Compact Threefolds Research Articles

Computational Mirror Symmetry

We present an efficient algorithm for computing the prepotential in compactifications of type II string theory on mirror pairs of Calabi-Yau threefolds in toric varieties. Applying this method, we exhibit the first systematic computation of genus-zero Gopakumar-Vafa invariants in compact threefolds with many moduli, including examples with up to 491 vector multiplets.

Quantum Bianchi-VII problem, Mathieu functions and arithmetic

The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII0 type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI0 case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.

BPS Spectra, Barcodes and Walls

BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper we approach this problem from the perspective of persistent homology. Persistent homology is at the base of topological data analysis, which aims at extracting topological features out of a set of points. We use these techniques to investigate the topological properties which characterize the spectra of several supersymmetric models in field and string theory. We discuss how such features change upon crossing walls of marginal stability in a few examples. Then we look at the topological properties of the distributions of BPS invariants in string compactifications on compact threefolds, used to engineer black hole microstates. Finally we discuss the interplay between persistent homology and modularity by considering certain number theoretical functions used to count dyons in string compactifications and by studying equivariant elliptic genera in the context of the Mathieu moonshine.