Elliptic Calabi-Yau Threefolds with Z_3 x Z_3 Wilson Lines

A torus fibered Calabi-Yau threefold with first homotopy group Z_3 x Z_3 is constructed as a free quotient of a fiber product of two dP_9 surfaces. Calabi-Yau threefolds of this type admit Z_3 x Z_3 Wilson lines. In conjunction with SU(4) holomorphic vector bundles, such vacua lead to anomaly free, three generation models of particle physics with a right handed neutrino and a U(1)_{B-L} gauge factor, in addition to the SU(3)_C x SU(2)_L x U(1)_Y standard model gauge group. This factor helps to naturally suppress nucleon decay. The moduli space and Dolbeault cohomology of the threefold is also discussed.Comment: 51 pages, 13 figures; v2: references adde

Donagi-Markman cubic for the generalised Hitchin system

Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi\u2013Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system

Webs of Lagrangian Tori in Projective Symplectic Manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe

The class of the locus of intermediate Jacobians of cubic threefolds

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat

The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.Comment: Revised version, 26 page

G-flux in F-theory and algebraic cycles

We construct explicit G4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G4 as a coherent sheaf. In this description the close link between G4 fluxes and algebraic cycles is manifest.Comment: 55 pages, 1 figure; added refs, corrected typo

Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D5 del Pezzo surfaces and the occurrence of free Z2 quotients that lead to a new class of heterotic duals.Comment: 117 pages, 1 Postscript figur