Narrow Results

In this paper, we fix the complex structure and explore the moduli space of the heterotic system by considering two different yet “dual” deformation paths starting from a Kähler solution. They correspond to deformation along the Bott–Chern cohomology class and the Aeppli cohomology class, respectively. Using the implicit function theorem, we prove the stability of the existence of heterotic solutions under these two deformations and hence establish an initial step in constructing local moduli coordinates around a Kähler solution.

It was proposed that the Calabi–Yau geometry can be intrinsically connected with some new symmetries, some new algebras. In order to do so the Berger graphs corresponding to K3-fibre CY_d (d≥3) reflexive polyhedra have been studied in detail. These graphs can be naturally obtained in the framework of Universal Calabi–Yau algebra (UCYA) and decoded in an universal way by changing some restrictions on the generalized Cartan matrices associated with the Dynkin diagrams that characterize affine Kac–Moody algebras. We propose that these new Berger graphs can be directly connected with the generalizations of Lie and Kac–Moody algebras.

Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, as well as some nonrelativistic models. We find that for the Calabi–Yau models, the Lanczos coefficients grow slower than linearly for small $n$’s, consistent with the behavior of integrable models. On the other hand, for the nonrelativistic models, the Lanczos coefficients initially grow linearly for small $n$’s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope.

To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kähler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT constructions for the simplest known examples that are based in Calabi-Yau geometry, namely for the toroidal superconformal field theories and their ℤ₂-orbifolds. En route from geometry to CFT, I offer a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is recalled as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.