**1. Introduction**

Numerous research studies point toward a basic fact: topology and physics are destined to come closer to each other in the following decades [1–4]. This in itself is not totally new because several authors, for example, Henri Poincare, E. Cartan, and Hermann Weyl, had already advocated topological thinking in physics [5–7]. However, a salient feature of this convergence is the focus on material engineering applications, for example, metamaterials and topology-based devices. In this paper, we look into the general and rigorous foundations of the discipline behind these applications, namely the framework of *nonlocal continuum field theories* [8,9], with focus on explicating the generic multiscale topological

**Citation:** Mikki, S. On the Topological Structure of Nonlocal Continuum Field Theories. *Foundations* **2022**, *2*, 20–84. https://doi.org/10.3390/ foundations2010003

Academic Editor: Eugene Oks

Received: 21 October 2021 Accepted: 13 December 2021 Published: 31 December 2021

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structure of continua studied by such theories. We propose that in addition to the now mainstream approach to topological materials [10,11], where the emphasis is often laid on exploiting the global dependence of the wave function on momentum (Fourier) space, there is a need to consider how materials can be assigned an indirect structure indexed by parameters taken directly from the *spatial* side of the configuration space, i.e., either space–time or space–frequency.

Our key observation is that arriving at an adequate understanding and characterization of nonlocality in generic scenarios would naturally require gathering information at the microtopological level of what we dub *nonlocal microdomains* (the topological level of small regions around every point where the response is nonlocal), then collectively aggregating these microdomains in order to obtain the global topological structure (the macro-topological level). The fundamental insight coming from topology is precisely how this process of "moving from the local to the global" can be enacted. We have found that a very efficient method to do this is the natural formulation of the entire problem in terms of a fiber bundle superspace, where conventional spacetime or space–frequency are here understood as nothing but "index spaces" embedded into a larger (in our opinion more fundamental) fibered superspace characteristic of nonlocal continuum field theories. In other words, and in contrast to existing approaches to local field theories and topological materials, our strategy is not to first solve Maxwell's equations in order to find the state function as expressed within the Fourier **k**-space, after which one proceeds to study topology over momentum space; instead, we start in spacetime (or space–frequency), and then formulate the *extended* or superspace structure of a topology over a fiber bundle where the conventional position space of the nonlocal continuum, e.g., Euclidean space, would manifest itself merely as the index space of the fiber bundle superspace.

The principal conceptual and philosophical message behind this work is that spacetime (or space–frequency) is not adequate for formulating nonlocal continuum field theories, and that a more appropriate natural approach is the superspace formalism proposed below, which, in our case, is based on a specific fiber bundle construction taking into account the intricate physics-based microdomain structure of the generic nonlocal continuum. It is the hope of the author that by helping scientists generate new insights into their physics and models, this formalism may provide a rigorous approach complementing some of the exciting theories and researches currently addressing various topics in continuum field theories, nonlocal metamaterials, and topological materials, while possibly stimulating the creation of novel algorithms for the computation of suitable topological invariant characterizing complex material domains. Due to the wide scope and complexity of this work, we first provide in Section 2 a relatively lengthy overview on the our contribution, where high-level information about this work, in addition to a guide to the literature and how to read the present paper, are outlined before moving to the more technical treatments of the subsequent sections and appendices.

### **2. Preliminary Considerations**

While the essential idea of the superspace formalism introduced here will be valid for a generic nonlocal continuum field theory, it is much easier sometimes to work with a concrete example, especially in explaining what nonlocality is for someone who is coming to the subject for the first time. Therefore, in this preliminary section, we emphasize the special but very important case of *electromagnetic* nonlocality.

### *2.1. What Is Nonlocality?*

In classical electromagnetic (EM) theory, it is currently widely held that there are no nonlocal interactions or phenomena in vacuum because Maxwell's equations, which capture the ultimate content of the physics of electromagnetic fields, are essentially local differential equations [12]. In other words, an effect applied at point **r** in space will first be felt at the same location but then spread or propagate slowly into the *infinitesimally* immediate neighborhood. Long-term disturbances, such as electromagnetic waves, propagate through both vacuum and material media by cascading these infinitesimal perturbations in outward directions (rays or propagation paths) emanating from the time-varying point source that originated the whole process. However, if we leave behind vacuum electromagnetism and move into electromagnetically-responsive matter-filled space, then we note that *nonlocal* interactions in material domains differ fundamentally from the de facto local vacuumlike picture in allowing fields applied at position **r** to influence the medium at *different* location **r** [13]. That is, in the nonlocal material system, a location not *infinitesimally* close to the source position **r** can experience a nonvanishing effect emanating from the source location. While the "nonlocality scale" |**r** − **r** | tends to be quite small in most natural media (and certainly zero in vacuum), in some types of materials, the so-called *nonlocal media*, observable response can be found such that this "radius of nonlocality" |**r** − **r** | becomes appreciably different from zero [14–16].

The existence of multiple scales in the fundamental physics of nature is not really new. The scaling properties are important in Yang–Mills fields, the non-abelian field theory, and it has been recently used to propose the presence of fractal structures in the dynamical evolution of the fields. For example, one may consider the fractal structure of Yang–Mills fields [17] as an example of a multiple-scale effect in fundamental field theories.<sup>1</sup> In a more familiar setting, it is generally accepted that Aharonov–Bohm-type effects, which lead to observable nonlocal electrodynamic effects [18], have their origin in quantum physics. Bringing quantum physics into field theory can be shown to lead to intrinsically nonlocal effects since quantum field theory may be considered a fundamentally nonlocal theory due to, for example, entanglement effects [19,20]. However, in this paper, we focus on *classical* field theory realized through *phenomenological* models of the electromagnetic response of the material domain. The phenomenological model itself (the constitutive relations [9]) may have as its ultimate origin a purely quantum effect. For example, the main example considered in this paper, the nonlocal semiconductor material domain, has as its "origin of nonlocality" the essentially *quantum* process of exciton polariton coupling in solids (Section 7). It should be noted that in recent years some authors suggested that *classical* electromagnetism, under certain conditions, *may* induce nonlocal effects [21–23]; nevertheless, such scenarios are *outside* the scope of the physical paradigm treated in the present paper.

On the other hand, and interestingly enough for our purposes, Cvijanovich proposed several decades ago a theoretical model in which *vacuum* itself is modeled as a nonlocal constitutive *non*-material domain, where the standard Lorentzian spacetime manifold of general relativity is assumed here to play the role of the "medium" transmitting nonlocal actions [24]. Such proposal might be linked to field–matter interaction regimes where there is a strong coupling between gravitational fields and electromagnetic degrees of freedom. For flat spacetime, however, we already know from experiments that classical electromagnetism is strictly local. Nevertheless, it was discovered recently that classical electromagnetism can be made nonlocal if the photon mass is nonzero. More precisely, classical massive electromagnetism can be shown to arise in certain nonlocal (spatially dispersive) homogeneous domains [25]. Therefore, the statement that "classical electromagnetism is strictly local" should be qualified by allowing for the possibility that the photon mass might be proved experimentally to be non-vanishing, say in a future empirical research. In spite of all these interesting proposals on how to modify classical electromagnetic theory in order to make it compatible with nonlocality at the very fundamental level, the system of field theory treated in this paper is mainly classical, and the underlying spacetime structure is flat (the gravitational degrees of freedom are ignored).

The research field concerned with the study of the classical electromagnetism of nonlocal material domains is called *nonlocal electromagnetism/electromagnetics/electrodynamics.* This paper introduces a comprehensive general approach to this emerging discipline together with a series of selected applications. An extensive literature survey on past researches into nonlocal electromagnetism is given in Appendixes A.1 and A.2. The subject of nonlocal electromagnetism, here understood as the electromagnetism of nonlocal material

domains, is presently treated as a subdomain of the science of metamaterials. Historically, it has not been a well-defined direction of research, with researchers working on nonlocal structures often coming from very diverse and distinct fields, such as plasma physics, crystal optics, periodic structures, metasurfaces, and so on. One of the objectives of this article is to propose a coherent view of the inherently cross-disciplinary nonlocal materials research program, encompassing contributions coming from theoretical physics, applied physics, chemistry, engineering, with mathematical physics as the unifying framework of our inquiry.

#### *2.2. Key Contributions and Motivations in the Present Work*

Currently, there is an interest within applied physics and engineering in harnessing nonlocal media as a new generation of metamaterials for use in various settings, e.g., optical devices, energy control, antennas, circuit systems, etc., see Appendixes A.1 and A.3. The main goal of the present work is to explore, at a very general level, the conceptual and mathematical foundations of nonlocality in connection with applied electromagnetic metamaterials (MTMs). Our approach is conceptual and theoretical, with the main emphasis being laid on understanding the mathematical foundations of the subject and how they relate to the underlying physical bases of some illustrative examples. Indeed, while a massive amount of numerical and experimental data on all types of nonlocal materials abound in a literature that goes back to as early as the 1950s, the purpose of the present paper is attaining some clear understanding of the essentials of the subject, particularly in connection with the ability to build a very general superspace formalism for nonlocal continuum field theory without restricting the formalism first to particular classes of materials such as metals, plasma, or semiconductors.

The central theoretical idea in this work is the introduction of the superspace concept into the process of constructing a general formalism suitable for understanding, analyzing, and designing nonlocal material systems in classical field theory. The superspace formalism has a long history in physics, mathematical physics, and mathematics (see Appendix A.4). It will be shown below that nonlocal continuum field theory appears to lead very naturally to a reformulation of its essential configuration space by upgrading the conventional space– time or frequency space to a larger superspace in which the former spaces serve as base spaces for the new (larger) superspace. Such reconsideration of the fundamental structure of the problem may help foster future numerical methods and potential applications as will be discussed later, e.g., see Appendix A.11.2.

The key motivation behind the proposed superspace approach is explicating a subtle, but often overlooked, difference between two fundamental scales of interactions in nature:


We believe that this topological difference has not received the attention it deserves in the growing theoretical and methodological literature on nonlocal media. In particular, the author believes that a majority of present approaches to nonlocal metamaterials conflate the topologically local (but EM nonlocal) domain of small neighborhoods and global domains. However, general topology and much of modern mathematical physics is based on clearly distinguishing the last two topological levels. Explicating these subtle conceptual differences emanating from the existence of distinct types of spatial scales in field–matter interactions, while aided by a precise, rigorous, and powerful mathematical language, is one of the principal aims of this work. In fact, we believe that a complete understanding of material nonlocality in nature cannot be attained without relying on a fairy advanced mathematical apparatus such as the theory of smooth fiber bundles and infinite-dimensional manifolds developed below.

Let us give a brief summary of the main conceptual findings of this research. First, we highlight that the main idea of the superspace formalism is not restricted to electromagnetic theory, but applies to all types of nonlocal continuum theories, i.e., field theories in nonlocal continuous media. However, for concreteness, and in order to reduce the complexity of the mathematical formalism, we chose to work with a specific type of field theories, namely the classical field paradigm based on Maxwell's equations. As will be seen below, It turns out that the standard formalism of local field theory, which is based on spacetime points and their *differential* (but not topological) neighborhoods, viewed as the basic configuration space of the problem, is *not* the most natural or convenient framework for formulating field theory in nonlocal materials. This is mainly because the *physics*-based domain of nonlocality (to be defined precisely below), which captures the effective region of field–matter nonlocal interactions, is found to not always be naturally transportable into the mathematical formalism of boundary-value problems characteristic of classical field theory, as practiced in several domains, such as applied electromagnetism, heat transfer, hydrodynamics, etc. By investigating the subject from an alternative but enlarged and intrinsically broader perspective, it will be shown that a natural space for conducting nonlocal metamaterials research is the *vector bundle* structure, more specifically, a Banach bundle [26] where every element in the fiber superspace is a vector field on the entire domain of nonlocality.

The main result of this paper is that every generic nonlocal domain can be topologically described by a superspace comprised of a Banach (infinite-dimensional) vector bundle M. If two materials described by their corresponding vector bundles M1 and M2 are juxtaposed, then one may use topological methods to combine them and to compare their topologies. The present paper's focus is mainly on the first part, i.e., how to construct the material bundle M. That is, the derivation of the various vector bundle structures starting from a generic phenomenological model of electromagnetic nonlocality is the main contribution of the present work. It is hoped by the author that the superspace theory developed below will stimulate new approaches to computational field theories by adopting methods borrowed from or inspired by computational topology and differential topology to help supporting ongoing efforts to solve challenging problems in complex material domains as in nanoscale hydrodynamics, nonlocal optical materials, topological insulators, topological photonic devices, and other areas where nonlocality is currently important or expected to play an increasingly dominant role in the future.

### *2.3. An Outline of the Present Work*

Because of the considerable complexity of the present article, which is unavoidable in treatment of the subject of nonlocality in the continuum field theory at this broad theoretical level, and in order to help make our contribution accessible to a wider audience involving, for instance, physicists, engineers, and mathematicians, we have divided the argumen<sup>t</sup> into different stages with different flavors, as follows. First, Section 3 provides a general mathematical description of nonlocality in the continuum field theory, emphasizing the settings of the electromagnetic case. The key ingredients of nonlocal metamaterials/materials are illustrated in Section 3.1 using an abstract excitation-response model. This is followed in Section 3.2 by a more detailed description of the special but important case of spatial dispersion, which tends to arise naturally in many investigations of nonlocal metamaterials. In Section 4, we begin the elucidation of the main topological ideas behind electromagnetic nonlocality, most importantly, the concept of EM *nonlocality microdomains*, which provides the key link between physics, material engineering, and topology in this paper. The various physical and mathematical structures are spelled out explicitly, followed in Section 5 by a more careful construction of a natural fiber bundle superspace structure that appears to satisfy simultaneously both the physical and mathematical requirements of EM nonlocality (Sections 5.1 and 5.2). We then provide a key computational application of the proposed theory in Section 5.3, where it is shown that the material response function is representable as a special fiber bundle homomorphism over the metamaterial base space. In this way, a more general map than linear operators in local field theory is derived, providing solid

mathematical foundations for possible future computational topological methods where, for example, the bundle homomorphism itself might be discretized instead of the original spacetime-based linear operator. The fiber bundle superspace algorithm is summarized in Section 6, where it is highlighted that the main data needed are the physics-based (e.g., electromagnetic) nonlocality microdomains, which do not arise solely from purely mathematical considerations, but require some empirical input, for example the microscopic theory of materials, which ultimately would involve both electromagnetism and quantum mechanics. In this manner, the entire construction of the nonlocal metamaterial superspace may proceed as per the procedure outlined there. In order to illustrate how the above mentioned microdomain structure can be actually estimated in practice, in Section 7 we present a fairly detailed computational example based on nonlocal semiconductors, where we also explore in depth the physical origin of nonlocality in this particular setting. Insights into the lack of general EM boundary conditions in nonlocal EM are provided in Section 8 based on the superspace formalism.

This paper provides a series of technical appendices designed to provide necessary information to expand the scope of the treatment found in the main text. In Appendix A.1, we back up our major formulation as developed by introducing a general review of electromagnetic nonlocality targeting a wide audience of mathematicians, physicists, engineers, and applied scientists. This review does not restrict itself to specific types of materials, such as plasma, metals, and semiconductors, but aims at integrating the author's own understanding of the vast literature on the subject in a tentative and necessarily provisional, but somehow more coherent view. Because of the extreme importance of the special case of spatial dispersion for understanding nonlocality, we provide some brief historical remarks on this subject in a separate Appendix A.2. Some technical and historical explications of the concept of superspace, as needed and used in the main text, is given in Appendix A.4, which is not meant as a complete rigorous introduction to the concept of superspace in mathematics and theoretical physics, a topic far from being well-defined and focused. Instead, the goal of this appendix is to fix the very specific meaning we have in mind in this paper whenever we speak about superspace structures in order to avoid confusing our concept with other usages found in physics, such as in supersymmetry.

The Appendixes A.6–A.9 supply important technical information needed in order to fully comprehend the specific main example developed in this paper, to illustrate the use of the superspace formalism in actual real-life scenarios (the inhomogeneous nonlocal excitonic semiconductor material system of Section 7). We opted to separate the content of these appendices from the main text in order to simplify the presentation. The subject of nonlocal semiconductor metamaterials is already well-known in the specialized literature, but is also highly technical. In order to help keep the flow of the various ideas treated in the main text tightly focused on the conceptual and mathematical aspects of our proposed superspace theory, we relegated some background material, especially detailed derivations and explanations more related to semiconductor physics than the superspace formalism, to the three appendices mentioned above.

Some basic familiarity with vector bundles and Banach spaces is assumed, but essential definitions and concepts will be reviewed briefly within the main formulation and references where more background on vector bundles can be found will be pointed out. The paper intentionally avoids the strict theorem-proof format to make it accessible to a wider audience. Most of the time we give only proof sketches and leave out straightforward but lengthy computations. In general, just the very basic definitions of smooth manifolds, vector bundles, Banach spaces, etc., are needed to comprehend this theory (also see Appendix A.5 for a guide to the mathematical background.) The only place where the treatment is mildly more technical is in Section 5.3 when the bundle homomorphism is constructed using partition of unity technique as a detailed computational application of the superspace theory.

In Appendixes A.3 and A.11, various additional current and future applications to fundamental methods, applied physics, and engineering are outlined in brief form. Some of the applications mentioned there, for instance numerical methods and topological devices, appear to us to be directly relevant to the scope of a superspace extension of conventional nonlocal electromagnetic field continuum theory, such as the one attempted below within the main text. On the other hand, some of the other applications discussed there, e.g., digital communications and energy, are of a more general nature and belong to our broader tentative global review of the subject of nonlocality in nature and engineering attempted in the Appendix A sections of this paper. Finally we end with the conclusion.

### **3. The Nonlocal Continuum Response Model**

#### *3.1. A Generic Nonlocal Response Model in Inhomogeneous Continua*

In order to introduce the concept of nonlocality in the simplest way possible, let us first start with a scalar field theory setting. As mentioned in the introduction, vacuum classical fields cannot exhibit nonlocality, so in order to attain this phenomenon, one must consider fields in specialized domains. We therefore kick-start the technical mathematical treatment by reviewing the broad theory of such media. The goal is to outline the main ingredients of the spacetime-based *configuration space* on which such theories are often founded in literature. To further simplify the presentation, we work in the regime of *linear response theory*: i.e., all material media considered throughout this paper are assumed to be linear with respect to field excitation.

In detail, if the medium response and excitation fields are captured by the spacetime functions *<sup>R</sup>*(**<sup>r</sup>**, *t*) and *<sup>F</sup>*(**<sup>r</sup>**, *t*), respectively, then the most general response is given by an operator equation of the form [9]

$$R(\mathbf{r}, t) = \mathcal{L}\{F(\mathbf{r}, t)\},\tag{1}$$

where L is a linear operator describing the medium, and is ultimately determined by the laws of physics relevant to the structure under consideration [27–29].

Now, the entire physical process will occur in a spacetime domain. In a nonrelativistic formulation (like the one in the present work), we intentionally separate and distinguish space from time. Therefore, let us consider a process of field–matter interactions where *t* ∈ R, while we spatially restrict to a "small" region spanned by the position

$$
\mathbf{r} \in D \subset \mathbb{R}^3 \tag{2}
$$

where *D* is an open set containing **r**. (Throughout this paper, we assume the normal Euclidean topology on R<sup>3</sup> for all spatial domains.) Since the operator L is linear, one may argue (informally) that its associated *Green's function* or *kernel function*

$$\mathbb{K}(\mathbf{r}, \mathbf{r}'; t, t') \tag{3}$$

must exist. Strictly speaking, this is not correct in general and one needs to prove the existence of the Green's function for every given linear operator on a case by case basis by actually constructing one [30,31].<sup>2</sup> However, we will follow (for now) the common trend in physics and engineering by assuming that linearity alone is enough to justify the construction of Green's function. If this is accepted, then we can immediately infer from the very definition of the Green's function itself that [12,32]

$$\mathcal{R}(\mathbf{r},t) = \int\_{D} \int\_{\mathbb{R}} \mathbf{d}^3 r' \, \mathbf{d}t' \, \mathcal{K}(\mathbf{r}, \mathbf{r}'; t, t') F(\mathbf{r}', t') . \tag{4}$$

The relation (4) represents the most general response function of a (scalar) material medium valid for linear field–matter interaction regimes [37,38]. The kernel (Green) function *<sup>K</sup>*(**<sup>r</sup>**,**r** ; *t*, *t* ) is often called the *medium response function* [9,32,37].

If we further assume that all of the material constituents of the medium are time invariant (the medium is not changing with time), then the relation (4) maybe replaced by

$$R(\mathbf{r},t) = \int\_{D} \int\_{\mathbb{R}} \mathbf{d}^3 r' \, \mathbf{d}t' \, \mathbf{K}(\mathbf{r}, \mathbf{r}'; t - t') F(\mathbf{r}', t'), \tag{5}$$

where the only difference is that the kernel function's temporal dependence is replaced by *t* − *t* instead of two separated arguments. Such superficially small difference has nevertheless considerable consequences. Most importantly, by working with (5) instead of (4), it becomes possible to apply the Fourier transform method to simplify the timedependent formulation of the problem [39]. Indeed, taking the temporal Fourier transform of both sides of (5) leads to

$$R(\mathbf{r},\omega) = \int\_{D} \mathbf{d}^3 r' \mathcal{K}(\mathbf{r}, \mathbf{r}'; \omega) F(\mathbf{r}'; \omega),\tag{6}$$

where the Fourier spectra of the fields are defined by

$$\begin{split} F(\mathbf{r}; \omega) &:= \int\_{\mathbb{R}} \mathbf{d}t F(\mathbf{r}; t) e^{-i\omega t}, \\ R(\mathbf{r}; \omega) &:= \int\_{\mathbb{R}} \mathbf{d}t R(\mathbf{r}; t) e^{-i\omega t}. \end{split} \tag{7}$$

On the other hand, the medium response function's Fourier transform is given by the essentially equivalent formula

$$\mathbb{K}(\mathbf{r}, \mathbf{r}'; \omega) := \int\_{\mathbb{R}} \mathbf{d}(t - t') \mathbb{K}(\mathbf{r}, \mathbf{r}'; t - t') \varepsilon^{-i\omega \left(t - t'\right)}.\tag{8}$$

In this paper, we focus on time invariant material media and, hence, work exclusively with frequency domain expressions, such as (6), (7), and (8), though we often suppress the frequency dependence on *ω* in order to simplify the notation whenever no confusion would arise.

The generalization to the three-dimensional (full-wave) electromagnetic picture is straightforward when the dyadic formalism is employed [28,40]. The relation corresponding to (4) is

$$\mathbf{R}(\mathbf{r},t) = \int\_{D} \mathbf{d}^3 r' \int\_{\mathbb{R}} \mathbf{d}t' \,\overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; t - t') \cdot \mathbf{F}(\mathbf{r}', t'), \tag{9}$$

where we replaced the scalar fields *<sup>F</sup>*(**r**) and *<sup>R</sup>*(**r**) by vector fields **<sup>F</sup>**(**r**), **<sup>R</sup>**(**r**) ∈ R3. The kernel function *K*, however, must be transformed into a *dyadic* function (tensor of second rank) [14,28,41,42]:

$$
\mathbb{K}(\mathbf{r}, \mathbf{r}'; t - t'). \tag{10}
$$

In the (temporal) Fourier domain, (9) becomes

$$\mathbf{R}(\mathbf{r},\omega) = \int\_{D} \mathbf{d}^3 r' \, \mathbf{\overline{K}}(\mathbf{r}, \mathbf{r}'; \omega) \cdot \mathbf{F}(\mathbf{r}'; \omega) , \tag{11}$$

where

$$\overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; \omega) := \int\_{\overline{\mathbb{R}}} \mathbf{d}(t - t') \, \overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; t - t') e^{-i\omega(t - t')} \tag{12}$$

is the frequency domain response kernel, while

$$\begin{aligned} \mathbf{F}(\mathbf{r};\omega) &:= \int\_{\mathbb{R}} \mathbf{d}t \, \mathbf{F}(\mathbf{r};t) e^{-i\omega t}, \\ \mathbf{R}(\mathbf{r};\omega) &:= \int\_{\mathbb{R}} \mathbf{d}t \, \mathbf{R}(\mathbf{r};t) e^{-i\omega t} \end{aligned} \tag{13}$$

are the corresponding frequency domain excitation and response fields, respectively.

The essence of electromagnetic nonlocality can be neatly captured by the mathematical structure of the basic relation (9). It says that the field response **<sup>R</sup>**(**r**) is determined not only by the excitation field **<sup>F</sup>**(**r**) applied at location **r**, but at all points **r** ∈ *D*. Consequently, here we find that the following is true:

*In nonlocal continuum field theories, knowledge of the field response at a specific point r requires knowledge of the cause (excitation field) on an entire topological neighborhood set D r.*

On the other hand, if the medium is local, then the material response function can be written as

$$\overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; \omega) = \overline{\mathbf{K}}\_0(\omega) \delta(\mathbf{r} - \mathbf{r}'),\tag{14}$$

where **K**0 is a spatially constant tensor and *<sup>δ</sup>*(**r** − **r**) is the three-dimensional Dirac delta function. In this case, (11) reduces to [13]

$$\mathbf{R}(\mathbf{r};\omega) = \overline{\mathbf{K}}\_0(\omega) \cdot \mathbf{F}(\mathbf{r};\omega),\tag{15}$$

which is the standard constitutive relation of linear electromagnetic materials. Clearly, (15) says that only the exciting field **<sup>F</sup>**(**r**) data at **r** is needed in order to induce a response at the same location. In a nutshell, locality implies that the natural configuration space of the electromagnetic problem is just the point-like spacetime manifold *D* ⊂ R<sup>3</sup> or the entire Euclidean space R3.

**Remark 1** (**Infinitesimal domains**)**.** One may use the "infinitesimally immediate vicinity" of a given point **r**, where a response is sought, for computing that response itself, ye<sup>t</sup> while still remaining within the *local* regime of continuum field theory. Indeed, for the case of electromagnetic theory, we note that, according to the constitutive relation (15), while only the exciting field at **r** is required for computing the response, Maxwell's equations themselves, on the other hand, *still* must be coupled with the local constitutive relation model of the problem. Now, the fact that Maxwell's equations are *differential* equations implies that the "largest" domain beside the point **r** needed for carrying out the mathematical description of the details of the relevant field–matter interaction physics is just the region *infinitesimally* close to **r**. In other words, in continuum field theories, infinitesimal domains should be treated as neither topological domains nor neighborhoods. The infinitesimal belong to *any* type of continuum field theory built on the differential calculus and, hence, is not a criterion for distinguishing local and nonlocal theoretical structures.

Conventional boundary-value problems in applied electromagnetism are formulated in this manner, i.e., with a three-differential manifold as the main problem space on which spatial fields live [28,29,32,40,41,43–46]. Note that, strictly speaking, the full configuration space in local electromagnetism (also called *normal optics* [16]) is the four-dimensional manifolds *D* × R or R<sup>4</sup> since either time *t* or the (temporal) circular frequency *ω* must be included to engender a full description of electromagnetic fields. However, nonlocal materials are most fundamentally a *spatial* type of materials/metamaterials where it is the spatial structure of the field what carries most of the physics involved [32,47]. For that reason, *throughout this paper, we investigate the required configuration spaces with focus mainly on the spatial degrees of freedom.* This will naturally lead to the discovery of the fiber bundle structure of nonlocality, the main topic of the present work.

### *3.2. Spatial Dispersion in Homogeneous Nonlocal Material Domains*

Spatial dispersion is considered by some researchers as one of the most promising routes toward nonlocal metamaterials, e.g., see [16,47–49]. It is by large the most intensely investigated class of nonlocal media, receiving both theoretical and experimental treatments by various research groups since the early 1960s.<sup>3</sup> The basic idea is to restrict electromagnetism to the special, but important case of media possessing *translational symmetry*, an

important special scenario of material nonlocality that holds when the medium is *homogeneous*. In such situation, the material tensor function satisfies

$$
\overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; \omega) = \overline{\mathbf{K}}(\mathbf{r} - \mathbf{r}'; \omega). \tag{16}
$$

The spatial Fourier transforms are defined by

$$\overline{\mathbf{K}}(\mathbf{k},\omega) := \int\_{\mathbb{R}^3} \mathbf{d}^3(r - r') \, \overline{\mathbf{K}}(\mathbf{r} - \mathbf{r'}; \omega) e^{i\mathbf{k} \cdot (\mathbf{r} - \mathbf{r'})},\tag{17}$$

with

$$\begin{aligned} \mathbf{F}(\mathbf{k},\omega) &:= \int\_{\mathbb{R}^3} \mathbf{d}^3 r \, \mathbf{F}(\mathbf{r};t) e^{i\mathbf{k}\cdot\mathbf{r}}, \\ \mathbf{R}(\mathbf{k},\omega) &:= \int\_{\mathbb{R}^3} \mathbf{d}^3 r \, \mathbf{R}(\mathbf{r};t) e^{i\mathbf{k}\cdot\mathbf{r}}. \end{aligned} \tag{18}$$

After inserting (16) into (11), taking the spatial (three-dimensional) Fourier transform of both sides, the following equation is obtained:

$$\mathbf{R}(\mathbf{k},\omega) = \overline{\mathbf{K}}(\mathbf{k},\omega) \cdot \mathbf{F}(\mathbf{k},\omega). \tag{19}$$

The dependence of **<sup>K</sup>**(**<sup>k</sup>**, *ω*) on the wave vector ("spatial frequency") **k**, here added to the already existing temporal frequency *ω* dependence, is *the* signature of spatial dispersion. As a spectral transfer function of *a homogeneous* medium, **<sup>K</sup>**(**<sup>k</sup>**, *ω*) includes *all* the information needed to compute the nonlocal material domain's response to arbitrary spacetime field excitation functions **<sup>F</sup>**(**<sup>r</sup>**, *t*) (through the application of inverse four-dimensional Fourier transform [16]).

**Remark 2.** In several treatments of the subject within electromagnetic theory, the excitation field is taken as the electric field **<sup>E</sup>**(**<sup>r</sup>**, *t*), while the response function is **<sup>D</sup>**(**<sup>r</sup>**, *t*). In such formulation, the material tensor function **<sup>K</sup>**(**<sup>k</sup>**, *ω*) takes into account *both* electric and magnetic effects [14–16,37–39,50–54]. This is different from the permittivity tensor often invoked in local electromagnetism [28], which is ultimately based on the popular *multipole* model [43] of electromagnetic interactions in material media. A comparison between the two material response formalisms, the one based on **<sup>K</sup>**(**<sup>k</sup>**, *ω*) and the multipole model, is given in [32,47,53].

Complex heterogeneous arrangements of various nonlocal materials can be realized by juxtaposing several subdomains where each subunit is homogeneous, hence can be described by a spatial dispersion profile of the form **<sup>K</sup>**(**<sup>k</sup>**, *ω*) discussed above. The idea is that even materials that are inhomogeneous at a given spatial scale may become homogeneous at a different (less refined) spatial level, leading to a "grid-like" spatially dispersive cellular building blocks at the lower level. In Figure 1, we show a nonlocal metamaterial system with various multiscale structures. A large nonlocal domain, e.g., **<sup>K</sup>**3(**<sup>r</sup>**,**r**) in the figure, acts like a "substrate" holding together several other smaller material constituents, such as **<sup>K</sup>***n*(**<sup>r</sup>**,**r**), *n* = 1, 2, 4. We envision that each nonlocal subdomain may possess its own specially tailored nonlocal response function profile serving one or several applications.<sup>4</sup> By concatenating multiple regions, interfaces between subdomains with different material constitutive relations are created. We here show subdomains D*<sup>n</sup>*, *n* = 1, 2, 3, 4, while some of the possible intermaterial interfaces include D1/D2, D1/D3, D2/D3, D3/D4. More complex geometrical and topological interfaces than those shown in Figure 1 are possible where the topological type of the interface manifold can be controlled by introducing handles, holes, gluing, cutting, and so on.

Topological holes

**Figure 1.** A generic depiction of an electromagnetic nonlocal metamaterial system. Each of the domains D*n* is captured by a general linear nonlocal response function **<sup>K</sup>***n*(**<sup>r</sup>**,**r**).

Recall that in *local* electromagnetism each intermaterial interface should be assigned a special electromagnetic boundary condition in order to ensure the existence of a unique solution to the problem [9,41]. This, however, is not possible in nonlocal electromagnetism. Indeed, and as already mentioned earlier, nonlocal electromagnetism introduces several subtle issues that are absent in the local case: *additional boundary conditions* are often invoked to handle the transition of fields along barriers separating different domains, such as between two nonlocal domains, or even one nonlocal and another local domain [16,55,56]. The topological fiber bundle theory to be developed in Section 5 will provide a clarification of why this is so since it turns out that the traditional spacetime approach often employed in local electromagnetism is not necessarily the most natural one (see also Section 8). There is a need, then, to examine in a more in-depth fashion the detailed structural phenomena associated with the presence of *multiple topological scales* in nonlocal metamaterials. This paper will provide some new insights into these issues.

### *3.3. Preliminary Remarks on the Existence of Multiple Topological Scales in Nonlocal Continuum Field-Theoretic Structures*

For completeness and maximal clarity, we discuss here some of the directly observable topological scales in nonlocal continuum systems whose preliminary understanding at this stage of our presentation does not require the use of the quite elaborate mathematical apparatus to be carefully constructed in the remaining parts of this paper. We list the most important of these topological levels as follows:


The above topological levels are called "directly observable" because their determination does not require the use of abstract and advanced concepts from continuum field theory. This is in contrast to the more subtle distinction that will be discussed next.

In Remark 3, we discuss the very important conceptual distinction between topologybased and physics-based nonlocal domains, a demarcation between two concepts that has already been invoked several times above, and will also figure up repeatedly throughout the remaining parts of this article.

**Remark 3** (**Distinction between physics- and topology-based locality/nonlocality**)**.** The terms *local* and *global* possess two different senses, one physics-based, e.g., electromagnetic theory; the other is spatio-geometric in essence, belonging to the purely formal and mathematical dimensions of the structure of the nonlocal continuum theory of the material system. Elucidating this subtle interconnection between the two senses will be one of the main objectives of the present work but we will first need to introduce the various relevant microscale topological concepts to be given in Section 4 (see also Remark 17) For the time being, let the following be known:

	- (a) *Physics-based non-local level*: this includes how the response of the material continuum depends on locations **r** *not* infinitesimally close to the point **r** where the excitation field is applied. That is, **r** − **r** is nonzero but it is also not a differential. (On infinitesimal domains, see Remark 1.)
	- (b) *Physics-based local level*: this is the physical regime whose essence is captured by local constitutive relations of the form (15).
	- (a) *Topology-based non-local level*: this is the topologically global level, e.g., the entire topological manifold in contrast to the local description applicable only to a coordinate patch [57], and so on. At this level, the non-local-as-global is an emerging structure based on gluing together "smaller pieces" of the total manifold. We will see examples of processes occurring basically at this level when we use partition of unity methods.
	- (b) *Topology-based local level*: this is the topological layer associated with structures, such as open sets, topological neighborhoods, closed sets, and so on. A topological space is defined as a collection of all such local sets [58,59].

The two concepts outlined above interact with each. There is a subtle relation between physics and topology. This paper will address some of these delicate interrelations in subsequent sections.

**Remark 4** (**Electromagnetic Domains**)**.** For simplicity, in what follows we will occasionally use 'electromagnetic (EM) domain' and 'physics-based nonlocal domain' as interchangeable terms. It should be kept–in min–that the concept of physics-based nonlocality is broader than EM nonlocality. The former refers to a characteristic structural trait enjoyed by all nonlocal continuum field theories, while the latter is restricted to the realm of just one such theory, that of the electromagnetism of continuous media.

### **4. The Microscopic Topological Structure of Physics-Based Nonlocal Domains**
