Special Polynomial Rings, Quasi Modular Forms and Duality of Topological Strings

Abstract

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.

55 pages, 4 figures

References in corpus (8)

• S-duality in N=2 supersymmetric gauge theories
• On Rationally Parametrized Modular Equations
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions
• Topological Strings and (Almost) Modular Forms
• Lectures on Mirror Symmetry and Topological String Theory
• Topological Strings on Elliptic Fibrations
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Cited by in corpus (6)

• LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
• Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory
• Differential Rings from Special Kähler Geometry
• Topological String on elliptic CY 3-folds and the ring of Jacobi forms
• Algebraic structure of $t{t}^{\ast }$ equations for Calabi-Yau sigma models
• Algebra of Kodaira-Spencer Gravity and Deformation of Calabi-Yau Manifold