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Monad Bundles in Heterotic String Compactifications

Lara B. Anderson, Yang-Hui He, Andre Lukas

TL;DR

The paper presents a finite, classification-driven study of positive monad bundles on favourable CICYs in $E_8\times E_8$ heterotic compactifications, establishing anomaly-consistency and a practical stability check framework while enabling full spectrum calculations. By combining Koszul and exterior-power techniques, the authors compute bundle cohomology and show that vector-like family–anti-family pairs are absent for all analyzed models, with the number of families given by $\text{ind}(V)$. A comprehensive scan yields 7118 positive monads on 36 CICYs, which, under rudimentary three-family constraints, leaves only 21 promising models, signaling substantial pruning of the model space. The work provides algorithms for spectrum computation and motivates extending the analysis to non-positive monads to approach realistic heterotic vacua, with full stability proofs to follow in companion work.

Abstract

In this paper, we study positive monad vector bundles on complete intersection Calabi-Yau manifolds in the context of E8 x E8 heterotic string compactifications. We show that the class of such bundles, subject to the heterotic anomaly condition, is finite and consists of about 7000 models. We explain how to compute the complete particle spectrum for these models. In particular, we prove the absence of vector-like family anti-family pairs in all cases. We also verify a set of highly non-trivial necessary conditions for the stability of the bundles. A full stability proof will appear in a companion paper. A scan over all models shows that even a few rudimentary physical constraints reduces the number of viable models drastically.

Monad Bundles in Heterotic String Compactifications

TL;DR

The paper presents a finite, classification-driven study of positive monad bundles on favourable CICYs in heterotic compactifications, establishing anomaly-consistency and a practical stability check framework while enabling full spectrum calculations. By combining Koszul and exterior-power techniques, the authors compute bundle cohomology and show that vector-like family–anti-family pairs are absent for all analyzed models, with the number of families given by . A comprehensive scan yields 7118 positive monads on 36 CICYs, which, under rudimentary three-family constraints, leaves only 21 promising models, signaling substantial pruning of the model space. The work provides algorithms for spectrum computation and motivates extending the analysis to non-positive monads to approach realistic heterotic vacua, with full stability proofs to follow in companion work.

Abstract

In this paper, we study positive monad vector bundles on complete intersection Calabi-Yau manifolds in the context of E8 x E8 heterotic string compactifications. We show that the class of such bundles, subject to the heterotic anomaly condition, is finite and consists of about 7000 models. We explain how to compute the complete particle spectrum for these models. In particular, we prove the absence of vector-like family anti-family pairs in all cases. We also verify a set of highly non-trivial necessary conditions for the stability of the bundles. A full stability proof will appear in a companion paper. A scan over all models shows that even a few rudimentary physical constraints reduces the number of viable models drastically.

Paper Structure

This paper contains 40 sections, 116 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Heterotic Calabi-Yau Model Building
  3. Complete Intersection Calabi-Yau Threefolds
  4. Configuration Matrices and Classification
  5. Favourable Configurations
  6. Line Bundles on CICYs
  7. The Monad Construction on CICYs
  8. Bundleness:
  9. Non-triviliaty:
  10. Positivity:
  11. Correct structure group:
  12. Anomaly cancellation/effectiveness:
  13. Classification of Positive Monads on CICYs
  14. Stability
  15. Step 1: Using the Koszul Sequence
  16. ...and 25 more sections

Figures (2)

  • Figure 1: (a) Histogram for the index, ${\rm ind}(V)$, of the 7118 positive monads found over 36 favourable CICYs: the horizontal axis is ${\rm ind}(V)$ and the vertical, the number of bundles; (b) the same data set, but only taking into account monads with ${\rm ind}(V) = 3k$ for some positive integer $k$, such that $k$ divides the Euler number of the corresponding CICY.
  • Figure 2: (a) Histogram for the index, ${\rm ind}(V)$, of the positive monads, 5680 of rank 3 (in red), 1334 of rank 4 (in blue), and 104 of rank 5 (in gray), found over 36 favourable CICYs: the horizontal axis is ${\rm ind}(V)$ and the vertical, the number of bundles; (b) the same data set, but only taking into account monads with ${\rm ind}(V) = 3k$ for some positive integer $k$, such that $k$ divides the Euler number of the corresponding CICY.