Joshua Celeste | Searcharxiv

arXiv2023

Cohomology of knotted semimetals in three dimensions

We extend the topological classification scheme of Weyl semimetals via cohomology and the Mayer-Vietoris sequence to account for nodal line semimetals with space-time inversion symmetry. These are semimetals where bands meet generally in 1-dimensional submanifolds, which can generally be knots in $\T^3$. These nodal loops have two charges, the quantized Berry phase and the $\Z_2$-monopole charge, the second related to linking numbers of nodal knots between bands. We provide a manifestly topological proof of the Weyl charge cancellation condition for the $\Z_2$ monopole charge, which is known to be the second Stiefel-Whitney class of a tubular neighbourhood surrounding a Weyl submanifold via the Mayer-Vietoris sequence.

math-ph↗

arXiv2022

The families Seiberg-Witten invariants of smooth families of Kähler surfaces

We consider a generalisation of the Seiberg-Witten invariant to the families Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres diffeomorphic to a 4-manifold $X$. Of particular interest is the special case when the family has a smoothly varying Kähler structure. We obtain a general computation of the invariants when $b_1=0$ in terms of characteristic classes of some vector bundles corresponding to the cohomology groups of holomorphic line bundles of the family. Finally, we apply the formula to some examples of Kähler families where some more further explicit computations can be made. We consider a family of $\mathbb{CP}^2$'s obtained from the projectivisation of a rank 3 complex vector bundle, a family of $\mathbb{CP}^1\times\mathbb{CP}^1$'s obtained as the fibre product of the projectivisation of two rank 2 complex vector bundles and a family with fibres being the blowup of a Kähler surface at a point known as the universal blowup family.

math.AG↗