Spectral flow of monopole insertion in topological insulators

# Spectral flow of monopole insertion in topological insulators

Alan L. Carey and Hermann Schulz-Baldes

Mathematical Sciences Institute, Australian National University, Canberra, Australia
and the School of Mathematics and Applied Statistics, University of Wollongong, Australia

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
###### Abstract

Inserting a magnetic flux into a two-dimensional one-particle Hamiltonian leads to a spectral flow through a given gap which is equal to the Chern number of the associated Fermi projection. This paper establishes a generalization to higher even dimension by inserting non-abelian monopoles of the Wu-Yang type. The associated spectral flow is then equal to a higher Chern number. For the study of odd spacial dimensions, a new so-called ‘chirality flow’ is introduced which, for the insertion of a monopole, is then linked to higher winding numbers. This latter fact follows from a new index theorem for the spectral flow between two unitaries which are conjugates of each other by a self-adjoint unitary.
Keywords: monopole, spectral flow, index pairings               MSC numbers: 58J30, 37B30

## 1 Overview

The motivation for this study is Laughlin’s thought experiment [23, 2]. It considers a Landau Hamiltonian describing a two-dimensional electron in a constant magnetic field in which a magnetic flux tube is inserted at some point. This produces supplementary discrete spectrum between Landau levels which flows through a given gap while pushing the flux through. The outcome is that the spectral flow is equal to the Chern number of the Fermi projection below the given gap. While the analysis in [23, 2] uses the particular form of the Landau operator, the equality of the spectral flow resulting from a flux insertion and the Chern number is a structural fact which can also be referred to as two-dimensional topological charge pump. In particular, no constant magnetic field and no translation invariance are needed for a non-trivial spectral flow, merely a non-vanishing Chern number. For example, a flux inserted into a disordered, but gapped Haldane model [18] leads to a unit spectral flow. These results were established in [12] for a gapped tight-binding Hamiltonian (based on ideas from [28]) and are recalled in Subsection 1.1, together with some background information on the quantum Hall effect.

This paper takes the following perspective on the Laughlin argument: it is a tool to test the topological nature of the ground state of the underlying non-interacting Fermionic system via the insertion of a flux tube. Indeed, a non-vanishing spectral flow during the flux insertion indicates a non-trivial topology of the associated Fermi projection, that is, a non-vanishing Chern number or equivalently a non-vanishing index pairing between the Fermi projection and the two-dimensional Dirac operator. This perspective naturally leads to the question: what replaces the Laughlin argument in spacial dimension different from two? Actually there are similar topological invariants in other dimensions that have played a prominent role in the theory of topological insulators [36, 21, 35]. The invariants of interest all stem from the -theory classes of the -dimensional torus. In the complex cases, namely if no symmetries invoking real structures are present (like time-reversal or particle-hole symmetry), all these invariants can be calculated by (non-commutative) differential topological tools as (higher) Chern number and (higher) winding numbers. Amongst all invariants in dimension there is one called the strong topological invariant. It is the -theory class of the -torus stemming from the -sphere and is the only invariant whose calculation requires the use of the derivatives in all spacial dimensions. For even this is a Chern number, while for odd and a chiral Hamiltonian it is a winding number. The definition of these objects in a classical differential topological setting is recalled in Subsection 2.7 below, for the non-commutative version required for the study of disordered systems the reader is referred to [35]. The reason why its precise definition is not relevant is that the strong invariant is related to an index pairing by an index theorem [5, 33, 34] and only this index pairing will enter in the main results below. As the Chern number in the classical Laughlin argument is the strong invariant, the above question can be reformulated as: what type of flux insertion allows one to calculate the -dimensional strong invariant as a spectral flow?

The answer provided below is the following. A non-abelian monopole in the Clifford degrees of freedom of the Dirac operator can be inserted and leads in even spacial dimension to a spectral flow that is equal to the strong invariant given by the Chern number. In odd spacial dimension, the monopole does not lead to a classical spectral flow of the Hamiltonian, but rather a new type of spectral flow that we term a ’chirality flow’ which in turn is then given by the strong invariant. In view of these results, the classical Laughlin argument is the special case in dimension . The case of dimension is somewhat special and discussed separately in the introduction to Section 5 which also serves as an elementary illustration of the chirality flow. In higher dimension, the main challenge is the construction of the monopoles and then their insertion in tight-binding lattice translations. The main inspiration for the first point comes from the Wu-Yang construction of a non-abelian monopole in dimension [41, 38]. A conceptual approach transposing to arbitrary dimension is developed in Section 2. Moreover, the new monopoles are thoroughly investigated, in particular, their rotational covariance properties and field strength, the associated Yang-Mills field equations as well as their topological charge. For the second point, namely the construction of non-abelian lattice translations, Section 3 follows the strategy outlined by Arai [1] for the construction of magnetic translations. Once these constructions are completed, it is relatively straightforward to insert the monopole in the topological tight-binding Hamiltonian and establish the above claims on the associated spectral and chirality flow (Sections 4 and 5).

Before completing this overview with a somewhat more technical description of the two-dimensional Laughlin argument and the chirality flow, let us stress that this work is not a mere mathematical generalization to higher dimension. Clearly, three-dimensional systems are physically relevant, but also effectively higher dimensional systems can be created in driven systems with one or several driving parameters. For example, there is a four-dimensional quantum Hall effect [44, 27]. The corresponding four-dimensional strong invariants (so-called second Chern numbers) have been shown to be relevant for magneto-electric effects [26] as well as for experimental set-ups in certain photonic crystals [43] and in atomic systems [40], and potentially also for non-adiabatic effects [22].

### 1.1 Review of the Laughlin argument for quantum Hall systems

Non-interacting quantum Hall systems are two-dimensional Fermionic systems having a topologically quantized Hall conductivity which is equal to the Chern number of the Fermi projection [5, 4, 6, 35]. The non-trivial topology can result from a constant magnetic field (as in the Landau operator) or a periodic magnetic field (as in the Haldane model [18]). Let us describe the relevant mathematical facts for a short-range tight-binding Hamiltonian on a lattice Hilbert space , or more precisely a covariant family of such operators indexed by a disorder configuration taken from a compact space on which is given a group action of and an invariant and ergodic probability measure (see the above references for precise discussions on all of these notions which, however, are not essential for the statements in the present paper). If lies in a spectral gap of and is the associated Fermi projection, the associated Chern number is defined as

 Ch(pμ)=2πıEP⟨0|pμ[ı[pμ,X1],ı[pμ,X2]]|0⟩,

where is the Dirac notation for a state localized at and is the origin, and denotes the average w.r.t. and finally are the two components of the unbounded position operator on defined by for . The Kubo formula shows that the zero-temperature Hall conductance is equal to [5, 6]. An index theorem furthermore shows that

 Ch(pμ)=Ind(pμVpμ),V=X1+ıX2|X1+ıX2|, (1)

where is a Fredholm operator on and the index on the r.h.s. is known to be almost surely constant. For reasons explained further below, is also called the Dirac phase. While the Chern number is only defined for covariant Hamiltonians, the index in (1) is defined for any local Hamiltonian with a gap at . Locality means by definition that decays sufficiently fast in . The Laughlin argument described next only uses this index and is hence a purely spectral-theoretic statement about one fixed Hamiltonian. Adapting Laughlin’s idea one adds a magnetic flux to just one specified cell of the lattice . This can be done by a rotationally symmetric gauge potential (for details see [12] or Section 3 below). This results in a one parameter family of bounded local Hamiltonians on with . The main results of [12] are then:

• is a compact operator, so that and have the same essential spectrum.

• where is as in (1).

• The spectral flow is equal to .

Items (i) and (ii) are linked to the rotationally symmetric gauge. For other choices of the gauge, one may not have compactness of even though and still have the same spectrum and the spectral flow is the same. The particular relation in (ii) is not essential, crucial is merely the norm continuity of and that the initial point and final point are unitarily equivalent. Item (iii) is the main result of [12] and the proof is essentially an application of Phillips’ results connecting spectral flow to an index (herein Theorem 3 recalled for the readers’ convenience in Appendix A). Let us stress that is not equal to where is the -th root. Such a unitary equivalence would imply that there is no spectral flow.

### 1.2 Generalization to higher even dimensions

In higher even dimension the strong invariant is given by the higher Chern number where is still the Fermi projection of a local possibly matrix-valued Hamiltonian on below a gap . The definition of the higher Chern number given in [33, 35] is irrelevant for the present purposes as is again linked to the index of a Fredholm operator by an index theorem completely analogously to (1), provided the unitary operator is chosen as follows. The -dimensional (dual) Dirac operator is given by

 D=d∑j=1γjXj, (2)

where are the components of the position operator and is an irreducible representation of the Clifford algebra acting on an auxiliary finite dimensional representation space (see [24, 35] or Section 2.2 for details on this representation). Note that upon discrete Fourier transform, becomes the standard first order Dirac operator on the -dimensional torus. Associated to the Dirac operator is also its selfadjoint Dirac phase by

 G=D|D|−1, (3)

In even dimension , the Dirac operator has a chiral symmetry for a suitable selfadjoint unitary matrix acting on the representation space. In the spectral representation of one has and therefore there is a unitary such that the Dirac phase is of the form

 G=(0V∗V0). (4)

Note that for , the matrices are given by the standard Pauli matrices and therefore is as in (1). Now with from (4) and , an index theorem similar to that in (1) still connects to [33, 35]. Again the index makes sense for any local Hamiltonian , and this is the only data needed for the higher dimensional argument.

The first major result of this paper (Theorem 1 in Section 4) states that items (i), a modified version of (ii) and (iii) in Subsection 1.1 also hold for matrix-valued Hamiltonians in higher even dimensions, provided that the path is obtained by inserting the higher dimensional analogue of a non-abelian Wu-Yang monopole [41, 38]. The construction of these monopoles and consequently their insertion in a lattice Hamiltonian take up a large part of the paper, notably Sections 2 and 3. Let us stress a major difference between the two and higher dimensional case: the monopoles in dimension satisfy the Yang-Mills equation only for , see Proposition 1.

### 1.3 Chirality flow for systems in odd dimension

The second major result of the paper (Theorem 2 in Section 5) shows what replaces the Laughlin argument for chiral local Hamiltonians in odd space dimension . If acts on , that is is even, the chiral symmetry operator is and the the chiral symmetry of reads The Fermi level in such systems is so that is invertible and the so-called flat band Hamiltonian is of the form

 h|h|−1=(0u∗u0), (5)

for a unitary operator on . This unitary is called the Fermi unitary [35] because it uniquely determines the Fermi projection of the chiral Hamiltonian. The Fermi unitary has a -dimensional winding number as a strong invariant which is also called an odd Chern number [34]. Again the reader is referred to [34, 35] for the definition because all that is relevant in the following is the link to the index of a Fredholm operator (Corollary 6.3.2 in [35]):

 Chd(u)=−Ind(QUQ),

where and . While is only defined for a covariant Hamiltonian, the index makes sense for every single chiral local Hamiltonian. The new contribution of the present paper is to calculate this index and thus the strong invariant as a suitably defined spectral flow. For this purpose, one inserts again a non-abelian monopole which in dimension is precisely the Wu-Yang monopole. This provides a path of chiral Hamiltonians on with . Generically, this path is invertible so that via (5) there are associated Fermi unitaries with . Typically, the spectrum of these unitaries fills the whole unit circle. The crucial facts, corresponding to those in the even dimensional case, are:

• is compact.

• where .

• .

Items (i) and (ii) follow again from the construction of the monopole. For , there is also a relation which corresponds to the relation in (ii), but this is of no importance for the definition of the spectral flow and the claim in (iii). Indeed, the path of unitaries connects two selfadjoint unitaries and with spectrum and, as also is compact, the above spectral flow counts the eigenvalues moving between them. Theorem 4 in the Appendix allows to show that this spectral flow is equal to the index in (iii). A precise statement of (iii) is given in Theorem 2 in Section 5. It is shows that the index in (iii)’ is equal to the spectral flow from to which justifies the terminology chiral flow. Let us also advertise that the introduction to Section 5 describes a one-dimensional version of this (insertion of a flux in the Su-Schrieffer-Heeger model).

### 1.4 Organization of the paper and omissions

As already stressed above, a large part of the paper is devoted to the construction and analysis of non-abelian monopoles (Section 2) and the associated non-abelian monopole translations on the lattice (Section 3). Based on this, the higher dimensional Laughlin argument is then proved in Section 4 for even and in Section 5 for odd . The latter section uses the notion of chirality flow which is introduced and studied in the Appendix. Sections 2 and 3 as well as the Appendix can be read independently of the remainder of the paper.

Let us add a short comment on what is omitted in this paper, but will be dealt with elsewhere. As shown in [12] for the two-dimensional case and in [8] for a particular one-dimensional model, one can implement real symmetries to the Hamiltonian or and then analyse the resulting symmetry properties of the spectral flow when a flux is inserted. These real symmetries are typically given by a time-reversal or a particle-hole symmetry. Such a symmetry analysis is also possible for higher dimensional models. On the index side of the equalities in (iii) and (iii), this was already carried out in [17]. When the monopole is inserted, the equation will then be relevant. In some situations, one can then prove the existence of bound states for half-flux , see the example in [12].

## 2 Non-abelian monopoles

This section presents a conceptional approach to the construction of static monopoles in classical non-abelian field theory over with structure group SU. Characteristic properties of such monopoles are a singularity at the origin of and a decaying field strength which has a rotational covariance property. For a particular value of the parameters (notably for half-flux), the monopole satisfies the Yang-Mills field equation and has a topological charge equal to . All these algebraic facts are proved in subsections below. In dimension and for half-flux, these monopoles reduce precisely to the Wu-Yang monopoles [41], see Subsection 2.8. Historically, Wu and Yang exhibited explicit solutions of the SU Yang Mills equations and their motivation came from the theory of isospin. More recently the Wu-Yang monopole has resurfaced in connection with the fractional Hall effect and a discussion with references is found in [13]. Further information on the Wu-Yang monopole can be found in the monograph [38]. Let us stress though that the constructions below work in arbitrary dimension. Interestingly, in even dimension the monopole acquires a supplementary chiral symmetry. This section is written such that it can be read independently of the rest of the paper and may be of interest in the context of classical non-abelian field theory. The covariant derivatives are analyzed as purely algebraic objects here and further functional analytic issues are then dealt with in Section 3. Throughout this section, we suppose that as the following formulas are not interesting in the case for which a separate treatment is given later.

The following has some similarities to the construction of instantons in even space dimension as presented by Fujii [14]. This will briefly be discussed at the end of Subsection 2.2.

### 2.1 General set-up for classical non-abelian gauge theory

To fix our notations, let us describe the framework and collect a few basic general facts on classical non-abelian gauge theory on or a subset of . We restrict to a SU-gauge theory with no external sources (not coupled to any other field so that one speaks of a pure gauge theory). The basic object is a non-abelian gauge potential which is a collection of functions in the Lie algebra , namely

 Ak(x)∗=−Ak(x),\rm Tr(Ak(x))=0,k=1,…,d. (6)

Below a particular gauge potential describing a monopole will be introduced. It will not be defined on all , but only for . Given , one next constructs covariant derivatives

 ∇k=∂k+Ak(x),k=1,…,d.

The non-abelian field strength tensor associated to is defined as the collection of operators

 Fk,l(x)=[∇k,∇l],k,l=1,…,d.

Actually, these operators are multiplication operators and can also viewed as matrix-valued functions on . They are given by the standard expression

 Fk,l(x)=∂kAl(x)−∂lAk(x)+[Ak(x),Al(x)]. (7)

Also in the present context the field strength is skewadjoint and antisymmetric in the tensor indices:

 Fk,l(x)∗=−Fk,l(x),Fk,l(x)=−Fl,k(x).

In particular, the field strength takes values in the Lie algebra su. The entries of the field strength are called the non-abelian magnetic fields (here is the completely antisymmetric tensor and the Einstein sum convention is used). The field strength is defined as a commutator of covariant derivatives. As these covariant derivatives satisfy a Jacobi identity, this leads to an equation for the field strength which called the second Bianchi identity:

 [∇n,Fk,l]+[∇k,Fl,n]+[∇l,Fn,k]=0.

Another set of equations that, however, a given field strength may or may not satisfy, are the Yang-Mills field equations (in absence of exterior currents). They read for , or equivalently

 d∑k=1(∂kFk,l+[Ak,Fk,l])=0,l=1,…,d. (8)

It will also be useful to slightly generalize the notations. For and another matrix-valued function , directional derivatives are introduced by

 Av(x)=d∑k=1vkAk(x)=⟨v|A(x)⟩,∂v=⟨v|∂⟩,

where here denotes the euclidean scalar product in , and . If and , one then also has Furthermore, setting for , one has

 Fv,w=∂vAw−∂vAw+[Av,Aw]=−Fw,v.

Finally let us associate differential forms to the gauge potential and field strength. The connection -form is a matrix-valued differential form on and the associated field strength is a matrix-valued curvature -form on . They are given by

 A=d∑k=1Akdxk,F=12d∑k,l=1Fk,l(x)dxk∧dxl, (9)

where the roman denotes exterior differentiation. The Cartan structure equation linking these forms , and also the Bianchi identity and Yang-Mills equation can be written out using these forms.

### 2.2 Construction of the monopole potential

Let be a faithful selfadjoint representation of the generators of the complex -dimensional Clifford algebra on the spinor space , namely for and and . This representation can be chosen irreducible if for odd and for even . Irreducibility is important for some, but not all results below. If is even, then there exists a grading operator on with and such that . For odd, the representation is chosen such that . Now introduce the (dual) Dirac operator defined by (2) as an unbounded, selfadjoint multiplication operator on with unitary, selfadjoint phase , see (3). As vanishes at the origin, one has to take care with the definition of . One first defines on , and then readily checks that the range of is dense. Hence is essentially selfadjoint and similarly one can check that . Upon Fourier transform, becomes a first-order differential Dirac operator on . Later on, we will also consider as an operator on and then the Fourier transform is the Dirac operator on the -torus which has compact resolvent. Let us further note that is only defined away from the origin and thus one has to choose on the lattice Hilbert space , see Proposition 6 below. If is even,

 ΓDΓ=−D. (10)

In the spectral representation of , this leads to, still only for even,

 Γ=(100−1),G=(0V∗V0), (11)

for a unitary which can also be expressed in terms of a lower-dimensional representation of . Now the monopole potential with flux is introduced by

 Aαk(x)=αG∗(∂kG),k=1,…,d, (12)

where are the partial derivatives (diagonally extended, namely is identified with ). Here the argument in indicates that we consider as a matrix-valued function on , but let us stress that it is also viewed as a multiplication operator on and as such part of the covariant derivatives. Indeed, can also be written as and hence the definition (12) can also be written in terms of the associated -form as

 Aα=αG∗(dG).

Let us also stress that is a pure gauge only for and (recall that a vector potential is called a pure gauge if it is a logarithmic derivative of a gauge transformation, that is, a unitary multiplication operator). As in Section 2.1, we also use the notation . By construction, the monopole potential is skew-adjoint and traceless, that is satisfies (6) and thus lies in the Lie algebra su of SU. Note that for even , one has so that is diagonal in the spectral representation (11) of .

Next let us derive alternative expressions for the monopole potential. This will use the radial operator

 R=(X21+…+X2d)12=|D|,

which is also a diagonal selfadjoint operator on . As , one has

 G[∂k,G]=G(γkR−XkDR3)=(DγkR2−XkD2R4)=Dγk−XkR2=12[D,γk]R2,

 Aαk(x)=α2[D,γk]R2=αDγk−xkR2=α∑j≠kγjγkxjR2. (13)

Furthermore, the monopole potential is divergence free away from the origin :

 d∑k=1∂kAαk(x)=α2d∑k=1([γk,γk]R2−2[D,γk]R3(2xk))=0.

The monopole potential is not rotationally invariant, but has a covariance property w.r.t. rotations and the spin representation, see (23) below.

As in Section 2.1, we similarly introduce the notation and . Then is a symmetry if :

 γ∗v=γv,γ2v=1.

The group is multiplicatively generated by the with [24]. Furthermore, the even elements in , namely those formed of an even number of ’s, form the subgroup [24]. Moreover, this notation allows to write

 D=⟨X|γ⟩=γX,Aαv(x)=α2[⟨x|γ⟩,⟨v|γ⟩]R2=α2[γx,γv]R2. (14)

A crucial property of the monopole potential is its radial scaling property:

 Aαv(rx)=1rAαv(x),r>0. (15)

In particular, it is completely determined by its values on the unit sphere . Furthermore, decays (in matrix operator norm) at infinity as

 ∥Aαv(x)∥≤|α|R. (16)

Let us conclude this section with a brief comment on the construction of instantons. These are everywhere defined (and smooth) gauge fields which decay and are of pure gauge type at infinity. Instantons also have a topological charge similar to the half-flux monopoles (12). However, the latter have a singularity and are not of pure gauge at infinity. In the present context, the construction of the gauge potential for instantons as given in [14] can be restated as

 A\rm\tiny instk(x)=x2x2+1G(x)∗∂kG(x),k=1,…,d.

For , this is the BPST instanton and in higher dimension it is connected to [14]. The topological charge (also called Pontryagin number or instanton number) can be calculated by similar techniques as in the subsections below.

### 2.3 Covariant derivatives of a monopole

The covariant derivative in the field of a monopole with flux is defined by

 ∇αk=∂k+Aαk(x),k=1,…,d. (17)

As in Section 2.1, we then also use the notations and for . As the are skew-adjoint, the operators are formally selfadjoint operators and it will be shown in Section 3.1 that they actually define selfadjoint operators on . An important algebraic relation is

 G∇αvG=∂v+G[∂v,G]+GAαv(x)G=∇1−αv. (18)

This follows from (13) combined with

 G12[D,γk]R2G=DR12[D,γk]R2DR=12R4D(Dγk−γkD)D=−12[D,γk]R2.

Let us also note that for even , one has so that is diagonal in the spectral representation of .

### 2.4 Field strength and Yang-Mills equations of a monopole

Associated to one now has via (7) the non-abelian field strength tensor of the -monopole. With some care, one can calculate the field strength explicitly from (13), namely from

 ∂kAαl(x)=αR4(R2γkγl−∑j≠l2xjxkγjγl)

and

 [Aαk(x),Aαl(x)]=2α2R4(−R2γkγl−∑j≠kxjxkγjγl+∑j≠lxjxlγjγk),

one deduces

 Fαk,l(x) =2(α2−α)R4(R2γlγk+D(xkγl−xlγk)). (19)

Therefore for and the field strength vanishes, in accordance with the fact noted above that and are pure gauge fields. Furthermore, in the same way as in (15), one has

 Fαk,l(rx)=1r2Fαk,l(x),r>0, (20)

and thus

 ∥Fαk,l(x)∥≤6|α2−α|R2.

For even dimension , Both sums in the Yang-Mills field equations (8) can be calculated explicitly for the monopole gauge potential :

###### Proposition 1

One has

 d∑k=1∂kFαk,l=2(α2−α)(d−2)R4(Dγl−xl),

and

 d∑k=1[Aαk,Fαl,k]=4α(α2−α)(d−2)R4(Dγl−xl).

In particular, for the Yang-Mills field equations (8) for hold if and only if .

Proof. Both identities follow from algebraic manipulations which require care and patience, but no creativity.

### 2.5 Action of the special orthogonal group on the Clifford algebra

The special orthogonal group naturally acts on the Clifford algebra via and the multiplicative extension to products of ’s and thus the whole Clifford algebra. This action is implemented by the restriction to the subgroup of the group action of on by adjunction [24], as will be explained in some detail next. The crucial fact, following from a straightforward calculation, is that

 γvγwγv=γRvw,

where and the reflection is defined by for and orthogonal to . Note that is orthogonal and squares to the identity, and has as a simple eigenvalue. Hence is a reflection in the conventional sense (leaving a hyperplane invariant). In order to further extend the action, let us recall next that the Clifford algebra is equipped with a linear anti-involution satisfying , , . This corresponds simply to the transpose (in a faithful matrix representation of the Clifford algebra). Then, given a sequence of unit vectors,

 (γv1⋯γvk)γwi(γv1⋯γvk)=γRv1⋯Rvkw.

On the other hand, every can be decomposed into an even number (more precisely, of them for even and for odd ) of reflections . Indeed, by the spectral theorem can be diagonalized into a block diagonal matrix with blocks given by rotations, and each such rotation block can be factorized into two reflections . In conclusion, for any one can set , which is a lift of into . Then the above reads

 γOw=gOγwi(gO). (21)

While all this is independent of the representation of , we here only wrote it out in one given representation on , so that is a unitary representation of on .

The group also naturally acts on and thus on by

 (O⋅ψ)(x)=ψ(O∗x),O∈\rm SO(d),ψ∈L2(Rd).

Without the monopole, that is , one can drop the fibre . Then one readily checks on the one parameter group that and so,

 O∇0vO∗=∇0Ov. (22)

Now let us extend the action of to by tensoring with the unitary representation . This action is denoted by . As is a multiplication operator on , one has

 ^gOAαv(x)^g∗O=^gOα2[γx,γv]R2^g∗O=α2[gOγO∗xg∗O,gOγvg∗O]R2=α2[γx,γOv]R2=AαOv(x), (23)

or alternatively . One deduces upon combination with (22)

 ^gO∇αv^g∗O=∇αOv. (24)

Also the field strength satisfies a similar covariance relation, namely

 ^gOFαv,w^g∗O=FαOv,Ow,Fαv,w(Ox)=gOFαO∗v,O∗w(x)g∗O. (25)

### 2.6 Total field strength

Let us introduce the differential -forms on

 Ωα2n=\rm Tr(G(Fα)∧n) (26)

where is the curvature -form associated to by (9). For even so that one also disposes of , let us set

 Ωα2n+1=\rm Tr(ΓAα∧(Fα)∧n), (27)

here with a connection -form defined by (9). These are -forms on . In this section, these forms are evaluated and then those of suitable degree are integrated over the unit sphere (or equivalently any homotopic surface). Towards the end of the section, it will be argued that these integrals are the total field strength. Let us begin by examining the behaviour of these forms under rotations and radial scaling. The equations (24) and (25) imply transformation laws for the connection -form and the curvature -form under the maps :

 O∗Aα=gOAαg∗O,O∗Fα=gOFαg∗O. (28)

Indeed, with matrix entries of ,

 (O∗Aα)(x)=d∑k,l=1gOAO∗ek(x)g∗OOk,ldxl=d∑k,l,m=1gOOk,mAm(x)g∗OOk,ldxl=gOAα(x)g∗O,

and similarly for the curvature form. From the invariance properties (28) of the connection -form and curvature -form combined with the invariance of the trace, one deduces

 O∗Ωα2n=Ωα2n,O∗Ωα2n+1=Ωα2n+1. (29)

As to radial scaling defined by for , the equations (15) and (20) imply and . As also , it follows that

 S∗rΩα2n=Ωα2n,S∗rΩα2n+1=Ωα2n+1. (30)

It is hence sufficient to restrict the forms on the unit sphere . For the form of maximal degree on , the rotational invariance (29) implies that the restriction is proportional to the volume form on . In particular, the form is closed on all . Let us evaluate the proportionality constant on .

###### Proposition 2

If the Clifford representation is irreducible, the restriction of to is given by

 Ωαd−1=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(2ı(α−α2))d−12(d−1)!νd−1,d odd,−2ıα(2ı(α−α2))d2−1(d−1)!νd−1,d even.

Proof. Due to the rotation invariance of , it is sufficient to evaluate at one point which we choose to be the unit vector . Starting from (19) with and , one first finds

 Fα(ed)=(α2−α)d∑k,l=1(γlγk+γd(δk,dγl−δl,dγk))dxk∧dxl=(α−α2)d−1∑k,l=1γkγldxk∧dxl.

Therefore for odd and thus , one has and thus

 Ωαd−1(ed) =(α−α2)d−12d−1∑k1,…,kd−1=1\rm Tr(γdγk1⋯γkd−1)