Delaunay Hodge star

Delaunay Hodge star

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for ...

The Hodge Star Operator

We’ll start out by defining the Hodge star operator as a map from ∧k(R) to ∧n−k(R). Here ∧k(R) denotes the vector space of alternating k-tensors on R. Later on, we will extend this definition to alternating tensors on a finite dimensional vector space that is equipped with an inner product. Let I = (i1, ..., ik) be some increasing multi-index of length k. That is i1 < i2 < i3 < .... Let J = (j1...

A Fermionic Hodge Star Operator

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in spacetimes of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field-theory, and induces a metric in the space of wave-function(al)s just as in exterior calculus. If made real (Hermitian), this induced metric turns out to be identi...

Hodge Groups of Hodge Structures with Hodge Numbers

This paper studies the possible Hodge groups of simple polarizable Q-Hodge structures with Hodge numbers (n, 0, . . . , 0, n). In particular, it generalizes earlier work of Ribet and MoonenZarhin to completely determine the possible Hodge groups of such Hodge structures when n is equal to 1, 4, or a prime p. In addition, the paper determines possible Hodge groups, under certain conditions on th...

Hodge loci and absolute Hodge classes