Advances in Discrete Differential Geometry pp 197-239 | Cite as

# Complex Line Bundles Over Simplicial Complexes and Their Applications

## Abstract

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise-constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension.

## 1 Introduction

Vector bundles are fundamental objects in Differential Geometry and play an important role in Physics [2]. The Physics literature is also the main place where discrete versions of vector bundles were studied: First, there is a whole field called Lattice Gauge Theory where numerical experiments concerning connections in bundles over discrete spaces (lattices or simplicial complexes) are the main focus. Some of the work that has been done in this context is quite close to the kind of problems we are going to investigate here [3, 4, 6].

Vector bundles make their most fundamental appearance in Physics in the form of the complex line bundle whose sections are the wave functions of a charged particle in a magnetic field. Here the bundle comes with a connection whose curvature is given by the magnetic field [2]. There are situations where the problem itself suggests a natural discretization: The charged particle (electron) may be bound to a certain arrangement of atoms. Modelling this situation in such a way that the electron can only occupy a discrete set of locations then leads to the “tight binding approximation” [1, 12, 15].

Recently vector bundles over discrete spaces also have found striking applications in Geometry Processing and Computer Graphics. We will describe these in detail in Sect. 2.

Let \({\tilde{{\mathrm E}}}\) be a smooth vector bundle over \({\tilde{{\mathrm M}}}\) of rank \(\mathfrak {K}\). Then we can define a discrete version \(\mathrm E\) of \({\tilde{{\mathrm E}}}\) by restricting \({\tilde{{\mathrm E}}}\) to the vertex set \(\mathcal {V}\) of the triangulation. Thus \(\mathrm E\) assigns to each vertex \(i\in \mathcal {V}\) the \(\mathfrak {K}\)-dimensional real vector space \(\mathrm E_i:={\tilde{{\mathrm E}}}_i\). This is the way vector bundles over simplicial complexes are defined in general: Such a bundle \(\mathrm E\) assigns to each vertex \(i\) a \(\mathfrak {K}\)-dimensional real vector space \(\mathrm E_i\) in such a way that \(\mathrm E_i\mathop {\cap }\mathrm E_j = \emptyset \) for \(i\ne j\).

So far the notion of a discrete vector bundle is completely uninteresting mathematically: The obvious definition of an isomorphism between two such bundles \(\mathrm E\) and \(\hat{\mathrm E}\) just would require a vector space isomorphism \(f_i:\mathrm E_i\rightarrow \hat{\mathrm E}_i\) for each vertex \(i\). Thus, unless we put more structure on our bundles, any two vector bundles of the same rank over a simplicial complex are isomorphic.

Let us look at the special case of a rank \(2\) bundle \(\mathrm E\) that is oriented and comes with a Euclidean scalar product. Then the \(90^{\circ }\)-rotation in each fiber makes it into \(1\)-dimensional complex vector space, so we effectively are dealing with a hermitian complex line bundle. If \({i\!jk}\) is an oriented face of our simplicial complex, the monodromy \(P_{\partial \,{i\!jk}}:\mathrm E_i\rightarrow \mathrm E_i\) around the triangle \({i\!jk}\) is multiplication by a complex number \(h_{{i\!jk}}\) of norm one. Writing \(h_{{i\!jk}}=e^{\imath \alpha _{{i\!jk}}}\) with \(-\pi <\alpha _{{i\!jk}}\le \pi \) we see that this monodromy can also be interpreted as a real curvature \(\alpha _{{i\!jk}}\in (-\pi ,\pi ]\). It thus becomes apparent that the information provided by the connection \(\eta \) cannot encode any curvature that integrated over a single face is larger than \(\pm \pi \). This can be a serious restriction for applications: We effectively see a cutoff for the curvature that can be contained in a single face.

Remember however our starting point: We asked for structure that can be naturally transferred from the smooth setting to the discrete one. If we think again about a triangulated smooth manifold it is clear that we can associate to each two-dimensional face \({i\!jk}\) the integral \(\varOmega _{{i\!jk}}\) of the curvature \(2\)-form over this face. This is just a discrete \(2\)-form in the sense of discrete exterior calculus [5]. Including this discrete curvature \(2\)-form with the parallel transport \(\eta \) brings discrete complex line bundles much closer to their smooth counterparts:

### Definition

*A hermitian line bundle with curvature over a simplicial complex*\(\mathcal {X}\)

*is a triple*\((\mathrm E,\eta ,\varOmega )\).

*Here*\(\mathrm E\)

*is complex hermitian line bundle over*\(\mathcal {X}\),

*for each edge*\({ij}\)

*the maps*\(\eta _{{ij}}:\mathrm E_i\rightarrow \mathrm E_j\)

*are unitary and the closed real-valued*\(2\)

*-form*\(\varOmega \)

*on each face*\({i\!jk}\)

*satisfies*

In Sect. 8 we will define for hermitian line bundles with curvature a degree (which can be an arbitrary integer) and we will prove a discrete version of the Poincaré-Hopf index theorem concerning the number of zeros of a section (counted with sign and multiplicity).

Finally we will construct in Sect. 10 for each hermitian line bundle with curvature a piecewise-smooth bundle with a curvature \(2\)-form that is constant on each face. Sections of the discrete bundle can be canonically extended to sections of the piecewise-smooth bundle. This construction will provide us with finite elements for bundle sections and thus will allow us to compute the Dirichlet energy on the space of sections.

## 2 Applications of Vector Bundles in Geometry Processing

Several important tasks in Geometry Processing (see the examples below) lead to the problem of coming up with an optimal normalized section \(\phi \) of some Euclidean vector bundle \(\mathrm E\) over a compact manifold with boundary \(\mathrm M\). Here “normalized section” means that \(\phi \) is defined away from a certain singular set and where defined it satisfies \(|\phi |=1\).

In all the mentioned situations \(\mathrm E\) comes with a natural metric connection \(\nabla \) and it turns out that the following method for finding \(\phi \) yields surprisingly good results:

*Among all sections* \(\psi \) *of* \(\mathrm E\) *find one which minimizes* \(\int _{\mathrm M} |\nabla \psi |^2\) *under the constraint* \(\int _{\mathrm M} |\psi |^2=1\). *Then away from the zero set of* \(\psi \) *use* \(\phi =\psi /|\psi |\).

The term “optimal” suggests that there is a variational functional which is minimized by \(\phi \) and this is in fact the case. Moreover, in each of the applications there are heuristic arguments indicating that \(\phi \) is indeed a good choice for the problem at hand. For the details we refer to the original papers. Here we are only concerned with the Discrete Differential Geometry involved in the discretization of the above variational problem.

### 2.1 Direction Fields on Surfaces

### 2.2 Stripe Patterns on Surfaces

*stripe pattern*on a surface \(\mathrm M\) is a map which away from a certain singular set assigns to each point \(p\in \mathrm M\) an element \(\phi (p)\in \mathbb S=\{z\in \mathbb C||z|=1\}\). Such a map \(\phi \) can be used to color \(\mathrm M\) in a periodic fashion according to a color map that assigns a color to each point on the unit circle \(\mathbb S\). Suppose we are given a \(1\)-form \(\omega \) on \(\mathrm M\) that specifies a desired direction and spacing of the stripes, which means that ideally we would wish for something like \(\phi =e^{i\alpha }\) with \(d\alpha =\omega \). Then the algorithm in [9] says that we should use a \(\phi \) that comes from taking \(\mathrm E\) as the trivial bundle \(\mathrm E=\mathrm M\times \mathbb C\) and \(\nabla \psi =d\psi -i\omega \psi \). Sometimes the original data come from an unoriented direction field and (in order to obtain the \(1\)-form \(\omega \)) we first have to move from \(\mathrm M\) to a double branched cover \({\tilde{{\mathrm M}}}\) of \(\mathrm M\). This is for example the case in Fig. 3.

### 2.3 Decomposing Velocity Fields into Fields Generated by Vortex Filaments

### 2.4 Close-To-Conformal Deformations of Volumes

## 3 Discrete Vector Bundles with Connection

An *(abstract) simplicial complex* is a collection \(\mathcal {X}\) of finite non-empty sets such that if \(\sigma \) is an element of \(\mathcal {X}\) so is every non-empty subset of \(\sigma \) [14].

An element of a simplicial complex \(\mathcal {X}\) is called a *simplex* and each non-empty subset of a simplex \(\sigma \) is called a *face of* \(\sigma \). The elements of a simplex are called *vertices* and the union of all vertices \(\mathcal {V} = \mathop {\cup }_{\sigma \in \mathcal {X}} \sigma \) is called the *vertex set of* \(\mathcal {X}\). The *dimension of a simplex* is defined to be one less than the number of its vertices: \(\dim \sigma := |\sigma |-1\). A simplex of dimension \(k\) is also called a \(k\)-*simplex*. The *dimension of a simplicial complex* is defined as the maximal dimension of its simplices.

To avoid technical difficulties, we will restrict our attention to *finite* simplicial complexes only. The main concepts are already present in the finite case. Though, the definitions carry over verbatim to infinite simplicial complexes.

### Definition 3.1

*discrete*\(\mathbb F\)

*-vector bundle*of rank \(\mathfrak {K}\in \mathbb N\) over \(\mathcal {X}\) is a map \(\pi :\mathrm E \rightarrow \mathcal {V}\) such that for each vertex \(i\in \mathcal {V}\) the

*fiber over*\(i\)

Most of the time, we slightly abuse notation and denote a discrete vector bundle over a simplicial complex schematically by \(\mathrm E \rightarrow \mathcal {X}\).

The usual vector space constructions carry fiberwise over to discrete vector bundles. So we can speak of tensor products or dual bundles. Moreover, the fibers can be equipped with additional structures. In particular, a real vector bundle whose fibers are Euclidean vector spaces is called a *discrete Euclidean vector bundle*. Similarly, a complex vector bundle whose fibers are hermitian vector spaces is called a *discrete hermitian vector bundle*.

So far discrete vector bundles are completely uninteresting from the mathematical viewpoint—the obvious definition of an isomorphism \(f\) between two discrete vector bundles \(\mathrm E\) and \({\tilde{{\mathrm E}}}\) just requires isomorphisms between the fibers \(f_i:\mathrm E_i \rightarrow {\tilde{{\mathrm E}}}_i\). Thus any two bundles of rank \(\mathfrak {K}\) are isomorphic. This changes if we connect the fibers along the edges by isomorphisms.

Let \(\sigma =\{i_0,\ldots ,i_k\}\) be a \(k\)-simplex. We define two orderings of its vertices to be equivalent if they differ by an even permutation. Such an equivalence class is then called an *orientation* of \(\sigma \) and a simplex together with an orientation is called an *oriented simplex*. We will denote the oriented \(k\)-simplex just by the word \(i_0\cdots i_k\). Further, an oriented \(1\)-simplex is called an *edge*.

### Definition 3.2

*discrete connection on*\(\mathrm E\) is a map \(\eta \) which assigns to each edge \({ij}\) an isomorphism \(\eta _{{ij}}:\mathrm E_i \rightarrow \mathrm E_j\) of vector spaces such that

### Remark 3.3

Here and in the following a morphism of vector spaces is a linear map that also preserves all additional structures—if any present. E.g., if we are dealing with hermitian vector spaces, then a morphism is a complex-linear map that preserves the hermitian metric, i.e. it is a complex linear isometric immersion.

### Definition 3.4

*morphism of discrete vector bundles with connection*is a map \(f:\mathrm E\rightarrow \mathrm F\) between discrete vector bundles \(\mathrm E\rightarrow \mathcal {X}\) and \(\mathrm F\rightarrow \mathcal {X}\) with connections \(\eta \) and \(\theta \) (resp.) such that

- (i)
for each vertex \(i\) we have that \(f(\mathrm E_i)\subset \mathrm F_i\) and the map \(f_i=\left. f\right| _{\mathrm E_i}:\mathrm E_i \rightarrow \mathrm F_i\) is a morphism of vector spaces,

- (ii)
for each edge \({ij}\) the following diagram commutes:

i.e. \(\theta _{{ij}}\circ f_i= f_j\circ \eta _{{ij}}\).

An *isomorphism* is a morphism which has an inverse map, which is also a morphism. Two discrete vector bundles with connection are called *isomorphic*, if there exists an isomorphism between them.

*trivial*, if it is isomorphic to the

*product bundle*

It is a natural question to ask how many non-isomorphic discrete vector bundles with connection exist on a given simplicial complex \(\mathcal {X}\).

### Remark 3.5

## 4 Monodromy—A Discrete Analogue of Kobayashi’s Theorem

*edge path*\(\gamma \) is a sequence of successive edges \((e_1, \ldots , e_\ell )\), i.e. \(s(e_{k+1})=t(e_k)\) for all \(k=1,\ldots ,\ell -1\), and will be denoted by the word:

*connected*, if any two of its vertices can be joined by an edge path. From now on we will only consider connected simplicial complexes.

*parallel transport*\(P_\gamma :\mathrm E_i \rightarrow \mathrm E_j\)

*along*\(\gamma \) by

*inverse path*\(\gamma ^{-1}\). If \({\tilde{\gamma }} = e_m\cdots e_{\ell +1}\) starts where \(\gamma \) ends, we can build the

*concatenation*\({\tilde{\gamma }}\gamma \): With the notation \({ij}^{-1} := ji\), we have

*edge loops*, i.e. edge paths starting and ending a given

*base vertex*\(i\) of \(\mathcal {X}\), where two such loops are identified if they differ by a sequence of

*elementary moves*[16]:

### Proposition 4.1

*monodromy*of the discrete vector bundle \(\mathrm E\).

### Proposition 4.2

Isomorphic discrete vector bundles with connection have isomorphic monodromies.

*structure group of*\(\mathrm E\).

Let \(\mathfrak V_\mathbb F^\mathfrak {K}(\mathcal {X})\) denote the *set of isomorphism classes of* \(\mathbb F\) *-vector bundles of rank* \(\mathfrak {K}\) *with connection over* \(\mathcal {X}\) and let \(\mathrm {Hom}\bigl (\pi _1(\mathcal {X}^1,i),\text {GL}(\mathfrak {K},\mathbb F)\bigr )/_\sim \) denote the *set of conjugacy classes of group homomorphisms from the fundamental group* \(\pi _1(\mathcal {X}^1,i)\) *into the structure group* \(\text {GL}(\mathfrak {K}, \mathbb F)\).

### Theorem 4.3

\(F:\mathfrak V_\mathbb F^\mathfrak {K}(\mathcal {X}) \rightarrow \mathrm {Hom}\bigl (\pi _1(\mathcal {X}^1,i),\text {GL}(\mathfrak {K},\mathbb F)\bigr )/_\sim \), \([\mathrm E]\mapsto [\mathfrak {M}]\) is bijective.

### Proof

By Proposition 4.2, \(F\) is well-defined. First we show injectivity. Consider two discrete vector bundles \(\mathrm E\), \({\tilde{{\mathrm E}}}\) over \(\mathcal {X}\) with connections \(\eta \), \({\tilde{\eta }}\) and let \(\mathfrak {M}\), \({\tilde{{\mathfrak {M}}}}\) denote their monodromies. Suppose that \([\mathfrak {M}]= [{\tilde{{\mathfrak {M}}}}]\). If we choose bases \(\{X^1_i,\ldots ,X^\mathfrak {K}_i\}\) of \(\mathrm E_i\) and \(\{\tilde{X}^1_i,\ldots ,\tilde{X}^\mathfrak {K}_i\}\) of \(\tilde{\mathrm E}_i\), then \(\mathfrak {M}\) and \(\tilde{\mathfrak {M}}\) are represented by group homomorphisms \(\rho ,\tilde{\rho }\in \mathrm {Hom}\bigl (\pi _1(\mathcal {X}^1, i), \text {GL}(\mathfrak {K},\mathbb F)\bigr )\) which are related by conjugation. Without loss of generality, we can assume that \(\rho =\tilde{\rho }\). Now, let \(\mathcal {T}\) be a spanning tree of \(\mathcal {X}\) with root \(i\). Then for each vertex \(j\) of \(\mathcal {X}\) there is an edge path \(\gamma _{\,i,j}\) from the root \(i\) to the vertex \(j\) entirely contained in \(\mathcal {T}\). Since \(\mathcal {T}\) contains no loops, the path \(\gamma _{\,i,j}\) is essentially unique, i.e. any two such paths differ by a sequence of elementary moves. Thus we can use the parallel transport to obtain bases \(\{X^1_j,\ldots ,X^\mathfrak {K}_j\}\subset \mathrm E_j\) and \(\{\tilde{X}^1_j,\ldots ,\tilde{X}^\mathfrak {K}_j\}\subset \tilde{\mathrm E}_j\) at every vertex \(j\) of \(\mathcal {X}\). With respect to these bases the connections \(\eta \) and \(\tilde{\eta }\) are represented by elements of \(\text {GL}(\mathfrak {K},\mathbb F)\). By construction, for each edge \(e\) in \(\mathcal {T}\) the connection is represented by the identity matrix. Moreover, to each edge \(e=jk\) not contained in \(\mathcal {T}\) there corresponds a unique \([\gamma _e]\in \pi _1(\mathcal {X}^1, i)\). With the notation above, it is given by \(\gamma _e = \gamma ^{-1}_{i,k}e\, \gamma _{\,i,j}\). In particular, on the edge \(e\) both connections are represented by the same matrix \(\rho ([\gamma _{e}])=\tilde{\rho }([\gamma _e])\). Thus, if we define \(f:\mathrm E \rightarrow \tilde{\mathrm E}\) by \(f(X^m_j):=\tilde{X}^m_j\) for \(m=1,\ldots ,\mathfrak {K}\), we obtain an isomorphism, i.e. \(\mathrm E\cong \tilde{\mathrm E}\). Hence \(F\) is injective.

To see that \(F\) is surjective we use \(\mathcal {T}\) to equip the product bundle \(\mathrm E:= \mathcal {V}\times \mathbb F^\mathfrak {K}\) with a particular connection \(\eta \). Let \(\rho \in \mathrm {Hom}\bigl (\pi _1(\mathcal {X}^1, i),\text {GL}(\mathfrak {K},\mathbb F)\bigr )\). If \(e\) lies in \(\mathcal {T}\) we set \(\eta _e = \text {id}\) else we set \(\eta _e:= \rho ([\gamma _e])\). By construction, \(F([\mathrm E])=[\rho ]\). Thus \(F\) is surjective. \(\square \)

### Remark 4.4

Note that Theorem 4.3 can be regarded as a discrete analogue of a theorem of S. Kobayashi [10, 13], which states that the equivalence classes of connections on principal \(G\)-bundles over a manifold \(M\) are in one-to-one correspondence with the conjugacy classes of continuous homomorphisms from the *path group* \(\Phi (\mathrm M)\) to the structure group \(G\). In fact, the fundamental group of the \(1\)-skeleton is a discrete analogue of \(\Phi (\mathrm M)\).

## 5 Discrete Line Bundles—The Abelian Case

In this section we want to focus on *discrete line bundles*, i.e. discrete vector bundle of rank \(1\). Here the monodromy descends to a group homomorphism from the closed \(1\)-chains to the multiplicative group \(\mathbb F_*:= \mathbb F\setminus \{0\}\) of the underlying field. This leads to a description by discrete differential forms (Sect. 6).

*group of isomorphism classes of*\(\mathbb F\)

*-line bundles over*\(\mathcal {X}\) by \(\mathcal {L}_{\mathcal {X}}^\mathbb F\).

The map \(F:\mathcal {L}_{\mathcal {X}}^\mathbb F\rightarrow \mathrm {Hom}\bigl (\pi _1(\mathcal {X}^1, i), \mathbb F_*\bigr )\), \([\mathrm L]\mapsto [\mathfrak {M}]\) is a group homomorphism. By Theorem 4.3, \(F\) is then an isomorphism.

*abelianization*

*group of*\(k\)

*-chains*\(\mathrm {C}_k(\mathcal {X}, \mathbb Z)\) is defined as the free abelian group which is generated by the \(k\)-simplices of \(\mathcal {X}\). More precisely, let \(\mathcal {X}^{or}_k\) denote the

*set of oriented*\(k\)

*-simplices of*\(\mathcal {X}\). Clearly, for \(k>0\), each \(k\)-simplex has two orientations. Interchanging these orientations yields a fixed-point-free involution \(\rho _k :\mathcal {X}_k^{or} \rightarrow \mathcal {X}_k^{or}\). The group of \(k\)-chains is then explicitly given as follows:

*elementary*\(k\)

*-chain*, i.e. the chain which is \(1\) for \(\sigma \), \(-1\) for the oppositely oriented simplex and zero else. With this identification a \(k\)-chain \(c\) can be written as a formal sum of oriented \(k\)-simplices with integer coefficients:

*boundary operator*\(\partial _k:\mathrm {C}_k(\mathcal {X},\mathbb Z) \rightarrow \mathrm {C}_{k-1}(\mathcal {X},\mathbb Z)\) is then the homomorphism which is uniquely determined by

*simplicial Homology groups*\(\mathrm {H}_k (\mathcal {X},\mathbb Z)\) may be regarded as a measure for the deviation of exactness:

*-cycles*, those of \(\mathrm {im}\,\partial _{k+1}\) are called \(k\)

*-boundaries*.

### Theorem 5.1

The isomorphism of Theorem 5.1 can be made explicit using discrete \(\mathbb F_*\)-valued \(1\)-forms associated to the connection of a discrete line bundle.

## 6 Discrete Connection Forms

Throughout this section \(\mathcal {X}\) denotes a connected simplicial complex.

### Definition 6.1

*group of*\(\mathfrak G\)

*-valued discrete*\(k\)

*-forms*is defined as follows:

*discrete exterior derivative*\(d_k\) is then defined to be the adjoint of \(\partial _{k+1}\), i.e.

*discrete de Rahm complex with coefficients in*\(\mathfrak G\):

*th de Rahm Cohomology group*\(\mathrm {H}^k (\mathcal {X},\mathfrak G)\)

*with coefficients in*\(\mathfrak G\) is defined as the quotient group

*closed*, those in \(\mathrm {im}\,d_{k-1}\) are called

*exact*.

*space of connections*on the discrete \(\mathbb F\)-line bundle \(\mathrm L \rightarrow \mathcal {X}\):

*base connection*\(\beta \in \mathfrak C_{\mathrm L}\) establishes an identification

### Remark 6.2

Note that each discrete vector bundle admits a trivial connection. To see this choose for each vertex a basis of the corresponding fiber. The corresponding coordinates establish an identification with the product bundle. Then there is a unique connection that makes the diagrams over all edges commute.

### Definition 6.3

Let \(\eta \in \mathfrak {C}_{\mathrm L}\). A *connection form representing the connection* \(\eta \) is a \(1\)-form \(\omega \in \varOmega ^1(\mathcal {X}, \mathbb F_*)\) such that \(\eta = \omega \beta \) for some trivial base connection \(\beta \).

Clearly, there are many connection forms representing a connection. We want to see how two such forms are related.

### Theorem 6.4

The map \(F:\mathcal {L}_{\mathcal {X}}^\mathbb F\rightarrow \varOmega ^1(\mathcal {X}, \mathbb F_*)/d\varOmega ^0(\mathcal {X}, \mathbb F_*)\), \([\mathrm L]\mapsto [\omega ]\), where \(\omega \) is a connection form of \(\mathrm L\), is an isomorphism of groups.

### Proof

### Lemma 6.5

Let \(\mathcal {X}\) be a simplicial complex and \(\mathfrak G\) be an abelian group. Then the restriction map \(\Phi :\varOmega ^k(\mathcal {X}, \mathfrak G) \rightarrow \mathrm {Hom}(\mathrm {ker}\,\partial _k, \mathfrak G),\,\omega \mapsto \left. \omega \right| _{\mathrm {ker}\,\partial _k}\) is surjective.

### Proof

If we choose an orientation for each simplex in \(\mathcal {X}\), then \(\partial _k\) is given by an integer matrix. Now, there is a unimodular matrix \(U\) such that \(\partial _k U = (0| H)\) has Hermite normal form. Write \(U = (A| B)\), where \(\partial _k A =0\) and \(\partial _k B =H\) and let \(a_i\) denote the columns of \(A\), i.e. \(A=(a_1,\ldots ,a_\ell )\). Clearly, \(a_i\in \mathrm {ker}\,\partial _k\). Moreover, if \(c\in \mathrm {ker}\,\partial _k\), then \(0= \partial _k c= (0|H) U^{-1}c\). Hence \(U^{-1}c= (q,0)^\top \), \(q\in \mathbb Z^\ell \), and thus \(c = Aq\). Therefore \(\{a_i\mid i=1,\ldots ,\ell \}\) is a basis of \(\mathrm {ker}\,\partial _k\). Now, let \(\mu \in \mathrm {Hom}(\mathrm {ker}\,\partial _k, \mathbb Z)\). A homomorphism is completely determined by its values on a basis. We define \(\omega = (\mu (a_1),\ldots ,\mu (a_\ell ),0\ldots ,0)U^{-1}\). Then \(\omega \in \varOmega ^k(\mathcal {X} , \mathbb Z)\) and \(\omega A = (\mu (a_1),\ldots ,\mu (a_\ell ))\). Hence \(\Phi (\omega )= \mu \) and \(\Phi \) is surjective for forms with coefficients in \(\mathbb Z\). Now, let \(\mathfrak G\) be an arbitrary abelian group. And \(\mu \in \mathrm {Hom}(\mathrm {ker}\,\partial _k, \mathfrak G)\). Now, if \(a_1,..,a_\ell \) is an arbitrary basis of \(\mathrm {ker}\,\partial _k\), then there are forms \(\omega _1,\ldots ,\omega _\ell \in \varOmega ^k(\mathcal {X}, \mathbb Z)\) such that \(\omega _i(a_j)=\delta _{{ij}}\). Since \(\mathbb Z\) acts on \(\mathfrak G\), we can multiply \(\omega _i\) with elements \(g\in \mathfrak G\) to obtain forms with coefficients in \(\mathfrak G\). Now, set \(\omega = \sum _{i=1}^\ell \omega _i\cdot \mu (a_i)\). Then \(\omega \in \varOmega ^k(\mathcal {X}, \mathfrak G)\) and \(\omega (a_i)= \mu (a_i)\) for \(i=1,\ldots ,\ell \). Thus \(\Phi (\omega )= \mu \). Hence \(\Phi \) is surjective for forms with coefficients in arbitrary abelian groups. \(\square \)

*integration along paths*: Fix some vertex \(i\). Then

### Theorem 6.6

The map \(F:\varOmega ^1(\mathcal {X}, \mathbb F_*)/d\varOmega ^0(\mathcal {X}, \mathbb F_*)\rightarrow \mathrm {Hom}(\ker \partial _1 , \mathbb F_*)\) given by \([\omega ]\mapsto \left. \omega \right| _{\ker \partial _1}\) is an isomorphism of groups.

### Theorem 6.7

## 7 Curvature—A Discrete Analogue of Weil’s Theorem

In this section we describe complex and hermitian line bundles by their curvature. For the first time we use more than the \(1\)-skeleton.

### Definition 7.1

*-curvature*of a discrete \(\mathbb F\)-line bundle \(\mathrm L \rightarrow \mathcal {X}\) is the discrete \(2\)-form \(\varOmega \in \varOmega ^2(\mathcal {X}, \mathbb F_*)\) given by

### Remark 7.2

Note that \(\varOmega \) just encodes the parallel transport along the boundary of the oriented \(2\)-simplices of \(\mathcal {X}\)—the “local monodromy”.

From the definition it is obvious that the \(\mathbb F_*\)-curvature is invariant under isomorphisms. Thus, given a prescribed \(2\)-form \(\varOmega \in \varOmega ^2(\mathcal {X}, \mathbb F_*)\), it is a natural question to ask how many non-isomorphic line bundles have curvature \(\varOmega \).

Actually, this question is answered easily: If \(d[\omega ]=\varOmega =d[{\tilde{\omega }}]\), then the difference of \(\omega \) and \({\tilde{\omega }}\) is closed. Factoring out the exact \(1\)-forms we see that the space of non-isomorphic line bundles with curvature \(\varOmega \) can be parameterized by the first cohomology group \(\mathrm {H}^1(\mathcal {X}, \mathbb F_*)\). Furthermore, the existence of a line bundle with curvature \(\varOmega \in \varOmega ^2(\mathcal {X}, \mathbb F_*)\) is equivalent to the exactness of \(\varOmega \).

### Example 7.3

Consider a triangulation \(\mathcal {X}\) of the real projective plane \(\mathbb R\mathrm {P}^2\). The zero-chain is the only closed \(2\)-chain and hence each \(\mathbb Z_2\)-valued \(2\)-form vanishes on every closed \(2\)-chain. But \(\mathrm {H}^2(\mathcal {X}, \mathbb Z_2)= \mathbb Z_2\) and hence there exists a non-exact \(2\)-form.

In the following we will see that this cannot happen for fields of characteristic zero or, more generally, for groups that arise as the image of such fields.

### Lemma 7.4

### Remark 7.5

Note, that for boundary cycles the condition is nothing but the closedness of the form \(\omega \). Thus Lemma 7.4 states that a closed form \(\omega \in \varOmega ^k(\mathcal {X}, \mathbb F)\) is exact if and only if the integral over all homology classes \([c]\in \mathrm {H}_k (\mathcal {X}, \mathbb Z)\) vanishes.

*-th fundamental sequence of forms with coefficients in*\(\mathfrak G\):

Combining Lemmas 6.5 and 7.4 we obtain that the fundamental sequence with coefficients in a field \(\mathbb F\) of characteristic zero is exact for all \(k>1\). This serves as an anchor point. The exactness propagates under surjective group homomorphisms.

### Lemma 7.6

Let \(\mathfrak A \xrightarrow {f} \mathfrak B \rightarrow 0\) be a an exact sequence of abelian groups. Then, if the \(k\)-th fundamental sequence of forms is exact with coefficients in \(\mathfrak A\), so it is with coefficients in \(\mathfrak B\).

### Proof

### Remark 7.7

The map \(f:\mathbb C\rightarrow \mathbb C,\, z\mapsto \exp (2\pi \imath \, z)\) provides a surjective group homomorphism from \(\mathbb C\) onto \(\mathbb C_*\), and similarly from \(\mathbb R\) onto \(\mathbb S\). Hence the \(k\)-th fundamental sequence of forms is exact for coefficients in \(\mathbb C_*\) and in the unit circle \(\mathbb S\).

### Remark 7.8

The \(k\)-th fundamental sequence with coefficients in an abelian group \(\mathfrak G\) is exact if and only if \(\varOmega ^k(\mathcal {X}, \mathfrak G)/d\varOmega ^{k-1}(\mathcal {X}, \mathfrak G)\cong \mathrm {Hom}(\ker \partial _k, \mathfrak G)\). The isomorphism is induced by the restriction map \(\Phi _k\).

The following corollary is a consequence of Remark 7.7. It nicely displays the fibration of the complex line bundles by their \(\mathbb C_*\)-curvature.

### Corollary 7.9

### Definition 7.10

Let \(\varOmega ^*\in \varOmega ^k(\mathcal {X}, \mathbb S)\). A real-valued form \(\varOmega \in \varOmega ^2(\mathcal {X}, \mathbb R)\) is called *compatible with* \(\varOmega ^*\) if \(\varOmega ^*= \exp \bigl (\imath \varOmega \bigr )\). A *discrete hermitian line bundle with curvature* is a discrete hermitian line bundle \(\mathrm L\) with connection equipped with a closed \(2\)-form compatible with the \(\mathbb S\)-curvature of \(\mathrm L\).

### Theorem 7.11

### Proof

Conversely, Corollary 7.9 yields a discrete version of a theorem of André Weil [11, 18], which states that any closed smooth integral \(2\)-form on a manifold \(\mathrm M\) can be realized as the curvature of a hermitian line bundle. This plays a prominent role in the process of prequantization [17].

### Theorem 7.12

If \(\varOmega \in \varOmega ^2(\mathcal {X}, \mathbb R)\) is integral, then there exists a hermitian line bundle with curvature \(\varOmega \).

### Proof

Consider \(\varOmega ^*:= \exp (i\varOmega )\). Since \(\varOmega \) is integral, \(\langle \varOmega ^*,c\rangle = 1\) for all \(c\in \mathrm {ker}\,\partial _2\). By Corollary 7.9, there exists \(r\in \varOmega ^1(\mathcal {X}, \mathbb S)\) such that \(dr=\varOmega ^*\). This in turn defines a hermitian line bundle with curvature \(\varOmega \). \(\square \)

### Remark 7.13

Moreover, Corollary 7.9 shows that the connections of two such bundles differ by an element of \(\mathrm {H}^1(\mathcal {X}, \mathbb S)\). Thus the space of discrete hermitian line bundles with fixed curvature \(\varOmega \) can be parameterized by \(\mathrm {H}^1(\mathcal {X}, \mathbb S)\).

## 8 The Index Formula for Hermitian Line Bundles

*rotation form*\(\xi ^\psi \)

*of*\(\psi \). This form is defined as follows:

### Theorem 8.1

### Proof

*degree of*\(\mathrm L\):

### Theorem 8.2

Now, let us consider the discrete case. In general, a *section* of a discrete vector bundle \(\mathrm E\rightarrow \mathcal {X}\) with vertex set \(\mathcal {V}\) is a map \(\psi :\mathcal {V} \rightarrow \mathrm E\) such that the following diagram commutes

i.e. \(\pi \circ \psi = id\). As in the smooth case, the space of sections of \(\mathrm E\) is denoted by \(\varGamma (\mathrm E)\).

*rotation form*\(\xi ^\psi \) of \(\psi \) is then defined as follows:

### Remark 8.3

Equation (4) can be interpreted as the condition that no zero lies in the \(1\)-skeleton of \(\mathcal {X}\) (compare Sect. 11). Actually, given a consistent choice of the argument on each oriented edge, we could drop this condition. Figuratively speaking, if a section has a zero in the \(1\)-skeleton, then we decide whether we push it to the left or the right face of the edge.

Now we can define the *index form* of a discrete section:

### Definition 8.4

*index form of*\(\psi \) by

### Theorem 8.5

The index form of a nowhere-vanishing discrete section is \(\mathbb Z\)-valued.

### Proof

*degree of*\(\mathrm L\) just as in the smooth case:

### Corollary 8.6

The discrete Poincaré-Hopf index theorem follows easily from the definitions:

### Theorem 8.7

### Proof

## 9 Piecewise-Smooth Vector Bundles over Simplicial Complexes

### Definition 9.1

- (a)
for each \(\sigma \in \mathcal {X}\) the restriction \(\mathrm E_\sigma := \left. \mathrm E\right| _\sigma \) is a smooth vector bundle over \(\sigma \),

- (b)
for each face \(\sigma ^\prime \) of \(\sigma \in \mathcal {X}\), the inclusion \(\mathrm E_{\sigma ^\prime } \hookrightarrow \mathrm E_\sigma \) is a smooth embedding.

As a simplicial complex, \(\mathcal {X}\) has no tangent bundle. Nonetheless, differential forms survive as collections of smooth differential forms defined on the simplices which are compatible in the sense that they agree on common faces:

### Definition 9.2

### Remark 9.3

Since the pullback commutes with the wedge-product \(\wedge \) and the exterior derivative \(d\) of real-valued forms, we can define the wedge product and the exterior derivative of piecewise-smooth differential forms by applying it componentwise.

### Definition 9.4

All the standard properties of \(\wedge \) and \(d\) also hold in the piecewise-smooth case.

### Definition 9.5

*connection*on a piecewise-smooth vector bundle \(\mathrm E\) over \(\mathcal {X}\) is a linear map \(\nabla :\varGamma _{ps}(\mathrm E) \rightarrow \varOmega _{ps}^1(\mathcal {X}, \mathrm E)\) such that

Once we have a connection on a smooth vector bundle we obtain a corresponding exterior derivative \(d^\nabla \) on \(\mathrm E\)-valued forms.

### Theorem 9.6

The curvature tensor survives as a piecewise-smooth \(\mathrm {End}(\mathrm E)\)-valued \(2\)-form:

### Definition 9.7

## 10 The Associated Piecewise-Smooth Hermitian Line Bundle

The goal of this section will be to construct for each discrete hermitian line bundle with curvature a piecewise-smooth hermitian line bundle with piecewise-constant curvature which in a certain sense naturally contains the discrete bundle. We first prove two lemmata.

### Lemma 10.1

### Proof

### Lemma 10.2

On the star of a simplex each closed piecewise-smooth form is exact.

Now we are ready to prove the main result of this section.

### Theorem 10.3

Let \(\mathrm L \rightarrow \mathcal {X}\) be a discrete hermitian line bundle with curvature \(\varOmega \) over a simplicial complex and let \({\tilde{\varOmega }}\) be the piecewise-smooth \(2\)-form associated to \(\varOmega \). Then there is a piecewise-smooth hermitian line bundle \(\tilde{\mathrm L}\rightarrow \mathcal {X}\) with connection \(\tilde{\nabla }\) of curvature \({\tilde{\varOmega }}\) such that \(\tilde{\mathrm L}_i = \mathrm L_i\) for each vertex \(i\) and the parallel transports coincide along each edge path. The bundle \(\tilde{\mathrm L}\) is unique up to isomorphism.

### Proof

For \(p\in S_i\), let \(f(p):= \int _{\gamma _i^{\,p}}\tilde{\omega }_i\), where \(\gamma _i^{\,p}\) denote the linear path from the vertex \(i\) to the point \(p\). Then \(\omega _i := {\tilde{\omega }}_i - df \) is a piecewise-smooth potential of \(\left. \varOmega \right| _{S_i}\) and vanishes on radial directions. For the uniqueness, let \(\hat{\omega }_i\) be another such potential. Then, the difference \(\omega _i-\hat{\omega }_i\) is closed and hence exact on \(S_i\), i.e. there is \(f:S_i \rightarrow \mathbb R\) such that \(df= \omega _i-\hat{\omega }_i\). Since \(df\) vanishes on radial directions \(f\) is constant on radial lines starting at \(i\) and hence constant on \(S_i\). Thus \(\omega _i=\hat{\omega }_i\).

*compatible*, i.e., wherever both are defined,

Now suppose there are two such piecewise-smooth bundles \(\tilde{\mathrm L}\) and \(\hat{\mathrm L}\) with connection \(\tilde{\nabla }\) and \(\hat{\nabla }\), respectively. We want to construct an isomorphism between \(\tilde{\mathrm L}\) and \(\hat{\mathrm L}\). Therefore we again use local systems. Explicitly, we choose a discrete direction field \(X \in \mathrm L\). This yields for each vertex \(i\) a vector \(X_i \in \tilde{\mathrm L}_i = \hat{\mathrm L}_i\) which extends by parallel transport along rays starting at \(i\) to a local sections \(\tilde{\phi }_i\) of \(\tilde{\mathrm L}\) and, similarly, to a local section \(\hat{\phi }_i\) of \(\hat{\mathrm L}\) defined on \(S_i\).

## 11 Finite Elements for Hermitian Line Bundles with Curvature

In this section we want to present a specific finite element space on the associated piecewise-smooth hermitian line bundle of a discrete hermitian line with curvature. They are constructed from the local systems that played such a prominent role in the proof of Theorem 10.3 and the usual piecewise-linear hat function.

### Definition 11.1

The *space of piecewise-linear sections* is given by \(\varGamma _{pl}(\tilde{\mathrm L}) := \mathrm {im}\,\iota \).

Thus we identified each section of a discrete hermitian line bundle with curvature with a piecewise-linear section of the associated piecewise-smooth bundle. This allows to define a discrete hermitian inner product and a discrete Dirichlet energy on \(\varGamma (\mathrm L)\), which is a generalization of the well-known cotangent Laplace operator for discrete functions on triangulated surfaces. Before we come to the Dirichlet energy, we define Euclidean simplicial complexes.

*piecewise-smooth*\(k\)

*-tensor*is a collection \(T=\{T_\sigma \}_{\sigma \in \mathcal {X}}\) of smooth contravariant \(k\)-tensors \(T_\sigma \) on \(\sigma \) such that

*Riemannian simplicial complex*is then a simplicial complex \(\mathcal {X}\) equipped with a

*piecewise-smooth Riemannian metric*, i.e. a piecewise-smooth positive-definite symmetric \(2\)-tensor \(g\) on \(\mathcal {X}\).

The following lemma tells us that the space of constant piecewise-smooth symmetric tensors is isomorphic to functions on \(1\)-simplices.

### Lemma 11.2

### Proof

### Definition 11.3

A *Euclidean simplicial complex* is a simplicial complex \(\mathcal {X}\) equipped with a *discrete metric*, i.e. a map \(\ell \) that assigns to each \(1\)-simplex \(e\) a length \(\ell _e > 0\) such that for each simplex \(\sigma \) the symmetric tensor \(S_{\sigma }^\ell \) is positive-definite.

*integral over*\(\mathcal {X}\) as follows:

Now we want to compute this metric explicitly in terms of given discrete data.

### Definition 11.4

A piecewise-linear section \(\tilde{\psi }\in \varGamma _{pl}(\tilde{\mathrm L})\) is called *concentrated at a vertex* \(i\), if it is of the form \(\tilde{\psi }= \iota (\psi _i)\) for some vector \(\psi _i\in \mathrm L_i\).

It is basically enough to compute the product of two such concentrated sections. Therefore, let \(\psi _i\in \mathrm L_i\) and \(\psi _j\in \mathrm L_j\) and let \(\tilde{\psi }^i\) and \(\tilde{\psi }^j\) denote the corresponding piecewise-linear concentrated sections.

As the next proposition shows, the identification of discrete and piecewise-linear sections perfectly fits together with the definitions in Sect. 8.

### Proposition 11.5

### Proof

The claim follows easily by expressing \(\tilde{\psi }\) with respect to some non-vanishing parallel section along the edge \({ij}\). \(\square \)

In particular, by Theorem 8.1, the index form of a non-vanishing section of a discrete hermitian line bundle with curvature counts the number of (signed) zeros of the corresponding piecewise-linear section of the associated piecewise-smooth bundle.

Let us continue with the computation of the metric on \(\varGamma (\mathrm L)\). To write down the formula we give the following definition.

### Definition 11.6

### Remark 11.7

Note that the functions \(\Theta _{\sigma ,i,j}^\varOmega \) are indeed well-defined. On a simplex, any two such measures induced by a discrete metric differ just by a constant.

With Definition 11.6 and Eq. (9) we obtain the following form of the metric:

### Theorem 11.8

Note that \(\Theta _{\sigma ,i,j}^\varOmega (k,l)\), and hence \(\mu _\varOmega ^{{ij}}\), can be computed explicitly using Fubini’s theorem and the following small lemma, which can be shown by induction.

### Lemma 11.9

*Dirichlet energy*of a section \(\tilde{\psi }\in \varGamma _{pl}(\tilde{\mathrm L})\), i.e.

*Dirichlet product*. Clearly, like the metric, the Dirichlet product is completely determined by the values it takes on concentrated sections.

### Lemma 11.10

### Proof

This immediately follows from two basic facts: First, \(dx_i (v_j-v_0) = \delta _{{ij}}\) for \(i,j> 0\). Second, \(h_i = \langle v_0-v_i, N_i\rangle \). \(\square \)

Lemma 11.10 yields almost immediately a higher dimensional analogue of the well-known cotangent formula for surfaces.

### Theorem 11.11

### Proof

### Definition 11.12

### Remark 11.13

Just like the functions \(\Theta _{\sigma ,i,j}^\varOmega \), the values \(\varLambda _{\sigma ,i,j}^\varOmega \) and the functions \(\Xi _{\sigma ,i,j}^\varOmega \) and are well-defined (compare Remark 11.7).

Now, with these definitions, we can summarize the above discussion by the following theorem.

### Theorem 11.14

*Dirichlet product*on \(\varGamma (\mathrm L)\) induced by the associated piecewise-smooth hermitian line bundle is given as follows: If \(\phi =\sum _i \phi _i\) and \(\psi = \sum _i \psi _i\) are two discrete sections,

## 12 Discrete Energies on Surfaces—An Example

While the computation of the Dirichlet product \(\langle \!\langle .,.\rangle \!\rangle _{\!D}\) and the metric \(\langle \!\langle .,.\rangle \!\rangle \) of discrete sections is quite complicated and tedious for higher dimensional simplicial manifolds, it is manageable for the \(2\)-dimensional case. We are going to compute it explicitly.

Throughout this section let \(\mathrm L\) denote a discrete hermitian line bundle with curvature \(\varOmega \) over a Euclidean simplicial surface \(\mathcal {X}\) and let \(\sigma =\{i,j,k\}\) be one of its triangles.

## Notes

### Acknowledgments

This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.

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