Explicit slope-stable hidden-sector bundle with vanishing five-brane class for the minimal heterotic standard model

The minimal heterotic standard model realizes, in the observable sector of an $E_8\times E_8$ heterotic string compactification on a $\mathbb Z_3\times\mathbb Z_3$ quotient of the Schoen Calabi–Yau threefold, exactly the MSSM particle content with a slope-stable $SU(4)$ bundle. Completing the vacuum requires a non-trivial hidden-sector vector bundle $V'$ solving the Green–Schwarz anomaly cancellation condition; with vanishing five-brane class this forces $c_2(V')=11\tau_1^2+8\tau_2^2-4\tau_1\tau_2$, and $V'$ must be slope-stable (or, more generally, poly-stable, that is, a direct sum of slope-stable bundles of equal slope) with vanishing slope on a Kähler class for which the observable bundle is also stable. The source paper argued, via the necessary Bogomolov inequality, that such bundles should exist in the overlap of the two stability regions but constructed no example. A 2013 follow-up by three of the same authors built explicit supersymmetric hidden sectors only with non-zero effective five-brane classes, and the subsequent strongly coupled line-bundle program likewise always retains a five-brane. The explicit vanishing-five-brane-class construction — or a proof of impossibility — remains open.

Background

Propagating the $E_8\times E_8$ heterotic superstring on a Calabi–Yau threefold $X$ (a compact Kähler manifold of complex dimension three with trivial canonical bundle and $h^{2,0}=0$) yields a four-dimensional $\mathcal N=1$ supersymmetric theory whose gauge group and massless spectrum are determined by a holomorphic vector bundle carried by $X$. Supersymmetry requires the internal gauge field to solve the Hermitian Yang–Mills equations (vanishing field strength components of type $(0,2)$ and $(1,1)$, the latter proportional to the Kähler form); by the Donaldson–Uhlenbeck–Yau theorem this holds precisely when the bundle is poly-stable with vanishing slope — the slope of a bundle is its first Chern class wedged with the square of the Kähler form and integrated over $X$, normalized by the rank, and a bundle is slope-stable when every proper coherent subsheaf has strictly smaller slope. Consistency of the ten-dimensional theory further requires the Green–Schwarz anomaly cancellation condition, which in cohomology reads $c_2(TX)-c_2(V)-c_2(V')-[\mathcal W]=0$, where $TX$ is the tangent bundle, $V$ and $V'$ are the gauge bundles of the two $E_8$ factors (the observable and hidden sectors), and $[\mathcal W]$ is the homology class of possible five-branes wrapped on holomorphic curves, which must be effective; such an effective class need not vanish. Braun, He, Ovrut and Pantev (JHEP 05 (2006) 043) constructed the minimal heterotic standard model on this basis: $X$ is the smooth quotient of the Schoen Calabi–Yau threefold $\tilde X=B_1\times_{\mathbb P^1}B_2$ (a fiber product of two rational elliptic surfaces, each a del Pezzo surface $dP_9$, that is, the complex projective plane blown up at nine points) by a free $\mathbb Z_3\times\mathbb Z_3$ action, so $h^{1,1}=h^{2,1}=3$ and $\pi_1(X)=\mathbb Z_3\times\mathbb Z_3$, which enables Wilson lines (constant flat gauge backgrounds valued in the fundamental group). The observable bundle is a slope-stable $SU(4)$ bundle $V$ — stability proven in a companion paper (Braun, He, Ovrut, JHEP 06 (2006) 032) on a region $\mathcal K^s$ of the Kähler cone containing $\omega=3(2\tau_1+3\tau_2+\phi)$ — which, together with $\mathbb Z_3\times\mathbb Z_3$ Wilson lines, gives gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{B-L}$, three families with right-handed neutrinos, one Higgs–Higgs-conjugate pair, and no exotic or vector-like matter. The $\mathbb Z_3\times\mathbb Z_3$-invariant part of $H^2$ is spanned by classes $\tau_1,\tau_2,\phi$ with relations $\tau_1^3=\tau_2^3=\phi^2=0$, $\tau_1\phi=3\tau_1^2$, $\tau_2\phi=3\tau_2^2$, and one computes $c_2(TX)=12(\tau_1^2+\tau_2^2)$ and $c_2(V)=\tau_1^2+4\tau_2^2+4\tau_1\tau_2$. Choosing the trivial hidden bundle makes $[\mathcal W]$ non-effective, so $V'$ must be non-trivial. For the simplest case $[\mathcal W]=0$, anomaly cancellation forces $c_2(V')=11\tau_1^2+8\tau_2^2-4\tau_1\tau_2$. A necessary condition for slope-stability is the Bogomolov inequality $\int_X c_2(V')\wedge\omega>0$, which for $\omega=x_1\tau_1+x_2\tau_2+y\phi$ defines the region $\mathcal K^B\colon 4x_1+7x_2-12y>0$; the overlap $\mathcal K^s\cap\mathcal K^B$ is a non-empty finite three-dimensional subcone of the Kähler cone, so both $V$ and $V'$ could in principle be slope-stable simultaneously. The authors argued this makes existence likely, but constructed no example and left the explicit construction as the stated open problem.

Problem Statement

Construct an explicit non-trivial holomorphic vector bundle $V'$ on $X=\tilde X/(\mathbb Z_3\times\mathbb Z_3)$, the quotient of the Schoen Calabi–Yau threefold used in the minimal heterotic standard model, with structure group a subgroup of the hidden $E_8$ (for example $SU(n)$) and $c_1(V')=0$, such that: (i) the Green–Schwarz anomaly cancellation condition holds with vanishing five-brane class, $c_2(V')=c_2(TX)-c_2(V)=11\tau_1^2+8\tau_2^2-4\tau_1\tau_2$ in $H^4(X,\mathbb Z)$, where $\tau_1,\tau_2,\phi$ span the invariant second cohomology; and (ii) $V'$ is slope-stable — or, more generally, poly-stable, that is, a direct sum of slope-stable bundles of equal slope — with vanishing slope with respect to at least one Kähler class $\omega\in\mathcal K^s\cap\mathcal K^B$ (that is, $4x_1+7x_2-12y>0$ for $\omega=x_1\tau_1+x_2\tau_2+y\phi$), so that by the Donaldson–Uhlenbeck–Yau theorem its connection solves the Hermitian Yang–Mills equations while the observable $SU(4)$ bundle is simultaneously stable. Both properties must be proven — the Bogomolov inequality is only a necessary condition and does not by itself establish stability. A rigorous proof that no such bundle exists would equally resolve the question, refuting the existence expectation stated in the source paper.

References

1. Volker Braun, Yang-Hui He, Burt A. Ovrut, Tony Pantev, The exact MSSM spectrum from string theory, J. High Energy Phys. 05 (2006) 043, DOI 10.1088/1126-6708/2006/05/043, arXiv:hep-th/0512177, https://arxiv.org/abs/hep-th/0512177
2. Volker Braun, Yang-Hui He, Burt A. Ovrut, Stability of the minimal heterotic standard model bundle, J. High Energy Phys. 06 (2006) 032, DOI 10.1088/1126-6708/2006/06/032, arXiv:hep-th/0602073, https://arxiv.org/abs/hep-th/0602073
3. Volker Braun, Yang-Hui He, Burt A. Ovrut, Yukawa couplings in heterotic standard models, J. High Energy Phys. 04 (2006) 019, DOI 10.1088/1126-6708/2006/04/019, arXiv:hep-th/0601204, https://arxiv.org/abs/hep-th/0601204
4. Volker Braun, Yang-Hui He, Burt A. Ovrut, Supersymmetric hidden sectors for heterotic standard models, J. High Energy Phys. 09 (2013) 008, DOI 10.1007/JHEP09(2013)008, arXiv:1301.6767, https://arxiv.org/abs/1301.6767
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7. Anthony Ashmore, Sebastian Dumitru, Burt A. Ovrut, Hidden sectors from multiple line bundles for the B−L MSSM, Fortschritte der Physik 70 (2022) 2200071, DOI 10.1002/prop.202200071, arXiv:2106.09087, https://arxiv.org/abs/2106.09087
8. Sebastian Dumitru, Burt A. Ovrut, Moduli and hidden matter in heterotic M-theory with an anomalous U(1) hidden sector, arXiv:2201.01624 (2022), https://arxiv.org/abs/2201.01624
9. Cédric Deffayet, Burt A. Ovrut, Paul J. Steinhardt, Stable vacua with realistic phenomenology and cosmology in heterotic M-theory satisfying Swampland conjectures, J. High Energy Phys. 07 (2024) 288, DOI 10.1007/JHEP07(2024)288, arXiv:2401.04828, https://arxiv.org/abs/2401.04828