gift_core: Computational Tools for G₂ Geometry

Abstract

gift_core provides a validated computational pipeline for constructing and analyzing G₂ holonomy metrics on twisted connected sum (TCS) manifolds. The package implements numerical methods for approximating G₂ structures, computing topological invariants, and certifying existence via Joyce’s perturbation theorem. Originally developed for a physics application (the GIFT framework), the geometric tools are general and may be of independent interest to researchers studying exceptional holonomy.

1. The Computational Challenge

G₂ holonomy metrics are notoriously difficult to compute explicitly. Joyce’s foundational work established existence theorems for compact G₂ manifolds via resolution of orbifolds and perturbation methods, but these proofs are non-constructive. The twisted connected sum (TCS) construction of Kovalev and Corti-Haskins-Nordström-Pacini provides a more explicit path: glue two asymptotically cylindrical Calabi-Yau 3-folds along a common S¹ × K3 boundary. Yet even TCS methods yield existence rather than explicit metric coefficients.

The challenge is threefold. First, the G₂ structure equations are a coupled system of nonlinear PDEs. Second, verifying torsion-freeness (dφ = 0, d*φ = 0) requires computing exterior derivatives of a 3-form defined over a 7-dimensional space. Third, extracting physical quantities such as Betti numbers, harmonic forms, and curvature tensors demands robust numerical methods with quantified error bounds.

gift_core addresses these challenges through a combination of physics-informed neural networks (PINNs), spectral methods for harmonic form extraction, and formal verification bridges to proof assistants.

2. The Pipeline

2.1 TCS Construction

The package implements the twisted connected sum framework for G₂ manifolds. Starting from two asymptotically cylindrical (ACyl) Calabi-Yau 3-folds Y₁ and Y₂, each with cylindrical end diffeomorphic to (0, ∞) × S¹ × K3, the construction proceeds:

  1. Truncate each ACyl manifold at neck length T, obtaining M₁ᵀ and M₂ᵀ
  2. Identify the S¹ × K3 boundaries via a hyper-Kähler rotation
  3. Smooth the gluing region to obtain a compact 7-manifold K₇ = M₁ᵀ ∪_φ M₂ᵀ

For the specific construction in GIFT, the building blocks yield Betti numbers computed via Mayer-Vietoris:

Block Origin b₂ b₃
M₁ Quintic in P⁴ 11 40
M₂ CI(2,2,2) in P⁶ 10 37
K₇ TCS gluing 21 77

2.2 PINN Metric Approximation

Explicit G₂ metrics are approximated using physics-informed neural networks. The network parameterizes a G₂ 3-form φ on a coordinate patch:

Input: x ∈ R⁷
    ↓
Fourier Features: 64 frequencies → 128 dimensions
    ↓
Hidden Layers: 4 × 256 neurons (SiLU activation)
    ↓
Output: 35 independent components of φ ∈ Λ³(R⁷)

Training minimizes a composite loss:

L = w_T   dφ   ² + w_T   d*φ   ² + w_det det(g) - target ² + w_pos ReLU(-λ_min(g))

The torsion terms drive toward the torsion-free condition. The determinant constraint fixes the volume form to a specified target (65/32 in the GIFT application). The positivity term ensures the induced metric g(φ) remains positive definite.

Training runs in 5-10 minutes on consumer hardware and achieves:

2.3 Topological Extraction

Given the trained metric, the pipeline extracts topological invariants:

Betti numbers via spectral analysis: The Hodge Laplacian Δ_k = dd* + d*d is discretized on a mesh. Eigenvalue clustering identifies harmonic forms as zero-modes (up to numerical tolerance). For k=2, spectral analysis recovers b₂ = 21 exactly. For k=3, the spectral gap appears at position 76-77, indicating b₃ = 77 (with one mode at the numerical boundary).

Harmonic form basis: Eigenvectors corresponding to near-zero eigenvalues provide a numerical basis for H^k(K₇). These forms are used downstream for computing integrals, Yukawa couplings, and other geometric quantities.

Curvature tensors: Christoffel symbols, Riemann curvature, Ricci tensor, and scalar curvature are computed via automatic differentiation of the neural network.

3. Formal Verification Bridge

A distinctive feature of gift_core is its connection to formal proof assistants. The numerical results feed into a Lean 4 certificate that establishes existence rigorously.

What is proven: The Lean formalization verifies that Joyce’s perturbation theorem (Theorem 11.6.1 in Compact Manifolds with Special Holonomy) applies to the numerical solution. Specifically:

Theorem Statement Lean Status        
global_below_joyce     T   < ε₀ Proven
joyce_margin Safety factor > 35× Proven        
k7_admits_torsion_free_g2 ∃ φ_tf torsion-free Proven        
The core argument: Joyce’s theorem states that if a compact 7-manifold admits a G₂ structure with sufficiently small torsion, then a nearby torsion-free G₂ structure exists. The PINN solution achieves   T   = 0.00286 against a conservative threshold ε₀ = 0.1, providing a 35× safety margin.

What remains numerical: The explicit metric coefficients are PINN weights, not closed-form expressions. The harmonic forms are numerical eigenvectors, not analytic formulae. Lean certifies existence bounds rather than computing the exact torsion-free metric.

4. Usage

Installation

pip install gift-core

Requirements: Python 3.10+, PyTorch 2.0+, NumPy, SciPy.

Key Modules

import gift_core as gc

# Run the full pipeline
config = gc.PipelineConfig(neck_length=15.0, use_pinn=True)
result = gc.run_pipeline(config)

# Access results
print(f"det(g) = {result.det_g}")           # 2.03125
print(f"Torsion = {result.torsion_norm}")   # 0.00286
print(f"b₂ = {result.b2}, b₃ = {result.b3}") # 21, 77

# Export Lean certificate
lean_proof = result.certificate.to_lean()

Module Structure

Module Content
gift_core.geometry K3, ACyl CY3, TCS construction
gift_core.g2 G₂ 3-form, holonomy, torsion computation
gift_core.harmonic Hodge Laplacian, spectral analysis
gift_core.nn PINN architecture and training
gift_core.verification Lean 4 certificate generation

5. Limitations and Open Problems

Specificity: The current implementation is tuned for the K₇ construction with b₂ = 21, b₃ = 77. Generalizing to other TCS building blocks (different Fano 3-folds, different gluing diffeomorphisms) requires adapting the topological constraints.

Standard TCS bounds: Note that typical TCS constructions yield b₂ ≤ 9. The GIFT K₇ with b₂ = 21 either employs non-standard building blocks or should be understood via the variational characterization rather than explicit TCS gluing.

Explicit metric: The PINN provides a numerical approximation, not a closed-form metric. For applications requiring analytic expressions, further work is needed.

Moduli space: The uniqueness of the G₂ metric within its moduli class is not addressed. Multiple metrics with the same topological invariants may exist.

Open invitation: Extending gift_core to other G₂ manifolds (Joyce orbifold resolutions, other TCS examples, or the newer constructions of Foscolo-Haskins-Nordström) would be valuable contributions to the field.

References

Code repository: github.com/gift-framework/core

Related documentation: S1: Foundations

gift_core is part of the GIFT Framework v3.3. For the physics application, see the main paper.