an ingredient for open/closed mirror symmetry for these hypersurfaces.

The Yau-Zaslow conjecture determines the reduced genus 0 Gromov-Witten invariants of K3 surfaces in terms of the Dedekind eta function. Classical intersections of curves in the moduli of K3 surfaces with Noether-Lefschetz divisors are related to 3-fold Gromov-Witten theory via the K3 invariants. Results by Borcherds and Kudla-Millson determine the classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

This work is concerned with branes and differential equations for one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes we derive the inhomogeneous Picard--Fuchs equations satisfied by the brane superpotential. In this way we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.

We study aspects of worldsheet instantons relevant to a heterotic standard model. The non-simply connected Calabi-Yau threefold used admits Z_3 x Z_3 Wilson lines, and a more detailed investigation shows that the homology classes of curves are H_2(X,Z)=Z^3+Z_3+Z_3. We compute the genus-0 prepotential, this is the first explicit calculation of the Gromov-Witten invariants of homology classes with torsion (finite subgroups). In particular, some curve classes contain only a single instanton. This ensures that the Beasley-Witten cancellation of instanton contributions cannot happen on this (non-toric) Calabi-Yau threefold.

We discuss the resolution of toroidal orbifolds. For the resulting smooth Calabi-Yau manifolds, we calculate the intersection ring and determine the divisor topologies. In a next step, the orientifold quotients are constructed.

As a first step towards computing instanton-generated superpotentials in heterotic standard model vacua, we determine the Gromov-Witten invariants for a Calabi-Yau threefold with fundamental group pi_1(X)=Z_3 x Z_3. We find that the curves fall into homology classes in H_2(X,Z)=Z^3+(Z_3+Z_3). The unexpected appearance of the finite torsion subgroup in the homology group complicates our analysis. However, we succeed in computing the complete genus-0 prepotential. Expanding it as a power series, the number of instantons in any integral homology class can be read off. This is the first explicit calculation of the Gromov-Witten invariants of homology classes with torsion. We find that some curve classes contain only a single instanton. This ensures that the contribution to the superpotential from each such instanton cannot cancel.

We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z(3) circle plus Z(3) Wilson lines. As we found in Part A [1], the integral homology group H-2 (X, Z) - Z(3) circle plus Z(3) circle plus Z(3) contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the self-mirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds.

As a first step towards studying vector bundle moduli in realistic heterotic compactifications, we identify all holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. Computing the homology, we find that H_2(X,Z)=Z^3+Z_3+Z_3. The torsion curves complicate our analysis, and we develop techniques to distinguish the torsion part of curve classes and to deal with the non-toric threefold X. In this paper, we use direct A-model computations to find the instanton numbers in each integral homology class, including torsion. One interesting result is that there are homology classes that contain only a single instanton, ensuring that there cannot be any unwanted cancellation in the non-perturbative superpotential.