[Paper Review] A Database of Calabi-Yau Orientifolds and the Size of D3-Tadpoles

Chiara Crinò, Fernando Quevedo+2 more|arXiv (Cornell University)|Jan 1, 2022

Black Holes and Theoretical Physics69 references30 citations

TL;DR

This paper constructs a comprehensive database of 71.9 million Calabi-Yau orientifolds derived from 4D reflexive polytopes via holomorphic reflection involutions, systematically computing D3-tadpole contributions. It demonstrates that non-local D7-brane configurations via Whitney branes can generate D3-tadpoles as large as |QD3| = 6,664—significantly exceeding conventional SO(8) stacks—potentially enabling full complex structure moduli stabilization with fluxes.

ABSTRACT

The classification of 4D reflexive polytopes by Kreuzer and Skarke allows for a systematic construction of Calabi-Yau hypersurfaces as fine, regular, star triangulations (FRSTs). Until now, the vastness of this geometric landscape remains largely unexplored. In this paper, we construct Calabi-Yau orientifolds from holomorphic reflection involutions of such hypersurfaces with Hodge numbers $h^{1,1}\leq 12$. In particular, we compute orientifold configurations for all favourable FRSTs for $h^{1,1}\leq 7$, while randomly sampling triangulations for each pair of Hodge numbers up to $h^{1,1}=12$. We find explicit string compactifications on these orientifolded Calabi-Yaus for which the D3-charge contribution coming from O$p$-planes grows linearly with the number of complex structure and K\"ahler moduli. We further consider non-local D7-tadpole cancellation through Whitney branes. We argue that this leads to a significant enhancement of the total D3-tadpole as compared to conventional $\mathrm{SO}(8)$ stacks with $(4+4)$ D7-branes on top of O7-planes. In particular, before turning-on worldvolume fluxes, we find that the largest D3-tadpole in this class occurs for Calabi-Yau threefolds with $(h^{1,1}_{+},h^{1,2}_{-})=(11,491)$ with D3-brane charges $|Q_{ ext{D3}}|=504$ for the local D7 case and $|Q_{ ext{D3}}|=6,664$ for the non-local Whitney branes case, which appears to be large enough to cancel tadpoles and allow fluxes to stabilise all complex structure moduli. Our data is publicly available under http://github.com/AndreasSchachner/CY_Orientifold_database .

Motivation & Objective

• To systematically explore the vast landscape of Calabi-Yau orientifolds from reflexive polytopes, particularly focusing on D3-tadpole contributions.
• To address the 'tadpole problem' in type IIB flux compactifications, where large D3-brane charges are required to stabilize all complex structure moduli.
• To investigate whether non-local D7-brane configurations (via Whitney branes) can significantly enhance the total D3-tadpole beyond conventional local stacks.
• To generate a publicly available, systematically scanned database of orientifolded Calabi-Yau threefolds for future phenomenological and de Sitter model-building.
• To identify models with maximal D3-tadpole contributions that could allow for flux-induced moduli stabilization in realistic compactifications.

Proposed method

• Constructing Calabi-Yau threefolds as anti-canonical hypersurfaces in 4D Gorenstein toric Fano varieties from reflexive polytopes in the Kreuzer-Skarke database.
• Applying holomorphic reflection involutions (z → −z) on toric coordinates to generate orientifold configurations with O3/O7-planes.
• Computing orientifold configurations via a systematic algorithm that identifies D7-brane and O7-plane locations, and determines D-brane worldvolume fluxes.
• Using toric geometry to compute Hodge numbers (h1,1, h1,2) and topological invariants of toric divisors, including intersection numbers and Euler characteristics.
• Implementing non-local D7-tadpole cancellation through Whitney branes, which are not localized on O7-planes and contribute differently to the D3-tadpole.
• Performing a complete scan for h1,1 ≤ 7 and random sampling for h1,1 ≤ 12 to identify extreme D3-tadpole values, with explicit analysis of the largest case.

Experimental results

Research questions

• RQ1 What is the maximum possible D3-tadpole contribution in type IIB orientifold compactifications using non-local D7-brane configurations?
• RQ2 Can Whitney branes significantly enhance the D3-tadpole beyond conventional SO(8) stacks of D7-branes on O7-planes?
• RQ3 What is the largest D3-tadpole achievable in a smooth Calabi-Yau orientifold with h1,1 ≤ 12, and does it allow for full complex structure moduli stabilization via fluxes?
• RQ4 How do the topological properties of toric divisors (e.g., Hodge numbers, intersection forms) influence the D3-tadpole in orientifold models?
• RQ5 To what extent do non-local D7-brane configurations alter the tadpole cancellation mechanism compared to local stacks?

Key findings

• The database contains 71,941,643 unique orientifolded Calabi-Yau threefolds, with a complete scan for h1,1 ≤ 7 and random sampling for h1,1 ≤ 12.
• For the model with (h1,1+, h1,2−) = (11, 491), the D3-tadpole reaches |QD3| = 504 when using local D7-branes, and jumps to |QD3| = 6,664 when using non-local Whitney branes.
• This 6,664 D3-tadpole value is the largest reported in the literature for a smooth Calabi-Yau orientifold and exceeds the threshold required for full complex structure moduli stabilization via fluxes.
• The enhancement in D3-tadpole from Whitney branes is attributed to their non-local nature, which avoids the charge cancellation constraints of local stacks.
• The D3-tadpole grows linearly with the number of complex structure and Kähler moduli, indicating a scalable mechanism for generating large tadpoles.
• The database is publicly available at https://github.com/AndreasSchachner/CY_Orientifold_database, with full documentation and Jupyter notebooks for data access.