Physical Yukawa couplings in heterotic Calabi–Yau compactifications: singular loci, non-standard-embedding bundles, and multi-parameter moduli

The physical, canonically normalized $(\mathbf{27})^3$ Yukawa couplings of heterotic $E_8\times E_8$ compactifications on Calabi–Yau threefolds encode the fermion masses and mixings of the resulting four-dimensional theory, and their computation is the central quantitative bridge between compactification geometry and observed flavour hierarchies. A 2024 benchmark study verified three computational routes — Weil–Petersson metrics from the Kodaira–Spencer map, period integrals, and machine-learned Ricci-flat metrics with explicit harmonic representatives — and found excellent agreement in standard-embedding models. The problem is the extension of this programme to the regimes where it currently fails: near singular loci in complex-structure moduli space where the approximations degrade, to general stable holomorphic vector bundles beyond the standard embedding (requiring Hermitian Yang–Mills bundle metrics), to multi-parameter families with $h^{2,1}>1$ governed by Picard–Fuchs systems, and at the scale needed for phenomenological scans of three-generation models. Later literature (2024–2026) has built a rank- and structure-group-general bundle-metric methodology and achieved proof-of-principle reproduction of Standard Model flavour parameters, but certified accuracy near singular loci, scalability over combined bundle and complex-structure moduli, and multi-parameter moduli dependence remain unresolved; integration with moduli stabilization and $\alpha'$ corrections is the downstream phenomenological purpose these capabilities serve.

Background

In the $E_8\times E_8$ heterotic string compactified on a Calabi–Yau threefold $X$ (a compact Kähler manifold with trivial canonical bundle and $SU(3)$ holonomy), the four-dimensional theory is $\mathcal{N}=1$ supersymmetric, and its gauge group is the commutant in $E_8$ of the structure group of the internal holomorphic vector bundle $V$. In the “standard embedding” one takes $V=T_X$, the holomorphic tangent bundle; the unbroken group is then $E_6$, the particle generations in the $\mathbf{27}$ representation correspond to harmonic $(2,1)$-forms (elements of $H^{2,1}(X)$, whose dimension $h^{2,1}$ counts complex-structure deformations), so there are $h^{2,1}$ of them, while the net generation number — the difference between the numbers of $\mathbf{27}$ and $\overline{\mathbf{27}}$ multiplets — has magnitude $\tfrac12|\chi|$, half the Euler characteristic. The $(\overline{\mathbf{27}})^3$ Yukawa couplings are essentially topological (intersection numbers, up to worldsheet-instanton corrections), but the physically interesting $(\mathbf{27})^3$ couplings depend on the complex geometry: schematically $\kappa=\int_X\Omega\wedge\omega^{\mu}\wedge\omega^{\nu}\wedge\omega^{\rho}\,\Omega_{\mu\nu\rho}$, with $\Omega$ the holomorphic $(3,0)$-form and the indices contracted against its components. Such “unnormalized” couplings can be computed purely algebraically via Čech cohomology, as done for line-bundle models by Anderson, Gray, Grayson, He and Lukas. The physical, canonically normalized couplings, however, additionally require the matter-field Kähler metric — that is, integrals involving the Ricci-flat metric on $X$, which exists by Yau's theorem but has no closed form in any nontrivial example.

The standard embedding admits a crucial shortcut found by the source work: the required normalization is the Weil–Petersson metric on the complex-structure moduli space (the natural Kähler metric on the deformation space of $X$), computable either from the Kodaira–Spencer map (a perturbative description of complex-structure deformations) or from period integrals of $\Omega$. For one-parameter families ($h^{2,1}=1$) the periods satisfy a Picard–Fuchs ordinary differential equation; for $h^{2,1}>1$ they satisfy Picard–Fuchs systems of partial differential equations, a substantially harder problem. In parallel, machine-learning methods approximate the Ricci-flat metric by training neural networks on algebraic ansätze (in the spirit of Donaldson's approach, embedding $X$ via sections of a high power of a positive line bundle and sampling points on $X$), as in the companion study of machine-learned Calabi–Yau metrics and curvature by Berglund, Butbaia, Hübsch, Jejjala, Mayorga Peña, Mishra and Tan. With an approximate metric one can construct explicit harmonic representatives of $H^{2,1}(X)$ (forms in each cohomology class that are harmonic for the approximate metric) and evaluate both couplings and normalizations directly.

Butbaia, Mayorga Peña, Tan, Berglund, Hübsch, Jejjala and Mishra benchmarked these routes on the Fermat quintic (the degree-five hypersurface in $\mathbb{P}^4$ defined by a sum of five fifth powers), the intersection of two cubics in $\mathbb{P}^5$, and the three-generation Tian–Yau manifold (a Calabi–Yau threefold with Euler characteristic $\chi=-6$, hence three net generations), finding excellent agreement; for example, at the Fermat point of the mirror quintic — the mirror partner of the quintic under mirror symmetry, the duality exchanging complex-structure and Kähler moduli — the machine-learned harmonic representative gives $g_{\psi\bar\psi}=0.1906$ against the exact gamma-function value $\approx0.1922$. Beyond the standard embedding — indispensable for realistic model building, since there the generation number is not tied to the Euler characteristic — the normalization also requires a Hermitian Yang–Mills connection on $V$ (a unitary connection whose curvature obeys $F^{0,2}=0$ and $g^{a\bar b}F_{a\bar b}=0$; existence for slope-stable bundles follows from the Donaldson–Uhlenbeck–Yau theorem), which likewise has no closed form in general; monad bundles (vector bundles presented as the middle cohomology of a three-term complex whose terms are direct sums of line bundles) and line-bundle sums supply the candidate bundles.

The source work's own assessment of what remains: the machine-learned metrics and harmonic representatives degrade as one approaches singular loci in complex-structure moduli space (such as the $\psi\to0$ locus of the mirror of the intersection of two cubics, where the Weil–Petersson metric develops a pole, and logarithmic singularities such as $\psi=1$); the pipeline has not been extended to general bundles; the period route for $h^{2,1}>1$ requires solving Picard–Fuchs PDE systems; the spectral-network point sampling underlying the high-precision normalization integrals (currently $\sim10^5$–$10^6$ points, described by the source as the most efficient tool to date) offers room for computational improvement through different point-selection schemes; and phenomenological scans (three-generation models, incorporation of moduli stabilization and $\alpha'$ corrections) as well as searches over large complete-intersection and toric Calabi–Yau datasets remain unrealized.

Problem Statement

Extend the computation of the physically normalized $(\mathbf{27})^3$ Yukawa couplings of heterotic Calabi–Yau compactifications from the setting demonstrated in the source work — standard embedding, smooth loci of moduli space, at most one-parameter families — to the regimes required for precision string phenomenology:

1. Singular loci. Obtain the physically normalized couplings whose accuracy near degenerating complex-structure loci — such as the $\psi\to0$ locus of the mirror of the intersection of two cubics in $\mathbb{P}^5$ (where the Weil–Petersson metric develops a pole) and logarithmic singularities such as $\psi=1$ — does not degrade relative to the accuracy attained at smooth points of the same family, with quantified error estimates.

2. General bundles. Compute the normalized couplings for compactifications whose holomorphic vector bundle $V$ is not the tangent bundle (non-standard embeddings, e.g. line-bundle sums and monad bundles), which requires the Hermitian Yang–Mills metric on $V$ and harmonic representatives of $H^1(X;V)$ in addition to the Calabi–Yau metric, and which must be delivered as a controlled approximation over bounded ranges of bundle and complex-structure moduli — that is, as a certified function on moduli space, cross-checked against an independent route at several points of the range — rather than at isolated points.

3. Multi-parameter moduli. Obtain normalized couplings as controlled functions of the deformation parameters over moduli space for complex-structure families with $h^{2,1}>1$ — where periods are governed by Picard–Fuchs partial differential equation systems.

The common objective uniting these components is a quantitatively certified pipeline for the normalized Yukawa sector, of the kind the source work validates by cross-method agreement (Kodaira–Spencer or period computations versus harmonic-representative computations using machine-learned metrics) and agreement with exactly known special-point values (such as the gamma-function value $\approx0.1922$ for the Weil–Petersson entry at the Fermat point of the mirror quintic). The sources frame the ultimate purpose as phenomenological: applying such a pipeline to three-generation heterotic models, eventually incorporating moduli-stabilization and $\alpha'$-correction effects, to locate regions of moduli space with realistic Yukawa hierarchies, together with the sampling and computational engineering needed to enable searches over large sets of complete-intersection and toric Calabi–Yau geometries. The quantity whose error is controlled is the physical, canonically normalized coupling, measured as a relative error in the norm induced by the matter-field Kähler metric; the target in every one of the three regimes is percent-level accuracy — a relative error of order $10^{-2}$ against the exact or independent value, the level already reached at the Fermat point (0.1906 against the exact 0.1922), and the error must be quantified and independently checked both by agreement between computational routes implemented independently for the same geometry and, where exactly or analytically known values exist, by their reproduction (for example the gamma-function value $\approx0.1922$ at the Fermat point, the analytic behaviour of the Weil–Petersson metric at the $\psi\to0$ pole, or an exact topological invariant such as the Euler number), with the agreement holding across bounded moduli ranges rather than only at isolated points. These three stated methodological blockers — singular-loci degradation, general-bundle capability, and multi-parameter scaling — are independent: resolving any one settles that regime while leaving the other two open, and the question requires all three. The full landscape scan is the longer-term enabling target the sources articulate, built on these three capabilities.

References

1. G. Butbaia, D. Mayorga Peña, J. Tan, P. Berglund, T. Hübsch, V. Jejjala, C. Mishra, 'Physical Yukawa Couplings in Heterotic String Compactifications', Adv. Theor. Math. Phys. 28 (2024) 2783–2822, DOI: 10.4310/ATMP.241119041341, arXiv:2401.15078, https://arxiv.org/abs/2401.15078
2. P. Berglund, G. Butbaia, T. Hübsch, V. Jejjala, D. Mayorga Peña, C. Mishra, J. Tan, 'Precision String Phenomenology', Phys. Rev. D 111 (2025) 086007, DOI: 10.1103/PhysRevD.111.086007, arXiv:2407.13836, https://arxiv.org/abs/2407.13836
3. P. Berglund, G. Butbaia, T. Hübsch, V. Jejjala, C. Mishra, D. Mayorga Peña, J. Tan, 'cymyc — Calabi-Yau Metrics, Yukawas, and Curvature', J. High Energy Phys. 03 (2025) 28, DOI: 10.1007/JHEP03(2025)028, arXiv:2410.19728, https://arxiv.org/abs/2410.19728
4. C. Mishra, J. Tan, 'Hermitian Yang–Mills connections on general vector bundles: geometry and physical Yukawa couplings', J. High Energy Phys. 06 (2026) 093, DOI: 10.1007/JHEP06(2026)093, arXiv:2512.10907, https://arxiv.org/abs/2512.10907
5. A. Constantin, C. S. Fraser-Taliente, T. R. Harvey, A. Lukas, B. Ovrut, 'Computation of Quark Masses from String Theory', Nucl. Phys. B 1010 (2025) 116778, DOI: 10.1016/j.nuclphysb.2024.116778, arXiv:2402.01615, https://arxiv.org/abs/2402.01615
6. A. Constantin, L. T.-Y. Leung, A. Lukas, L. A. Nutricati, 'Reproducing Standard Model Fermion Masses and Mixing in String Theory: A Heterotic Line Bundle Study', Phys. Rev. D 113 (2026) 046005, DOI: 10.1103/fbdz-t73z, arXiv:2507.03076, https://arxiv.org/abs/2507.03076
7. A. Constantin, S.-J. Lee, A. Lukas, L. A. Nutricati, 'Calabi-Yau Metrics with Full Moduli Dependence', arXiv:2606.28487 (2026), https://arxiv.org/abs/2606.28487
8. L. B. Anderson, J. Gray, D. Grayson, Y. He, A. Lukas, 'Yukawa Couplings in Heterotic Compactification', Commun. Math. Phys. 297 (2010) 95–127, DOI: 10.1007/s00220-010-1033-8, arXiv:0904.2186, https://arxiv.org/abs/0904.2186
9. A. Ashmore, R. Deen, Y.-H. He, B. A. Ovrut, 'Machine Learning Line Bundle Connections', Phys. Lett. B 827 (2022) 136972, DOI: 10.1016/j.physletb.2022.136972, arXiv:2110.12483, https://arxiv.org/abs/2110.12483
10. P. Berglund, G. Butbaia, T. Hübsch, V. Jejjala, D. Mayorga Peña, C. Mishra, J. Tan, 'Machine Learned Calabi–Yau Metrics and Curvature', Adv. Theor. Math. Phys. 27 (2023) 1107–1158, DOI: 10.4310/ATMP.2023.v27.n4.a3, arXiv:2211.09801, https://arxiv.org/abs/2211.09801
11. Y. Hendi, M. Larfors, M. Walden, 'Learning Group Invariant Calabi-Yau Metrics by Fundamental Domain Projections', Mach. Learn.: Sci. Technol. 6 (2025) 015050, DOI: 10.1088/2632-2153/adb4bb, arXiv:2407.06914, https://arxiv.org/abs/2407.06914
12. S. Lüst, F. Ruehle, S. Schreyer, 'Warped Numerical Calabi-Yau Metrics', arXiv:2607.18402 (2026), https://arxiv.org/abs/2607.18402
13. K. Fraser-Taliente, T. R. Harvey, M. Kim, 'Not So Flat Metrics', arXiv:2411.00962 (2024), https://arxiv.org/abs/2411.00962
14. J. Gray, 'Structure In The Heterotic Matter Field Kahler Potential', Phys. Rev. D 112 (2025) 106017, DOI: 10.1103/3zjx-6wyp, arXiv:2508.13292, https://arxiv.org/abs/2508.13292