## Abstract

Using a sheaf-theoretic extension of conventional principal bundle theory, the Dirac monopole is formulated as a spherically symmetric model free of singularities outside the origin such that the charge may assume arbitrary real values. For integral charges, the construction effectively coincides with the usual model. Spin structures and Dirac operators are also generalized by the same technique.

## 1. Introduction

In his classical paper on quantization of magnetic poles, Dirac (1931) remarked that ‘Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalisation of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation’. In accordance with this principle, the present paper further extends the work of Dirac by exploring the ‘dequantization’ of magnetic poles.

Diverse and numerous versions of the magnetic pole construction and the associated charge quantization condition of Dirac (1931, 1948) have appeared in the literature, but the basic model can be most concisely and accurately formulated in terms of the Hopf fibration (Wu & Yang 1975; Trautman 1977; Yang 1977). From a mathematical viewpoint, the rationale is as follows. The monopole potential is modelled as a connection *A* on a nontrivial principal *U*(1)-bundle *P* over a subset of Minkowski space. A charged matter field interacting with the monopole is accordingly modelled as a section of a vector bundle *E*_{n} associated with *P* via the representation *ρ*(*e*^{iθ})*ψ*=e^{inθ}*ψ* of *U*(1) on a complex vector space. Because *A* then induces the connection *n**A* on *E*_{n}, the integer *n* can be identified with the electric charge of the matter field (see Urbantke 2003 for a detailed account of the description of a magnetic monopole in the language of the Hopf fibration).

However, following the spirit of Dirac, if we describe the monopole by a more general mathematical scheme, then the interaction of matter fields and magnetic poles *with arbitrary real charges* can also be modelled in a consistent manner. The present paper introduces a new sheaf-theoretic framework permitting an explicit construction of arbitrarily charged magnetic monopoles. This framework is likewise applied to generic *U*(1)-bundles and also yields, as a by-product, the notion of a quasi-spin structure defined on arbitrary space-times. The results suggest that topological quantization in general can be viewed from a more flexible perspective.

Although magnetic monopoles have not been observed experimentally, one important physical consequence of the present model is that their detection would not necessarily imply the quantization of electric charge. Likewise, an observed violation of charge quantization would not necessarily imply the nonexistence of magnetic monopoles. Furthermore, grand unified theories require the existence of magnetic monopoles, and, according to conventional field theory described in terms of manifolds this necessarily implies charge quantization; hence, the detection of nonintegral electric charge would indicate that the present scheme, based on sheaves rather than manifolds, may be physically more realistic. Also, a sheaf construction permits the global description of fermion fields on Lorentzian manifolds possessing no conventional spin structure, as an alternative to the cumbersome machinery of Kähler fermions.

The paper is organized as follows. In §2 we present a very brief sketch of sheaf theory for the benefit of readers less acquainted with the subject. This is intended to provide the bare minimum of information necessary for following the ensuing discussion; for further details on the sheaf theory, see Bredon (1997), Kultze (1970), or Wells (2008). In §3 we develop the basic mathematical machinery used in later sections. The key idea here is the construction of a principal -sheaf bundle that generalizes the conventional notion of a principal bundle. In §4 we prove that equivalence classes of principal *G*-bundles can, under certain hypotheses, be mapped injectively into equivalence classes of -sheaf bundles. This implies that the Dirac sheaf bundle constructed in §5 does indeed constitute a generalization of the conventional Dirac monopole. The spherically symmetrical connection and curvature of the Dirac sheaf bundle are constructed in §6, demonstrating that the magnetic charge of this model can assume arbitrary real values. The interaction of the generalized monopole with a charged matter field is considered in §7. Dequantization of more general *U*(1) bundles is considered in §8, with particular attention to gravitational and electromagnetic instantons. In §9 the basic machinery developed earlier is applied to spin structures and Dirac operators. In §10 the paper concludes with a brief discussion of possible implications in diverse areas of physics.

## 2. Elements of sheaf theory

The concept of a sheaf over a manifold *X* provides a way of interpolating local data and global data on X. We begin with the definition of a presheaf. A *presheaf* on *X* is a functor assigning, to each open , a group *F*(*U*), abelian or otherwise, such that for each the restriction map , , defines a homomorphism and such that for . An element *σ*∈*F*(*U*) is referred to as a *section* of *F*(*U*) over *U*. The restriction of *σ*∈*F*(*U*) on is thus given by .

The sections *σ*∈*F*(*U*) and τ∈*F*(*V*) are said to be *equivalent* at if there is a neighbourhood *W* of *x* such that . The equivalence class containing *σ*∈*F*(*U*) is called the *germ* of *σ* at *x*, and the set of all such germs for any fixed *x* is denoted by .

The disjoint union of all the sets provides local information about the structure of . However, information concerning the global structure has been lost because we have discarded relations between the for varying *x*. To retrieve some global structure, we introduce a topology in the following manner. For a fixed *σ*∈*F*(*U*) the set of all germs for *x*∈*U* is taken to be an open set in , and the topology of is defined as that generated by these open sets.

The projection mapping into *x* has the property that for any point with *π*(*t*)=*y* there is a neighbourhood *N*={*σ*_{x}|*x*∈*U*} for *σ*∈*F*(*U*) and *σ*_{y}=*t* such that the restriction *π*|_{N} is a homeomorphism onto a neighbourhood of *y*. These ideas can be summarized as follows:

## Definition 2.1.

A *sheaf* of groups on *X* is a pair such that

is a topological space (in general, not Hausdorff),

*π*is a local homeomorphism of onto*X*,each ,

*x*∈*X*, is a group called the*stalk*of at*x*, andthe group operations are continuous with respect to the relative topology on the subset of .

For , a continuous map such that *π**σ*(*x*)=*x* is called a *section* of over *U*. The totality of such *σ* will be denoted . Note that every element of the group *F*(*U*) specified by the presheaf functor naturally determines an element of , but the converse is only true locally.

Some of the key properties of a sheaf are as follows. If {*U*_{α}}_{α∈Λ} is an open covering of an open set , and if *σ*,*σ*′∈*F*(*U*) are such that *σ*|_{Uα}=*σ*′|_{Uα} for all *α*, then *σ*=*σ*′. Furthermore, if *σ*_{α}∈*F*(*U*_{α}) are such that for all *α*,*β*, then there exists an element *σ*∈*F*(*U*) with *σ*|_{Uα}=*σ*_{α} for all *α*. The first property implies that if the restrictions of a pair of sections always agree, then the two sections are identical—thus a section over *U* is determined by the totality of its restrictions to subsets of *U*. The second property, somewhat complementary to the first, implies that if pairs of sections always agree on their overlapping regions, then a global section can be constructed from the local data—thus a section over *U* may be assembled from consistent local sections on subsets of *U*.

A sheaf contains localized information concerning the topological space *X*. Global information about *X* can then be extracted from by consideration of exact sequences, quotients, and so on. Given two sheaves and over *X*, a sheaf homomorphism is a continuous map such that the stalk map is a homomorphism of into for each *x*∈*X*. A sequence of sheaf homomorphisms of the form
such that the corresponding sequence of stalk maps is exact for all *x*∈*X* is called a short exact sequence of sheaves. Evidently, exactness is a local property. Given a short exact sequence of sheaves, the induced sequence
is exact at and , but in general not at . That is to say, the local exactness of a sequence does not imply exactness with respect to the global sections over *X*. The measure of inexactness at can then be characterized by cohomology.

Recall that in the cohomology theories of *X* one computes *H*^{i}(*X*,*G*), where *G* is an abelian group. In sheaf cohomology the coefficients are not elements of a fixed group *G* but are, rather, local sections of some sheaf over *X*. More precisely, let be an open covering of *X*. For any *U*=(*U*_{1},…,*U*_{q+1}) such that , we define the set of *q*-cochains by . For any , define the coboundary operator *δ* by
2.1
where *U*_{i}=(*U*_{1},…,*U*_{i−1},*U*_{i+1},…,*U*_{q+1}) and is the sheaf restriction map. These coboundary operators define a complex
and the cohomology groups of this complex are then defined in the usual manner:
2.2
Passing to a direct limit over progressively finer coverings, we obtain the sheaf cohomology groups .

## 3. Basic machinery

The reader will notice that the definitions in this section run parallel, *mutatis mutandis*, to the conventional definitions in the theory of fibre bundles. The -sheaf plays the role of a trivial and the -sheaf bundle that of a generally nontrivial fibre bundle.

## Definition 3.1.

Let *X* and *F* be topological spaces and a sheaf of groups over *X*. A *-sheaf* over *X* is a triple such that

*π*is a local homeomorphism of*F*onto*X*,for each

*x*∈*X*, the stalk operates (by left action) upon , andif is equipped with the relative topology in

*F*×*G*, then the mapping*k*:*F*Δ*G*→*F*defined by*k*(*f*,*g*)=*g**f*is continuous.

For any subset , the obvious restriction maps define a -sheaf over *A*. In the sequel, when *A* is clearly understood, we shall, for brevity, use the term -sheaf in place of -sheaf. We call a *principal -sheaf* if and operates by left translation.

## Definition 3.2.

Given two -sheaves and over *X*, a continuous map *ϕ*:*F*→*F*′ will be called a *-sheaf map* of into provided

*π*′*ϕ*=*π*andfor each

*x*∈*X*, the induced map satisfies 3.1 for all and .

Again, if , then a -sheaf map will, for brevity, be called a -sheaf map when the restriction is clearly understood. A *-sheaf isomorphism* is a bijective -sheaf map.

## Definition 3.3.

A *-sheaf bundle* over a topological space *X* is defined by the following data:

an open covering of

*X*,for each

*α*∈Λ, a -sheaf over*U*_{α}, andfor each nonempty intersection , a -sheaf isomorphism 3.2 such that the cocycle condition 3.3 is satisfied on .

We call a *principal -sheaf bundle* if each is a principal -sheaf. Two -sheaf bundles and over *X*, defined in terms of the respective open coverings and , are *equivalent* provided that the -sheaf bundles and induced by the respective restriction maps on some common refinement of and are such that there exist -sheaf isomorphisms for each *α*∈Λ satisfying
3.4
for all *α*, *β*∈Λ.

## Remark 3.4.

Of academic interest is the fact that, given a -sheaf bundle , one can, from the presheaf defined by local sections of , define a -sheaf , which in turn uniquely determines the structure of up to equivalence (Kashiwara & Schapira 2006). However, this fact will not be required in the sequel.

## Definition 3.5.

Let *X* be a smooth manifold, *G* a Lie group and *H* a closed central subgroup of *G*. For each open subset *U* of *X*, let *Γ*(*U*,*G*) denote the totality of smooth maps *U*→*G*, which forms a group under pointwise multiplication, i.e. (*σ**σ*′)(*x*):=*σ*(*x*)*σ*′(*x*). Let *Γ*_{c}(*U*,*H*) denote the totality of constant maps *U*→*H*, which forms a central subgroup of *Γ*(*U*,*G*). Assign to each open the quotient group
3.5
With the obvious restriction maps, this defines a presheaf; denote by the corresponding sheaf of groups over *X*, regarded as a principal -sheaf, and its restriction to a subset *A* of *X* by . A principal -sheaf bundle over *X* such that for some open covering {*U*_{α}}_{α∈Λ} of *X* will, for brevity, be called a * G_{H}-bundle*.

## Example 3.6.

With the notation of definition 3.3, let (*P*,*π*,*X*) be a principal *G*-bundle, and {*U*_{α}}_{α∈Λ} an open covering of *X* by trivializing neighbourhoods. On each nonempty intersection , the bundle transition function defines an element of and hence a continuous section of the sheaf . The germs of , acting by right multiplication, provide a -sheaf map , which is obviously bijective, i.e. is a -sheaf isomorphism. Moreover, the cocycle condition (3.3) follows immediately from the corresponding cocycle condition for the bundle transition functions. Thus, we obtain a *G*_{H}-bundle over *X*.

## 4. Classification of *G*_{H}-bundles

The following result shows that under a certain simple hypothesis the conventional principal *G*-bundles may be identified with a subset of the *G*_{H}-bundles over *X*. Throughout the sequel, we shall assume that *X* is paracompact, i.e. that every open covering of *X* has a locally finite subcovering.

## Proposition 4.1.

*For fixed H, the correspondence* *induces an injective map of equivalence classes of principal G-bundles into equivalence classes of* *-sheaf bundles over X, provided that the Čech cohomology group* .

We first prove three lemmata.

## Lemma 4.2.

*Let G be an arbitrary group. A mapping ϕ:G→G that commutes with all left translations is a right translation.*

## Proof.

Given that *ϕ*(*g*_{1}*g*_{2})=*g*_{1}*ϕ*(*g*_{2}) for all (*g*_{1},*g*_{2})∈*G*×*G*, let *g*_{2}=*e*. Then *ϕ*(*g*_{1})= *g*_{1}*ϕ*(*e*) for all *g*_{1}∈*G*. ▪

Recall that denotes the totality of continuous sections of a sheaf over a subset .

## Lemma 4.3.

*Let U be an open subset of the topological space X, G a group*, *a principal* *-sheaf over X and* *a* *-sheaf isomorphism. Then, for each point x*∈*U, there exists a neighbourhood V*_{x} *of x in U and a* *such that Tσ*(*y*)=*σ*(*y*)*g*(*y*) *for every section* *and all y*∈*V*_{x}.

## Proof.

By lemma 4.2 and condition (ii) of definition 3.2, for each *x*∈*U* there exists a such that *T*_{x}*f*_{x}=*f*_{x}*g*_{0}(*x*) for all . Now, fix *x*_{ 0}∈*U*, and choose any and any section with *V*_{x 0} open in *U*, *x*_{ 0}∈*V*_{x 0} and *σ*(*x*_{ 0})= *f*_{x 0}. As *T* is continuous and stalk-preserving, *T**σ*(*y*)=*σ*(*y*)*g*_{ 0}(*y*) (*y*∈*V*_{x 0}) defines a continuous section of , and by the continuity of the group operations *g*_{ 0}(*y*)=*σ*(*y*)^{−1}*T**σ*(*y*) (*y*∈*V*_{x 0}) is also a continuous section. For any other section , writing *σ*′(*y*)=*f*′_{y}, we have *T**σ*′(*y*)=*T*_{y}*f*′_{y}= *f*′_{y}*g*_{ 0}(*y*)=*σ*′(*y*)*g*_{ 0}(*y*). Hence, has the desired property. ▪

## Lemma 4.4.

*Let* *and* *be sheaves*, *an epimorphic sheaf map and* . *Then, for any x*∈*X, there exists a neighbourhood V*_{x} *of x and a* *such that ϕσ*=*σ*′|_{Vx}.

## Proof.

This is an elementary fact of the sheaf theory (Kultze 1970, Hilfsatz 3.4). ▪

## Proof of proposition 4.1.

Clearly, equivalent principal bundles give rise to equivalent sheaf bundles. Conversely, let (*P*,*π*,*X*) and (*P*′,*π*′,*X*) be principal *G*-bundles defined in terms of trivializing open coverings and , respectively. Choosing, if necessary, a common refinement, we may suppose that . Then, with the foregoing notation, iff there exist -sheaf isomorphisms , *α*∈Λ, such that
4.1
on (*α*,*β*∈Λ). The -sheaves and are identical, both arising from the presheaf functor , *V* open in *U*_{α}. Let denote the sheaf over *U*_{α} defined by the presheaf functor *V* →*Γ*(*V*,*G*). Then, the presheaf epimorphism induces an epimorphic sheaf map ; hence, by lemmata 4.3 and 4.4, for each *x*∈*U*_{α} there exist a neighbourhood and a section such that for all *σ*_{α}∈*Γ*(*V*_{x},*G*), where the square brackets denote cosets in , regarded as continuous sections in . Then, choosing the covering sufficiently fine, we may, by paracompactness, omit the superscript *x* and simply write *T*_{α}[*σ*_{α}]=[*σ*_{α}*g*_{α}] for all *σ*_{α}∈*Γ*(*U*_{α},*G*), with square brackets denoting cosets in , again regarded as elements of . Hence, recalling the definition of the *T*_{αβ} in example 3.6, relation (4.1) implies that
4.2
on . Because *σ*_{α} is arbitrary, this clearly means that
4.3
for certain constant functions
4.4
Applying the cocycle condition for principal bundles to both members of (4.3), we deduce that the constants *h*_{αβ} also satisfy the cocycle condition, and therefore define an element of . Hence, if the covering {*U*_{α}}_{α∈Λ} is chosen sufficiently fine, then, by hypothesis, there exists a 0-cochain {*h*_{α}}_{α∈Λ} such that . Defining , we find that equation (4.3) becomes
4.5
which is the condition for equivalence of *P* and *P*′. ▪

As used in the foregoing proof, lemmata 4.3 and 4.4 show that for a sufficiently fine covering , the transition sheaf isomorphisms of a *G*_{H}-bundle are of the form
4.6
for fixed and arbitrary . Likewise, the -sheaf isomorphisms *T*_{α} in the definition of -sheaf bundle equivalence are of the form
4.7
for fixed and arbitrary .

A standard theorem states that equivalence classes of principal *G*-bundles over *X* correspond biuniquely to elements of the sheaf cohomology set . In view of equation (4.6), the argument used in proving this theorem (see Lawson & Michelsohn 1989, appendix A) can also be applied in a straightforward manner to yield the following result.

## Corollary 4.5.

*The equivalence classes of G*_{H}*-bundles over X correspond biuniquely to elements of the sheaf cohomology set* . *For H*={*e*}, *this correspondence reduces to the standard theorem cited above*.

If *G* is abelian, then proposition 4.1 follows more directly from standard results as follows. Let denote the sheaf arising from the constant presheaf defined by *H*_{c}(*U*)=*Γ*_{c}(*U*,*H*)∼*H*. The short exact sequence
induces a sequence of sheaf maps
which is easily seen to be exact. This in turn induces a long exact cohomology sequence that includes, in particular, the segment
For a paracompact *G*-manifold *X*, the sheaf cohomology group is known to be isomorphic to the Čech cohomology group as well as the singular cohomology group *H*^{p}(*X*,*H*) with coefficients in *H* (Spanier 1994, Chapter 6). Hence, if , then the exactness of the above sequence implies that ** j*** is injective. Denoting the totality of equivalence classes of principal

*G*-bundles (resp.

*G*

_{H}-bundles) by

*P*

_{G}(

*X*) (resp. ), we note that the diagram where the upper arrow represents the correspondence and the vertical arrows the biunique correspondence of corollary 4.5, is commutative. The result follows.

Relations (4.6) and (4.7) also imply the following corollary.

## Corollary 4.6.

*For the special case H*={*e*} *(the identity of G), the G*_{H}*-bundles can be identified with conventional principal G-bundles, and* *-sheaf equivalence is merely conventional bundle equivalence.*

Thus, for *H*={*e*}, the present theory yields nothing new. However, for *H*≠{*e*} and , we obtain a nontrivial extension of the conventional theory of principal bundles, as we shall see, in particular, from the example in the next section, wherein *H*=*G*.

Note that the set of data (see definition 3.5 and equation (4.6)), subsequently referred to as a *presentation* of the *G*_{H}-bundle under consideration, plays a role analogous to that of a system of local trivializations in the conventional theory of principal bundles.

If is a refinement of , then induces, in the obvious manner, a presentation associated with . Recalling the definition of -sheaf equivalence (cf. (3.4)), relations (4.6) and (4.7) imply that two presentations and associated with the same covering {*U*_{α}}_{α∈Λ} define equivalent *G*_{H}-bundles iff there exist elements such that the respective transition elements and satisfy
4.8
for all *α*,*β*∈Λ and . If the associated open coverings and are different, then the condition for equivalence is given by (4.8) with respect to some common refinement of and .

Another immediate consequence of (4.6) is the fact that *G*_{H}-bundles, like conventional *G*-bundles, are functorial, i.e. a *G*_{H}-bundle over *X* and a continuous map *f*:*Y* →*X* naturally induce a *G*_{H}-bundle over *Y* , as the transition sections on *X* pull back to sections on *Y* which obviously satisfy the cocycle condition. Moreover, one can prove that if *X* and *Y* are compact Hausdorff spaces and the maps *f* and *f*′:*Y* →*X* are homotopic, then and are equivalent (cf. Lawson & Michelsohn 1989, appendix A).

## 5. Dirac sheaf bundles

Let *X*=*S*^{2} and *G*=*H*=*U*(1). We represent *S*^{2} by the unit sphere in , with spherical polar coordinates (*θ*,*ϕ*), (0≤*θ*≤*π*, 0≤*ϕ*<2*π*), and *U*(1) by *S*^{1}={e^{iγ}|0≤*γ*<2*π*}. For the open covering
5.1
of *S*^{2}, we have . Let be the sheaf of smooth *U*(1)-valued functions over *S*^{2}. Let denote the presheaf over *S*^{2} defined for each open *V* by
5.2
the associated -sheaf, and (*α*=1,2). If ν is an arbitrary real number, define a -sheaf isomorphism
5.3
by
5.4
for , where is the section of defined locally by
5.5
Here, the square brackets indicate the equivalence class modulo constant sections *c*_{h}(*θ*,*ϕ*)=e^{ih}. Denote the *G*_{H}-bundle so defined by . If ν=*n* is an integer, then is just the principal -sheaf bundle arising from the conventional *U*(1)-bundle *P*_{n} of charge *n*/2 (in the appropriate units), in accordance with the construction of example 3.6. In general, for an arbitrary abelian group *H*, the Čech cohomology groups are given by
5.6
(e.g. Spanier 1994). In particular, we have . Hence, by proposition 4.1, the sheaf bundles are mutually inequivalent. That ν≠ν′ implies for arbitrary real ν and ν′ will be proved in §6.

From the foregoing construction, the truth of the following proposition should be evident.

## Proposition 5.1.

*Let g*_{αβ} *be the transition functions for some trivializing atlas of a principal U(1)-bundle P over the base space X. Then, for any real number ν, the powers (g*_{αβ})^{ν} *define the transition isomorphisms of a U*(1)_{U(1)}*-bundle* *over X. If the Čech cohomology group* , *then the principal U*(1)*-bundles P*_{n} *defined by the transition functions (g*_{αβ})^{n}, *if mutually inequivalent, correspond biuniquely with mutually inequivalent U*(1)_{U(1)}*-bundles* .

The final assertion is merely an application of proposition 4.1.

## 6. Connections on *G*_{H}-bundles

The Lie algebra of *G* will be denoted by and the algebra of smooth -valued differential forms on a manifold *M* by .

## Definition 6.1.

A *connection* on a *G*_{H}-bundle given in terms of a presentation is specified by a family of -valued 1-forms satisfying1
6.1
for all *α*,*β*∈Λ and .

Both terms on the right-hand side of (6.1) are unambiguously defined as follows. The value of at is an element of , represented by for some neighbourhood *V*_{x} of *x*, and *f*_{αβ} is in turn represented by a section *g*_{αβ}∈*Γ*(*V*_{x},*G*). The first term on the right-hand side of (6.1) is then defined by *g*_{αβ}(*x*)*A*_{β}(*x*)*g*_{βα}(*x*), and the second by *g*_{αβ}(*x*) d*g*_{βα}(*x*). Any two possible choices *g*_{αβ} differ only by a constant factor in *H*, which affects neither term. Two such connections and defined in terms of the respective presentations and are equivalent, provided that, for some common refinement {*U*_{α}}_{α∈Λ} of the two respectively associated open coverings and with (cf. (4.8))
6.2
the relation
6.3
holds for all *α*∈Λ and *x*∈*U*_{α}. Here, the right-hand side of (6.3) is interpreted in the same manner as that of (6.1). By choosing a sufficiently fine covering, the above-mentioned neighbourhoods *V*_{x} may, by paracompactness, be identified with the trivializing neighbourhoods *U*_{α}, so that the representative sections *g*_{αβ} and *g*_{α} may be regarded as elements of and *Γ*(*U*_{α},*G*), respectively. The *g*_{αβ} then satisfy the cocycle condition modulo locally constant sections in *H*, which suffices to ensure the mutual consistency of the relations (6.1) as the indices *α* and *β* vary over Λ.

The *curvature* of such a connection is defined in the conventional manner, i.e. for each *U*_{α}, we have
6.4
By the same calculation as that used for conventional principal bundles, one finds, using (6.1), that
6.5
for . Furthermore, if *A*′_{α} and *A*_{α} are related by (6.3), then
6.6
If (*P*,*π*,*X*) is a principal *G*-bundle and *H* a central closed subgroup of *G*, then an ordinary connection *A* on *P* clearly gives rise to a connection on in the sense of definition 6.1, and the conventional local curvature forms then coincide with those of as defined in equation (6.4). By virtue of the relations (6.5) and (6.6), characteristic classes of *G*_{H}-bundles can be defined and shown to be independent of the choice of the presentation and the choice of the connection , as for conventional principal bundles.

We now define a connection on the Dirac sheaf bundle (). Using the notation of §5, let denote the presentation , and let
6.7
be the corresponding local 1-forms. On we have
6.8
in accordance with (6.1). The curvature is given by
6.9
and hence the first Chen2 class is −ν d*S*, where d*S* is the area form on the unit 2-sphere. This shows that the sheaf bundles for distinct ν are inequivalent, as claimed above.

For simplicity of exposition and to emphasize the analogy with the conventional Hopf fibration *S*^{1}→*S*^{3}→*S*^{2}, the Dirac sheaf bundle and the corresponding monopole connection have been constructed over *X*=*S*^{2} as base space. However, with a view to physical applications, the same construction applies verbatim if the base space is the subset of Minkowski space defined by . Again, expressing (*x*^{1},*x*^{2},*x*^{3}) by spherical polar coordinates, formulae (5.1), (5.2) and (5.6) remain unchanged. Thus, for arbitrary real ν, one obtains a *U*(1)_{U(1)}-bundle over with field strength (6.9).

## Remark 6.2.

According to the conventional bundle picture, there are no singularity-free gauge potentials {*A*_{α}}_{α=1,2} with the properties that (i) their curls are equal to the field, and that (ii) they are related by gauge transformations on the overlapping region , unless electromagnetic charges are quantized (Wu & Yang 1975, theorem 3). The above result demonstrates that this conclusion is not valid in the present generalized model.

## 7. Particle fields

Let *G* be a Lie group, *H* a closed subgroup of *G* and a representation of *G* on a (real or complex) vector space . Let *X* be a smooth paracompact manifold, and for each open , let denote the totality of smooth maps . If and *h*∈*H*, define *h*τ by (*h*τ)(*x*)=*h*(τ(*x*)). Denote by the quotient space of under this action of *H*. If , then the restriction of to obviously commutes with the action of *H*, and thus one obtains a restriction map . If , then , hence the system defines a presheaf , and a corresponding sheaf 3

Next, for *x*∈*X*, define a left action of the stalk of the sheaf (see definition 3.5) upon as follows. For some neighbourhood *U* of *x*, the germ is represented by an element and in turn by a section *g*∈*Γ*(*U*,*G*), while is represented by an element of and in turn by a section . Define as the element of represented by the section (*g**v*)(*x*)=*ρ*(*g*(*x*))*v*(*x*). One can readily check that this is independent of the choices of *g* and *v*.

## Definition 7.1.

Let be a *G*_{H}-bundle defined by a presentation . A *particle field* is a system of sections satisfying the condition
7.1
for every *α*,*β*∈Λ and . The totality of such particle fields will be denoted by .

Inspection of the foregoing construction clearly shows that for *H*={*e*} this definition effectively reduces to that of a conventional particle field (cf. corollary 4.6), i.e. a section of the associated bundle defined by a principal *G*-bundle and a representation . Also, note that if *H*≠{*e*}, then is *not* a vector space. Nevertheless, one can define covariant derivatives on , as is shown by the ensuing examples 7.2 and 7.4.

Let {*A*_{α}}_{α∈Λ} be the local potentials of a connection with respect to some presentation , as in definition 6.1. Consider a point *x*∈*U*_{α}, and choose local coordinates {*x*^{μ}} in a neighbourhood of *x* so that *A*_{α}=*A*_{αμ} d*x*^{μ} with . Let and choose a section *v*_{α}=*v*_{α}(*y*) representing in some neighbourhood *W*_{x} of *x*, with . Then define
7.2
where *A*_{αμ} acts upon *v*_{α} in accordance with the Lie algebra representation induced by *ρ*. This germ is clearly independent of the choice of *v*, and thus one obtains a well-defined section of . Moreover, for ,
7.3
that is, if and are related by equation (7.1), then so are their covariant derivatives. To establish (7.3) note that both sides of the equation are well-defined local sections of . Hence, we may choose arbitrary representatives *v*_{α}, *v*_{β} and *g*_{αβ} of , and , respectively, to perform the calculation. We choose representatives satisfying the relation *v*_{α}(*x*)=*ρ*(*g*_{αβ}(*x*))*v*_{β}(*x*), and by a routine computation, as for conventional minimal coupling, we verify that
7.4
which implies (7.3). Choosing a sufficiently fine open covering, we may, by paracompactness, identify the aforesaid open sets *W*_{x} with the *U*_{α}, and hence assume that . Accordingly, we let denote the element of determined by (7.3). For each *α*, one can define a 1-form
7.5
on *U*_{α} assuming values in the sheaf , and (7.3) shows that these combine to provide a global -valued form on *X*.

The quantization of the gauge and particle fields described above will be investigated elsewhere. Accordingly, we shall not attempt to define Lagrangians in the present paper. Rather, we shall merely postulate that the equations of motion derived in the conventional gauge field theory are also valid in the present context.

## Example 7.2 (Generalized interaction of Dirac monopole with charged scalar fieldGeneralized interaction of Dirac monopole with charged scalar field).

Consider the fundamental representation of *U*(1), that is, *ρ*(e^{iθ})*z*=e^{iθ}*z* for . We consider the Dirac sheaf bundle and connection4*A*_{ν} defined over the subset of Minkowski space, as described in §6. Let be a particle field associated with by the representation *ρ*, as in definition 7.1. The classical equation of motion for a spin-0 charged particle interacting with an electromagnetic potential *A* on Minkowski space is
7.6
where the charge factor i*e* has been absorbed into *A*. If *ϕ* is replaced by , then, by virtue of relation (7.3), both sides of the equation
7.7
are well-defined elements of .

## Remark 7.3.

A solution of equation (7.7) might be physically described as a ‘wave function’ determined only up to a locally constant phase factor. Current dogma holds that a globally constant phase factor is undetectable, but the detectability of a locally constant phase factor using a physical measuring apparatus also seems problematical. Hence, the model described in the present example appears to be physically plausible.

The use of induced bundles to describe conventional interactions between gauge fields and particle fields initially defined in terms of different principal bundles (Bleecker 1981) can be extended to *G*_{H}-bundles in a straightforward manner. If and are, respectively, *G*_{H}- and *G*′_{H′}-bundles defined over the same base space *X*, then, passing to a common refinement if necessary, we may assume that and are given in terms of presentations and over the same system of trivializing neighbourhoods {*U*_{α}}_{α∈Λ} with transition isomorphisms determined by and , respectively. Then, the pairs define a (*G*×*G*′)_{H×H′}-bundle over *X*, with a presentation . Furthermore, if and are the local 1-forms of connections and relative to the presentations and , respectively, then the local 1-forms determine a connection on relative to the presentation . Now, let and be linear representations on a vector space such that *ρ*(*g*)*ρ*′(*g*′)=*ρ*′(*g*′)*ρ*(*g*) for all (*g*,*g*′)∈*G*×*G*′. Then (*g*,*g*′)↦*ρ*(*g*)*ρ*′(*g*′) defines a linear representation . The elements of represent particle fields that interact with potentials defined on as well as those defined on .

## Example 7.4 (Generalized interaction of Dirac monopole and spinor fieldGeneralized interaction of Dirac monopole and spinor field).

Let *ρ* denote the representation of *U*(1) on defined by scalar multiplication *ρ*(e^{iθ})*ψ*=*e*^{iθ}*ψ*, and let , where *D*^{1/2,0}(*g*)=*g* and *D*^{0,1/2}(*g*)=(*g*^{†})^{−1}. Then *ρ* and *ρ*′ obviously commute in the foregoing sense, hence (*ρ*×*ρ*′)(e^{iθ},*g*)*ψ*=e^{iθ}*ρ*′(*g*)*ψ* defines a representation of on . Let be the Dirac *U*(1)_{U(1)}-bundle over , with connection forms *A*_{να} (*α*=1,2) relative to the presentation , as in (6.7), extended from *S*^{2} to in the obvious manner, as described in §6. Let be the trivial -bundle over , or essentially , with trivial connection forms *A*′_{α}=0 corresponding to the trivial presentation . Thus, in this case, the local 1-forms *A*_{α}⊕*A*′_{α}=*A*_{να}⊕0, so the action of *A*_{α}⊕*A*′_{α} upon the local section is simply pointwise multiplication by the imaginary number *A*_{να}(*x*). The conventional Dirac equation in the presence of an electromagnetic potential, transcribed in the present context, becomes
7.8
Again, by virtue of (7.3), both sides of (7.8) represent well-defined elements of .

## 8. Sheaf bundles for electromagnetic instantons

As indicated by the remarks in §1, the mechanism of charge quantization *à la* Dirac is not an exclusive feature of the classical Dirac monopole alone. The essential ingredient is a *nontrivial**U*(1) bundle over some space-time.5 Therefore, consideration of various other examples could be theoretically instructive as well as suggesting possible examples of applications to physical phenomena.

The foregoing Ansatz for the construction of the Dirac sheaf bundle can likewise be applied to cases where the underlying manifold *X* is not a 2-sphere. Recall that given a principal bundle *P* with abelian structure group *G* and transition functions *g*_{αβ} for some trivializing atlas , the powers for integral *n* are the transition functions of a principal *G*-bundle *P*_{n} with respect to the same atlas. Moreover, if {*A*_{α}} are the local 1-forms of a connection *A* on *P*, then one can easily check that *A*_{nα}:=*n**A*_{α} provide the local 1-forms of a connection *A*_{n} on *P*_{n}. Furthermore, if *F*=d*A*, then *F*_{n}=d*A*_{n}=*n**F* for the curvature (field strength) of *A*_{n}. This means that if *I*(*F*^{j}) is a characteristic class of *P*, then the corresponding class of *P*_{n} is just . In particular, this applies to the Chen classes.

Now, in accordance with the discussion in §6, we can proceed similarly in the context of *U*(1)_{U(1)}-bundles, replacing the integer *n* by the arbitrary real number ν, and deduce that if *c*_{j} is the *j*th Chen class of the circle bundle *P*, then ν^{j}*c*_{j} is the corresponding Chen class of the *U*(1)_{U(1)}-bundle . To illustrate, consider the canonical connection on a circle bundle over the complex projective plane, which defines a gravitational and electromagnetic instanton. As local minima of the Riemannian Hilbert–Maxwell action, such instantons provide significant contributions to the partition function in the joint path integral quantization of the gravitational and electromagnetic fields, thus playing a role analogous to that of the usual instantons in the pure Yang–Mills theory (cf. Eguchi & Hanson 1979; Gibbons & Hawking 1979).

## Example 8.1 (Gravitational and electromagnetic instantonGravitational and electromagnetic instanton).

Let and consider the canonical *U*(1) bundle over *X*. The complex projective plane with coordinates {*z*_{i}}_{i=1,2,3} satisfying is the quotient space of the 5-sphere *S*^{5} by the circle action *z*→e^{iϕ}*z*. We regard *S*^{5} as a subspace of :
8.1
The Hopf map is defined by . Parametrizing *S*^{5} by
8.2
where *θ*_{1},*θ*_{2}∈[0,*π*], , formula (8.2) becomes
8.3
As trivializing neighbourhoods, let . Writing (ζ^{1},ζ^{2})=(*z*_{1}/*z*_{3},*z*_{2}/*z*_{3}) for the inhomogeneous coordinates on *U*_{3}, a section *σ*_{3} over *U*_{3} is given by
8.4
Similarly, we have
8.5
for sections over *U*_{1} and *U*_{2}, respectively. By virtue of the relation *φ*_{1}+*φ*_{2}+*φ*_{3}=0, the transition function of the *U*(1) bundle on, say, is given by *σ*_{3}=e^{iφ1}*σ*_{1}. Similarly, we have *σ*_{2}=e^{iφ2}*σ*_{3} and *σ*_{1}=e^{iφ3}*σ*_{2}. Thus, the generic transition function is simply *g*_{αβ}=*e*^{iφ}. The canonical connection on the bundle is given by the Hermitian inner product . In terms of the local coordinates on *U*_{3} given by and , the local connection form is . A similar calculation shows that the local connection form on {*U*_{α}}_{α=1,2} reads and . Using the relation *φ*_{1}+*φ*_{2}+*φ*_{3}=0, we see at once that the local connection forms are related by the gauge transformation , where *g*_{12}=e^{iφ3}. Similarly, we have with *g*_{23}=e^{iφ2} and with *g*_{31}=e^{iφ1}. Calculating the field strength *F*=*d*ω on *U*_{3}, say, we obtain
8.6
This expression for *F* agrees with the one obtained by Trautman (1977); an alternative expression is given by Gibbons & Pope (1978) using different local coordinates. The field *F* is self-dual with vanishing energy–momentum tensor, and thus, along with the Fubini–Study metric on , solves the Einstein–Maxwell equation with cosmological constant. At this point, one can merely speculate upon the possible incorporation of the corresponding *U*(1)_{U(1)}-bundles into the above-mentioned path integral formalism.

## Example 8.2.

The field *F* defined on induces solutions to the Maxwell equation on analytic submanifolds of . Trautman (1977) considered an example given by the Veronese embedding (cf. Brody & Hughston 2001) of in . For *n*=2 this is the embedding , which defines a conic in . A short calculation shows that, in terms of the spherical polar coordinates (*θ*,*ϕ*) of , the local connection forms of the bundle on the two hemispheres {*U*_{α}}_{α=1,2} are given by . Comparison with equation (6.7) for ν=1 shows that ω_{α}=2*A*_{α}. Thus, the electromagnetic field induced on *S*^{2} corresponds to a magnetic pole of unit strength, which might appropriately be called the *Trautman monopole*. Another elementary example is the solution to the Maxwell equation arising from the Segré embedding of in ; this defines a quadric in . In terms of the spherical polar coordinates (*θ*_{1},*θ*_{2},*ϕ*_{1},*ϕ*_{2}) of , one finds at once that the local connection forms on are given by in the respective trivializing neighbourhoods {*U*_{α}}_{α=1,2}. Thus, we obtain a pair of disjoint Dirac monopoles. The potential physical significance of the extensions to nonintegral charges ν would, of course, be similar to that of the simple Dirac sheaf bundles described in §5.

## Remark 8.3.

A nontrivial example of a new solution to the Maxwell equation on a torus *T*^{2} can be constructed by pulling the field *F* back to *T*^{2} via the elliptic curve (*z*)↪(1,℘(*z*),℘′(*z*)) in . Here, *z* is the complex coordinate of the torus and ℘(*z*) denotes the Weierstraß ℘-function.

## Remark 8.4.

Because the equivalence classes of principal *U*(1)-bundles over *X* are indexed by (Lawson & Michelsohn 1989, appendix A), any realization of the usual hypothesis of charge quantization through a topological mechanism is excluded in cosmological models over a contractible base, such as the Schwarzschild universe (see also Trautman 1979). This constitutes an apparent contradiction with certain grand unified theories that require the existence of monopoles (’t Hooft 1974). The author is not aware of any Ansatz whereby this apparent contradiction could be resolved.

## 9. Quasi-spin structures and Dirac operators

Consider the short exact sequence of Lie groups
with *H* closed and central in *G*. As described in example 3.6, a principal *G*-bundle (*P*,*π*,*X*) canonically determines a *G*_{H}-bundle over *X*. In the special case where *H* is discrete, a principal *K*-bundle (*Q*,*π*,*X*) canonically determines a *G*_{H}-bundle over *X* as follows. Let {*k*_{αβ}} denote the transition functions of *Q* relative to a trivializing open covering of *X*. As *H* is discrete, the above epimorphism *ϕ*:*G*→*K* is a local homeomorphism; hence, for any , there exists a neighbourhood *V*_{x} of *x*, with , and a smooth mapping such that , and any two such and are related by for some constant *h*∈*H*. Therefore, uniquely defines an element of (cf. (3.5)) and hence a continuous section of the sheaf . For we have
9.1
hence, , and because *H* is discrete, this product must be constant in some neighbourhood of *x*. Thus,
9.2
in *G*_{H}(*W*_{x}), and we replace *V*_{x} by *W*_{x}. Passing to a suitable refinement of , we may, by paracompactness, identify the *W*_{x} with the intersections , discard the superscripts *x* from the and thereby obtain, as in example 3.6, transition sheaf isomorphisms
9.3
where , which satisfy the cocycle condition by virtue of (9.2). We denote the resulting *G*_{H}-bundle by .

In particular, if *P* is a principal *G*-bundle covering *Q* in the sense that the transition functions *g*_{αβ} of *P* with respect to some common trivializing open covering for *P* and *Q* satisfy the relation
9.4
then we may select in the foregoing construction, and thus coincides with the *G*_{H}-bundle described in example 3.6.

Let denote the Lie algebra of *K*, and let *A* be a conventional connection on *Q*. Relative to a given system of local trivializations over the open covering , with transition functions *k*_{αβ}, *A* is determined by a family of -valued 1-forms satisfying
9.5
for all . Relative to another system of local trivializations over , with transition functions , the same connection *A* is represented by the 1-forms
9.6
For a sufficiently fine covering, we can choose sections *g*_{α}∈*Γ*(*U*_{α},*G*) and such that *ϕ*(*g*_{α}(*x*))=*k*_{α}(*x*) and, as above, *ϕ*(*g*_{αβ}(*x*))=*k*_{αβ}(*x*). These choices are unique modulo constant multiplicative factors in *Γ*_{c}(*U*_{α},*H*) and , respectively. Furthermore, the covering map *ϕ* induces a Lie algebra isomorphism . Applying *ϕ*^{−1}_{*} to equations (9.5) and (9.6), we obtain
9.7
and
9.8
Let and denote the sections of and determined by *g*_{αβ} and *g*_{α}, respectively, and write , . Then equations (9.7) and (9.8) imply the relations
9.9
and
9.10
in accordance with the explanatory remarks following equations (6.1) and (6.3). Thus, the family of 1-forms , corresponding to the presentation , where, as usual, , determines a connection on the *G*_{H}-bundle in the sense of definition 6.1.

Of particular physical interest is the case where *G*=*Spin*^{e}(*r*,*s*), *K*=*SO*^{e}(*r*,*s*) (the superscript *e* denoting the component of the identity) and *ϕ* is the canonical homomorphism with kernel . Given a principal *K*-bundle *Q* over *X*, a well-known theorem (Lawson & Michelsohn 1989; Friedrich 2000) states that *Q* is covered by some principal *G*-bundle *P* in the above sense iff the second Stiefel class vanishes, and then the equivalence classes (i.e. bundle equivalence compatible with the covering map onto *Q*) of such coverings *P* (known as *spin structures*) are parametrized by the elements of . Inequivalent spin structures may (or may not) be inequivalent as abstract *G*-bundles over *X*. Nevertheless, this can entail no inconsistency with proposition 4.1, because if the hypothesis of that proposition is satisfied, then, by the above-cited theorem, there exists at most one spin structure *P* over *Q*, and hence at most one , which, if existent, coincides with . However, the *quasi-spin structure* exists in any case, even if does not.

Specifically, let *X* be a smooth Lorentzian manifold and *Q* the orthonormal frame bundle of *X*. The Levi–Civita connection *A*_{LC} on *Q* then determines, as above, a connection on , and the spin representation , where denotes the spin module, gives rise to a space of particle fields (see definition 7.1). Covariant differentiation in is then well defined, just as was indicated, using local coordinates, in examples 7.2 and 7.4.

Relative to a given trivialization over *U*_{α} in a sufficiently fine open covering, a quasi-spinor field is represented by a smooth section , determined up to a constant factor *h*∈*H*, that is, up to a sign ±1. Using the local 1-form (*A*_{LC})_{α} relative to the given trivialization, one obtains a conventional Dirac operator *D*/_{α} over each *U*_{α}, which, applied to *ψ*_{α}, yields a section *D*/_{α}*ψ*_{α}, likewise defined up to a sign. Let denote the element of determined by *ψ*_{α}. Because the Dirac operator behaves covariantly with respect to gauge transformations *g*∈*G*, the relations (7.1), with , imply corresponding relations for the sections *D*/_{α}*ψ*_{α}, which thus combine to yield an element .

According to a well-known theorem of Geroch (1968), the orthonormal frame bundle of a noncompact Lorentzian manifold *X* possesses a spin structure iff *X* is parallelizable. A specific example of a noncompact Lorentzian manifold not admitting a spin structure is described, e.g. in Clarke (1971). The present formalism permits the global description of fermionic fields on such space-times, and the local properties of quasi-spinor fields and quasi-Dirac operators in this context are clearly identical with those of their conventional counterparts. The foregoing considerations are likewise applicable to the construction of quasi-spinor fields and quasi-Dirac operators on oriented Riemannian manifolds, with *G*=*Spin*(*n*), *K*=*SO*(*n*) and . In the conventional theory, the index of the Dirac operator on a ‘spinnable’ manifold *X* is expressible, by the Atiyah–Singer index formula, in terms of the Chen classes of *X* (Lawson & Michelsohn 1989; Berline *et al*. 1992). Even if *X* is not spinnable, this expression is still well defined. On the other hand, as is not a vector space, the index of the quasi-Dirac operator cannot be defined in the conventional manner. Of course, one could, by fiat, simply define the index of the quasi-Dirac operator as the value of the Atiyah–Singer expression. However, a more interesting approach would consist in providing a general geometrical definition of the index for differential operators on , and proving that its value for the quasi-Dirac operator is given by the Atiyah–Singer formula. This appears to pose a not entirely trivial problem.

In the pseudo-Riemannian case, if *X* is not space and time orientable, then one deals with pin structures (see Chamblin & Gibbons (1997) for a discussion of physical applications). Because the homomorphism *Pin*(*r*,*s*)→*O*(*r*,*s*) is also a double covering, the foregoing construction is directly applicable, and one thus obtains *quasi-pin structures*, even if true pin structures fail to exist.

*Spin*^{c}-structures can also be generalized in a similar vein, but because in this case the central subgroup *H*=*U*(1) is not discrete, the situation is somewhat more complicated and will be dealt with elsewhere.

## 10. Discussion

Mathematically sophisticated expositions of gauge theory tend to define connections and curvature *ab initio* on the total space of a principal bundle before pulling these quantities down to the base space via local trivializations. Some readers may, perhaps, inquire why we have not adopted this more elegant approach. The reason is that our sheaf bundles are not Hausdorff spaces, hence not manifolds, so the tangent spaces are undefined. Consequently, we must adopt the orthodox physical practice of working in terms of local sections.

In the relevant literature, one occasionally observes heuristic statements to the effect that topological solitons already display certain quantal effects at the classical level. As charge renormalization and running coupling constants can only result from field quantization in the conventional theory, one could, in a similar vein, assert that the generalized solitons described above already display quantum field theoretic effects at the classical level. The coupling constants can walk even before they begin to run.

We have constructed a family of generalized Dirac monopoles that include the conventional as special cases. The interact with conventional wave functions *ψ*, i.e. sections of the associated vector bundles, and likewise, mathematical consistency in the description of interactions involving the requires the introduction of generalized wave functions Ψ that are sections of certain sheaves. Every conventional *ψ* defines a Ψ, but there also exist Ψ that do not arise from any *ψ*. For example, consider normalized wave functions defined over the circle *X*=*S*^{1}. In the conventional model, these include, e.g. for integral *n*, but not for arbitrary real ν. In our model, however, the (and more generally, , where *f*(*x*) is an ordinary normalized complex-valued function on *S*^{1}) represent well-defined sections of the sheaf , where *G*=*H*=*U*(1). Thus, rather than demanding knowledge of a global function on *X*, determined only up to a globally constant phase factor, we merely demand, for any point *x*∈*X*, the knowledge of a function on some neighbourhood *U* of *x*, and determined only up to a constant phase factor on *U*. The probabilistic interpretation of such a generalized wave function remains the same as in the conventional theory. Moreover, differentiation with respect to *x* and multiplication by ordinary functions *V* (*x*) are well defined, so we can apply Hamiltonian operators such as *H*=−d^{2}/d*x*^{2}+*V* (*x*).

Magnetic monopoles, of either integral charge or otherwise, have not yet been encountered in experimental studies (see Milton 2006 for the current status of the experimental limits). The discovery of either nonintegral magnetic monopoles or nonintegral electric charges would tend to indicate that the above-described model is not merely more general mathematically but also more realistic physically than the conventional principal bundle model. Such a discovery might therefore cast doubt upon the long-standing assumption that the venues for physical phenomena are necessarily manifolds, and suggest that sheaves, which provide a more local description, constitute the appropriate arenas for physical models.

The sheaf-theoretic dequantization of solitons and instantons for nonabelian gauge groups constitutes a challenging topic for research, and might have implications for grand unified theories. The above treatment of the Dirac sheaf bundle can be adapted to cases where the Dirac monopole is embedded within a nonabelian principal bundle, such as the ’t Hooft monopole.

The validity of the model constructed above would have certain obvious physical consequences. Firstly, the detection of a magnetic monopole would not necessarily imply the quantization of electromagnetic charges. Conversely, an observed violation of charge quantization would not necessarily imply the nonexistence of magnetic monopoles.

A definite conclusion concerning the actual physical quantization of electric charges remains elusive. For example, explosions of electron bubbles in liquid helium have been observed in the laboratory, suggesting that fractionally charged particles can possibly exist in isolation (Konstantinov & Maris 2003), although the interpretation of such experiments is controversial (Jackiw *et al*. 2001; Bender *et al*. 2005). Other physical observations apparently consistent with nonintegral charges include the fractional quantum Hall effect (Laughlin 1983) and geometric phase measurements in anisotropic spin systems (Bruno 2004). The possible variability of the fine structure constant over the cosmological timescale (Bekenstein 1982) and the postulated fractional charges of the quarks in the deconfinement phase could also be relevant to this issue. A sheaf-theoretic formulation of the underlying quantum theory along the foregoing lines might conceivably clarify some of these diverse phenomena.

## Acknowledgements

The author hereby expresses his gratitude to C. M. Bender, E. J. Brody, G. W. Gibbons, E. M. Graefe, L. P. Hughston, T. W. B. Kibble, B. Muratori, R. J. Rivers and R. P. W. Thomas for stimulating and informative discussions in connection with the foregoing material.

## Footnotes

↵1 For simplicity of notation, we assume that

*G*is a matrix group.↵2 I use the current officially valid romanization in place of the still prevalent ‘Chern’, which was officially discarded about 60 years ago.

↵3 In this context, all the definitions in §3 are to be interpreted with groups replaced by vector spaces and homomorphisms by linear transformations.

↵4 As explained in §8,

*A*_{ν}=ν*A*, where*A*is the connection form for the conventional Dirac monopole.↵5 For a trivial bundle, one need not satisfy any consistency (gauge invariance) relations between local connection forms, hence, the connection on an associated vector bundle could be any real multiple

*e**A*.- Received April 27, 2009.
- Accepted June 22, 2009.

- © 2009 The Royal Society