Complex line bundle Laplacians

Abstract

In the present work, we extend the theoretical and numerical discussion of the well-known Laplace–Beltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace–Beltrami operator as a special case.

To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finite-element discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians.

Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group H _m−1(M,∂M), we have the freedom to choose n phase shifts, one for each generator, resulting in a n-dimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes.

Employing our discretization technique, we provide and discuss several interesting computational examples highlighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.

Appendix: Algebraic topology

Let M be a path-connected topological space. A continuous map γ:[0,1]→M with γ(0)=γ(1)=p is called a loop with base point p. Two loops are called homotopic if one can be gradually deformed into the other. The set of equivalence classes of loops with a fixed base point is called the first fundamental group of M, denoted by π ₁(M,p).

A covering of M is a space $\widetilde{M}$ with a surjective local homeomorphism $\rho:\widetilde{M} \to M$. If $\widetilde{M}$ is simply connected, then ρ is called a universal covering of M. A deck transformation is a homeomorphism $h: \widetilde{M} \to \widetilde{M}$ such that ρ∘h=ρ. Using these notions, another geometric interpretation of the first fundamental group is given by the fact that it is isomorphic to the deck transformation group of the universal cover of M.

To visualize this construction, consider a topological torus M=S ¹×S ¹ as shown in Fig. 12. Its fundamental group is generated by two loops, denoted by a and b. Cutting the torus along these loops creates a rectangle, a so-called fundamental domain. The universal covering is the Euclidean plane $\widetilde{M}=\mathbb{R}^{2}$, which is tiled with copies of the fundamental domain. The fundamental group acts on the points $\widetilde{q} \in\widetilde{M}$ by translation, sending $\widetilde{q}$ to the corresponding point in another copy of the fundamental domain.

Fig. 12

Fundamental group of the torus acting on its universal covering space by deck transformations

Now, assume M is a manifold represented by a singular simplicial complex and let R be an arbitrary ring. A k-chain is a formal linear combination of oriented k-simplices with coefficients in the ring R. The set of k-chains form a group C _k under addition. The boundary operator ∂ _k :C _k →C _k−1 is a linear operator that maps any oriented simplex to the chain consisting of its appropriately signed oriented boundary simplices. A chain α∈C _k is called closed, or cycle, if ∂α=0 and it is called exact if it can be written as α=∂γ for some γ∈C _k+1. Any exact chain is closed as a consequence of the fact that the boundary of a boundary is empty, i.e. ∂ ²=0. Therefore, the sequence of chain groups C _k with the boundary operators in between form a chain complex.

Now let Z _k be the group of closed k-chains and let B _k be the group of exact k-cycles. Two cycles α,β∈Z _k are called homologous, if α−β∈B _k . The kth homology groups are the quotient groups H _k (M,R):=Z _k /B _k induced by this equivalence relation.

The set of homomorphisms from H _k to R form the k-th cohomology group H ^k(M,R).

For manifolds M with non-empty boundary ∂M, we will also need the so called relative homology groups H _k (M,∂M) which are obtained by modifying the homology equivalence relation to treat any chain on the boundary as zero. In this relative homology, two relative k-cycles a,β are called homologous if α−β=∂γ+δ for a (k+1)-chain γ and some k-chain δ contained in the boundary ∂M.

The well-known Lefschetz duality theorem guarantees the existence of an isomorphism between the cohomology group H ^k(M) and the relative homology group H _m−k (M,∂M) where m denotes the dimension of the manifold.

For the annulus above, the cycle α is a generator of H ₁(M). It is also a generator of the first fundamental group π ₁(M). The blue path β is not a cycle, since its boundary consists of two points. However, it is a relative cycle since these points lie on the boundary ∂M. In fact, β is a generator of H ₁(M,∂M).