Toric Geometry and String Theory

Abstract

In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We classify all tops and give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. Secondly, we compute all loop closed and open topological string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular, we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds. We determine the BPS structure of the amplitudes, and illustrate our general results in various examples with and without D-branes. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots. This thesis is based on hep-th/0303218 (with H. Skarke), hep-th/0405083 and hep-th/0411227 (with B. Florea and M. Marino).

Oxford University DPhil Thesis (Advisor: Philip Candelas), accepted October 2005, 152 pp., 43 figures

References in corpus (36)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Geometric Singularities and Enhanced Gauge Symmetries
• Black Hole Attractors and the Topological String
• Topological Strings and Integrable Hierarchies
• Quantum Foam and Topological Strings
• Disk Instantons, Mirror Symmetry and the Duality Web
• Maximally Supersymmetric String Theories in D<10
• String Theory on Calabi-Yau Manifolds
• Chern-Simons Theory and Topological Strings
• Moduli Space of CHL Strings
• The Space of Kähler metrics (II)
• Framed knots at large N
• The SO(32) Heterotic String on a K3 Surface
• Two Dimensional Yang-Mills, Black Holes and Topological Strings
• Type IIA-Heterotic Duals With Maximal Supersymmetry
• On the Classification of Reflexive Polyhedra
• Orientifolds, Mirror Symmetry and Superpotentials
• SO and Sp Chern-Simons at Large N
• On Type IIB Vacua With Varying Coupling Constant
• Localization and Gluing of Topological Amplitudes
• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry
• The Hitchin functionals and the topological B-model at one loop
• Topological Open String Amplitudes On Orientifolds
• Geometric Transitions
• $G_2$ Manifolds, Mirror Symmetry and Geometric Engineering
• Derivation of Calabi-Yau Crystals from Chern-Simons Gauge Theory
• Weight systems for toric Calabi-Yau varieties and reflexivity of Newton polyhedra
• A Note on the Geometry of CHL Heterotic Strings
• D-branes as Defects in the Calabi-Yau Crystal
• Orientifolds, Unoriented Instantons and Localization
• Orientifold and F-theory Duals of CHL Strings
• The Topological G2 String
• A Relation Between the Kauffman and the HOMFLY Polynomials for Torus Knots
• Simple-Current Symmetries, Rank-Level Duality, and Linear Skein Relations for Chern-Simons Graphs
• SO(N) Reformulated Link Invariants from Topological Strings
• A quantization of topological M theory

Cited by in corpus (4)

• Algebraic methods in random matrices and enumerative geometry
• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry
• Heterotic String Models on Smooth Calabi-Yau Threefolds
• Multi-qubits and Polyvalent Singularity in Type II Supestring Theory