Flops, Gromov-Witten invariants and symmetries of line bundle cohomology on Calabi-Yau three-folds
Journal of Geometry and Physics Elsevier 171 (2021) 104398
Abstract:
The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Many of these manifolds exhibit certain symmetries on the Picard lattice which preserve the zeroth cohomology.Swampland conjectures and infinite flop chains
Physical Review D American Physical Society 104:4 (2021) 46008
Abstract:
We investigate swampland conjectures for quantum gravity in the context of M-theory compactified on Calabi-Yau threefolds which admit infinite sequences of flops. Naively, the moduli space of such compactifications contains paths of arbitrary geodesic length traversing an arbitrarily large number of Kähler cones, along which the low-energy spectrum remains virtually unchanged. In cases where the infinite chain of Calabi-Yau manifolds involves only a finite number of isomorphism classes, the moduli space has an infinite discrete symmetry which relates the isomorphic manifolds connected by flops. This is a remnant of the eleven-dimensional Poincare symmetry and is consequently gauged, as it has to be, by the no-global symmetry conjecture. The apparent contradiction with the swampland distance conjecture is hence resolved after dividing by this discrete symmetry. If the flop sequence involves infinitely many nonisomorphic manifolds, this resolution is no longer available. However, such a situation cannot occur if the Kawamata-Morrison conjecture for Calabi-Yau threefolds is true. Conversely, the swampland distance conjecture, when applied to infinite flop chains, implies the Kawamata-Morrison conjecture under a plausible assumption on the diameter of the Kähler cones.Topological formulae for the zeroth cohomology of line bundles on del Pezzo and Hirzebruch surfaces
Complex Manifolds De Gruyter Open 8:1 (2021) 223-229
Abstract:
We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.Index formulae for line bundle cohomology on complex surfaces
Fortschritte der Physik / Progress of Physics Wiley 68:2 (2020) 1900086
Abstract:
We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equations for cohomology and they are a first step towards understanding similar formulae recently obtained for Calabi-Yau three-folds.Machine learning line bundle cohomology
Fortschritte der Physik Wiley 68:1 (2019) 1900087