Speaker
Description
We present a unified framework connecting Calabi--Yau geometry, singularity theory, and early-universe cosmology within heterotic and Type IIB string compactifications. Starting from the Tian--Yau threefold defined by its complete-intersection configuration matrix $P^3$ [3 0 1] ${P}^3$ [0 3 1], we demonstrate how a free $\mathbb{Z}_3$ quotient reduces the Euler characteristic to $\chi=-6$, yielding precisely three chiral generations. By embedding a stable $SU(4)$ vector bundle with $c_1(V)=0$ and $c_2(V)=12 \leq c_2(TX)=50$, we satisfy the Green--Schwarz anomaly cancellation condition while preserving $N=1$ supersymmetry and enabling Wilson-line breaking of $E_6$ to the Standard Model gauge group.
A central result is the identification of universal singularity structures governing vacuum transitions. Generic multi-field scalar potentials correspond to non-simple singularities with Milnor number $\mu=9$, but near critical degeneracy loci the Hessian vanishes and the system universally reduces to an $A_2$ cusp with $\mu=2$ [1]. This reduction controls the dynamics of first-order phase transitions in the early universe. We derive the stochastic gravitational-wave spectrum generated near these critical points, showing that potential flattening enhances the signal amplitude as $\Omega_{\text{GW}} \propto \delta^{-3}$, where $\delta=1-\hat{\xi}/\hat{\xi}_c$ measures the distance to criticality. The predicted peak frequencies naturally lie in the milli-Hertz band, and benchmark points satisfying strong first-order transition criteria yield $\Omega_{\text{GW}}h^2 \sim 10^{-12}\text{--}10^{-11}$, placing them directly within the sensitivity window of the LISA mission.
We systematically analyse the KKLT moduli stabilisation mechanism [2] incorporating leading $\alpha'^3$ corrections to the Kähler potential. The correction breaks the classical no-scale structure and generates three distinct vacuum regimes separated by a critical threshold $\hat{\xi}_c$. For $\hat{\xi} < \hat{\xi}_c$, controlled AdS minima persist and can be uplifted to metastable de~Sitter vacua; for $\hat{\xi} > \hat{\xi}_c$, the potential exhibits runaway behaviour characteristic of the Large Volume Scenario or decompactification. The conifold transition mechanism [3], where massless black holes resolve geometric singularities via flop transitions, provides a dynamical pathway connecting parent manifolds to the three-generation Tian--Yau/$\mathbb{Z}_3$ vacuum. We quantify the tension between shallow dS vacua near $\hat{\xi}_c$ and the de~Sitter swampland conjecture, mapping the parameter space where controlled string constructions remain consistent with quantum gravity constraints. Detailed derivations, explicit benchmark tables, and the full Hessian analysis are provided in [1]. Our results establish a direct observational bridge between compactification topology, singularity-driven cosmology, and gravitational-wave detectors.
Keywords: String compactification, Tian--Yau threefold, conifold transitions, singularity theory, Milnor number, KKLT mechanism, $\alpha'^3$ corrections, gravitational waves, LISA, de~Sitter swampland conjecture.
1.T. Obikhod, "Supersymmetry, Supergravity and Non--Perturbative Dynamics of Gauge Theories," arXiv:2604.19817v1 [hep-th] (2026).
2.S. Kachru, R. Kallosh, A. Linde, S.P. Trivedi, "De Sitter vacua in string theory," Phys. Rev. D68 (2003) 046005.
3.A. Strominger, "Massless black holes and conifolds in string theory," Nucl. Phys. B451 (1995) 96--108.
