English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We propose a new conjecture on the relation between the species doubling oflattice fermions and the topology of manifold on which the fermion action isdefined. Our conjecture claims that the maximal number of fermion species on afinite-volume and finite-spacing lattice defined by discretizing a$D$-dimensional manifold is equal to the summation of the Betti numbers of themanifold. We start with reconsidering species doubling of naive fermions on thelattices whose topologies are torus ($T^{D}$), hyperball ($B^D$) and theirdirect-product space ($T^{D} \times B^{d}$). We find that the maximal number ofspecies is in exact agreement with the sum of Betti numbers $\sum^{D}_{r=0}\beta_{r}$ for these manifolds. Indeed, the $4D$ lattice fermion on torus hasup to $16$ species while the sum of Betti numbers of $T^4$ is $16$. Thiscoincidence holds also for the $D$-dimensional hyperball and theirdirect-product space $T^{D} \times B^{d}$. We study several examples of latticefermions defined on discretized hypersphere ($S^{D}$), and find that it has upto $2$ species, which is the same number as the sum of Betti numbers of$S^{D}$. From these facts, we conjecture the equivalence of the maximal numberof fermion species and the summation of Betti numbers. We discuss a program forproof of the conjecture in terms of Hodge theory and spectral graph theory.

New conjecture on exact Dirac zero-modes of lattice fermions