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On Classifying the Divisor Involutions in Calabi-Yau Threefolds

Xin Gao, Pramod Shukla

TL;DR

The paper addresses the need for involutively odd moduli in type IIB Calabi–Yau compactifications by classifying CY threefolds with h^{1,1} ≤ 4 that admit divisor exchange involutions mapping identical divisors to each other. The authors implement a two-step procedure to identify Nontrivial Identical Divisors and test SR-Ideal-consistent exchanges, complemented by analyses of non-toric spaces under reflections. They provide explicit volume-form constructions for each divisor-exchange class, including strong/weak swiss-cheese structures and K3-fibered forms, highlighting how odd moduli enter the LVS framework. The resulting dataset and methods offer concrete backgrounds for constructing string vacua with involutively odd moduli, with implications for particle physics and cosmology, and they outline avenues to extend the work to higher h^{1,1} and CY fourfolds.

Abstract

In order to support the odd moduli in models of (type IIB) string compactification, we classify the Calabi-Yau threefolds with h^{1,1}<=4 which exhibit pairs of identical divisors, with different line-bundle charges, mapping to each other under possible divisor exchange involutions. For this purpose, the divisors of interest are identified as completely rigid surface, Wilson surface, K3 surface and some other deformation surfaces. Subsequently, various possible exchange involutions are examined under the symmetry of Stanley-Reisner Ideal. In addition, we search for the Calabi-Yau theefolds which contain a divisor with several disjoint components. Under certain reflection involution, such spaces also have nontrivial odd components in (1,1)-cohomology class. String compactifications on such Calabi-Yau orientifolds with non-zero h^{1,1}_-(CY_3/σ) could be promising for concrete model building in both particle physics and cosmology. In the spirit of using such Calabi-Yau orientifolds in the context of LARGE volume scenario, we also present some concrete examples of (strong/weak) swiss-cheese type volume form.

On Classifying the Divisor Involutions in Calabi-Yau Threefolds

TL;DR

The paper addresses the need for involutively odd moduli in type IIB Calabi–Yau compactifications by classifying CY threefolds with h^{1,1} ≤ 4 that admit divisor exchange involutions mapping identical divisors to each other. The authors implement a two-step procedure to identify Nontrivial Identical Divisors and test SR-Ideal-consistent exchanges, complemented by analyses of non-toric spaces under reflections. They provide explicit volume-form constructions for each divisor-exchange class, including strong/weak swiss-cheese structures and K3-fibered forms, highlighting how odd moduli enter the LVS framework. The resulting dataset and methods offer concrete backgrounds for constructing string vacua with involutively odd moduli, with implications for particle physics and cosmology, and they outline avenues to extend the work to higher h^{1,1} and CY fourfolds.

Abstract

In order to support the odd moduli in models of (type IIB) string compactification, we classify the Calabi-Yau threefolds with h^{1,1}<=4 which exhibit pairs of identical divisors, with different line-bundle charges, mapping to each other under possible divisor exchange involutions. For this purpose, the divisors of interest are identified as completely rigid surface, Wilson surface, K3 surface and some other deformation surfaces. Subsequently, various possible exchange involutions are examined under the symmetry of Stanley-Reisner Ideal. In addition, we search for the Calabi-Yau theefolds which contain a divisor with several disjoint components. Under certain reflection involution, such spaces also have nontrivial odd components in (1,1)-cohomology class. String compactifications on such Calabi-Yau orientifolds with non-zero h^{1,1}_-(CY_3/σ) could be promising for concrete model building in both particle physics and cosmology. In the spirit of using such Calabi-Yau orientifolds in the context of LARGE volume scenario, we also present some concrete examples of (strong/weak) swiss-cheese type volume form.

Paper Structure

This paper contains 12 sections, 48 equations, 7 tables.

Table of Contents

  1. Introduction
  2. The Geometry of Divisors
  3. The Classification of Calabi-Yau Spaces
  4. Typical Volume Forms for Divisor Exchange Involutions
  5. Exchange of completely rigid divisors
  6. Exchange of Wilson divisors with $h^{1,0}(D) \neq 0$
  7. Exchange of Deformation divisors with $h^{2,0}(D) \neq 0$
  8. $H^{1,1}(CY_3/ \sigma)$ Splitting in Non-toric Spaces
  9. Conclusions
  10. Computation tools
  11. List of $CY_3$ for $h^{1,1}_-\neq 0$ Under Divisor Exchange Involutions
  12. List of Non-toric Spaces