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The Geometer's Toolkit to String Compactifications

S. Reffert

TL;DR

These lecture notes present a geometry-first framework for string compactifications, centering on Calabi–Yau basics, toroidal orbifolds, and the powerful toolkit of toric geometry. The core approach is to construct smooth Calabi–Yau threefolds by desingularizing toroidal orbifolds via crepant (toric) resolutions, compute the full intersection ring and divisor topologies, and then extend to orientifold quotients to study how moduli and topological data behave under $\Omega I_6$. The work provides concrete, explicit recipes for patch gluing, inherited and exceptional divisors, and global linear relations that connect local toric data to the global geometry, including the impact on cohomology through $h^{(1,1)}_{-}$ in orientifolds. The resulting framework is essential for systematic model-building in string compactifications, offering a concrete bridge from combinatorial toric data to physically relevant geometric moduli and orientifold structures.$

Abstract

These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string theory compactifications. The emphasis is wholly on the geometry side, not on the physics. The treated topics include toroidal orbifolds, methods of toric geometry, desinglularization of toroidal orbifolds and their orientifold quotients.

The Geometer's Toolkit to String Compactifications

TL;DR

These lecture notes present a geometry-first framework for string compactifications, centering on Calabi–Yau basics, toroidal orbifolds, and the powerful toolkit of toric geometry. The core approach is to construct smooth Calabi–Yau threefolds by desingularizing toroidal orbifolds via crepant (toric) resolutions, compute the full intersection ring and divisor topologies, and then extend to orientifold quotients to study how moduli and topological data behave under . The work provides concrete, explicit recipes for patch gluing, inherited and exceptional divisors, and global linear relations that connect local toric data to the global geometry, including the impact on cohomology through in orientifolds. The resulting framework is essential for systematic model-building in string compactifications, offering a concrete bridge from combinatorial toric data to physically relevant geometric moduli and orientifold structures.$

Abstract

These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string theory compactifications. The emphasis is wholly on the geometry side, not on the physics. The treated topics include toroidal orbifolds, methods of toric geometry, desinglularization of toroidal orbifolds and their orientifold quotients.

Paper Structure

This paper contains 33 sections, 138 equations, 15 figures, 11 tables.

Table of Contents

  1. Calabi--Yau basics and orbifolds
  2. Calabi--Yau manifolds
  3. 1D Calabi--Yaus
  4. 2D Calabi--Yaus
  5. 3D Calabi--Yaus
  6. Orbifolds -- A simple and common example
  7. Point groups and Coxeter elements
  8. List of point groups and lattices
  9. Fixed set configurations and conjugacy classes
  10. Literature
  11. Toric Geometry
  12. The basics
  13. Resolution of singularities
  14. Mori cone and intersection numbers
  15. Divisor topologies
  16. ...and 18 more sections

Figures (15)

  • Figure 1: Fundamental region of a $T^2$
  • Figure 2: Schematic picture of the fixed set configuration of $\IZ_{6-I}$ on $G_2^2\times SU(3)$
  • Figure 3: Fans of projective spaces
  • Figure 4: Toric diagram of $\IC^3/\IZ_{6-I}$ and dual graph
  • Figure 5: Toric diagram of the resolution of $\IC^3/\IZ_{6-I}$ and dual graph
  • ...and 10 more figures