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

A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger

TL;DR

This work delivers a comprehensive framework for Calabi–Yau complete intersections in toric varieties, combining a complete mirror-pair catalog with methods to compute topological string amplitudes on compact multi-parameter CYs. It develops the multi-parameter B-model propagators, solves the holomorphic anomaly equations, and expresses topological partitions in terms of GV and DT invariants, enabling nonperturbative checks of string dualities (notably heterotic–type II). The study of K3-fibered CYs, free Z2 quotients, and their phase structures reveals both expected and novel duality phenomena, including S-duality tests in topological strings and nonuniqueness of nonperturbative heterotic completions across diffeomorphic or rationally equivalent models. The framework lays groundwork for systematic exploration of CY landscapes, their dual descriptions, and nonperturbative BPS spectra, with implications for threshold corrections and modular structures in string theory. All results are conveyed with explicit geometric data, modular forms, and higher-genus predictions that can guide future checks and refinements in string dualities.

Abstract

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.

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

TL;DR

This work delivers a comprehensive framework for Calabi–Yau complete intersections in toric varieties, combining a complete mirror-pair catalog with methods to compute topological string amplitudes on compact multi-parameter CYs. It develops the multi-parameter B-model propagators, solves the holomorphic anomaly equations, and expresses topological partitions in terms of GV and DT invariants, enabling nonperturbative checks of string dualities (notably heterotic–type II). The study of K3-fibered CYs, free Z2 quotients, and their phase structures reveals both expected and novel duality phenomena, including S-duality tests in topological strings and nonuniqueness of nonperturbative heterotic completions across diffeomorphic or rationally equivalent models. The framework lays groundwork for systematic exploration of CY landscapes, their dual descriptions, and nonperturbative BPS spectra, with implications for threshold corrections and modular structures in string theory. All results are conveyed with explicit geometric data, modular forms, and higher-genus predictions that can guide future checks and refinements in string dualities.

Abstract

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.

Paper Structure

This paper contains 29 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Toric CICYs, fibrations and torsion
  3. Complete intersections in toric varieties
  4. Free quotients
  5. Resolution of terminal singularities
  6. Fibrations
  7. The (2,30) example
  8. The geometry of toric CICYs
  9. Equivalence of different nef-partitions
  10. Equivalence of different polyhedra: the (2,30) model
  11. The Mori cone, topological data and fibration structure
  12. The first realization of the (2,30) model
  13. Other realizations of the (2,30) model
  14. The phase diagram
  15. Topological strings on compact Calabi--Yau threefolds
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Secondary fan of the facet $\langle 3456789\rangle$ of $\Delta^*_{(C)}$ with circuits relating its triangulations.
  • Figure 2: Secondary fan of the CICYs with $(h^{1,1},h^{2,1})=(2,30)$.