Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras

Abstract

Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and ${\mathbb{Z}}_2$-invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework.

1. Introduction

Understanding the emergence of order in matter when a large number of basic constituents interact with each other has always presented itself as a significant hurdle. This challenge arises particularly in situations where ions or electrons, for example, engage in interactions. In ordered phases that adhere to metric principles, such as crystals and magnets, the order is elucidated through the mechanism of symmetry breaking. For instance, the order in a crystal is achieved by breaking the symmetry of the Euclidean group, which encompasses rotations and translations in three-dimensional space, as ions arrange themselves periodically due to their electrostatic interactions. The prevailing belief was that all states of matter could be classified based on their broken symmetries, leading to distinctive local metrical order parameters until the experimental revelation of the quantum Hall effect. In the case of the quantum Hall state, however, this classification falls short, as it is predicated on a global topological invariant [1,2,3].

The quantum Hall effect is an intriguing phenomenon that occurs when a magnetic field permeates a two-dimensional conducting sample composed of electrons at low temperatures. It is particularly relevant due to its association with the quantization of the sample’s conductivity. It is noteworthy to mention that the quantum Hall state represents the first topological state of matter that has been experimentally observed. Topological states with different values for the global topological invariant can agree for all symmetries. Thus, topologically distinct matter states cannot be adiabatically, that is, preserving the gap among energy levels, deformed continuously to each other as long as they share the same symmetries. Therefore, the global topological invariant classifying these states of matter plays a protective role being totally unaffected by perturbations and deformations. Conclusively, there exists a topological type of order underlying the quantum Hall state of matter, which surpasses the old metrical paradigm.

One of the most significant discoveries in recent years is the occurrence of a similar type of topological order in materials, known as the quantum Hall spin states [4]. In this particular case, the magnetic field’s role is assumed by the mechanism of spin-orbit coupling, which is an intrinsic property of all solids. This remarkable topological state of matter exhibits the ability for materials to insulate internally while facilitating conduction externally. Therefore, these materials, although they insulate on the inside, are able to conduct on the outside. Moreover, the conducting electrons present on the surface demonstrate an organized alignment, with spin-up electrons traveling in one direction and spin-down electrons traveling in the opposite direction. This arrangement generates a unique “spin flow” that exhibits transport properties without any associated dissipation.

In this context, it is instructive to cite the following short excerpt from a recent review article on condensed matter physics [4]: “Condensed matter physics starts at microscopic scales with a clear notion of spacetime distance defined by the Minkowski metric of special relativity. It has been understood since the discovery of the integer quantum Hall effect (IQHE) that some phases of matter, known as topological phases, reflect the emergence of a different type of spacetime. The notion of a metric disappears entirely, and macroscopic physics is controlled by properties that are insensitive to spacetime distance and described by the branch of mathematics known as topology. The emergence of a topological description reflects a type of order in condensed matter physics that is quite different from conventional orders described in terms of symmetry breaking”.

The notion of global topological phase factors appears in the context of parametric dependence of a driving Hamiltonian when its eigenstates are localized over some parameter space, which is usually a topological space. By utilizing the implicit temporal dependence of the Hamiltonian, and thus of its eigenstates, from the underlying parameters, we may consider time-parameterized closed paths on the base space of the induced fibration probing, in this way, global invariant topological aspects, for instance, holes, gaps, or inaccessible regions. By lifting these loops to the sections of the fibration, that is, to the energy eigenfunctions, via a connection, for example, the standard adiabatic Chern–Berry connection, we obtain, by integration over these base loops, global phase factors encoding the invariant topological information of the base localization space. For example, in the usual cases considered in condensed matter and according to the assumptions of the Floquet–Bloch theory referring to the propagation of electrons across a periodic crystal lattice [5], the base localization space is the topological lattice of quasi-momentum variables, considered to be modular parameters. We stress that these global phase factors are called topological, because they depend solely on the topological and geometric aspects of the base parameter space pathway along which the state vector is transported. Therefore, they do not depend on either the metrical temporal duration of the evolution or the particular types of dynamics applicable to the system.

In a nutshell, topological states of matter can be classified only in terms of global topological invariants. According to the above, these global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. Theoretically, they are obtained if we consider the structure of a uniform homotopy fibration, which localizes the electron eigenstates of the driving Hamiltonian (energy eigenstates) with respect to some base space of the control parameters depending implicitly on time. In solid state and condensed matter physics, the localizing base space is considered to be the quasi-momentum lattice. More concretely, in solids, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the momentum in a certain direction, when the edge of the zone is reached we obtain a closed path. If we lift this loop from the base space to the sections of the fibration, that is, to the energy eigenfunctions by means of a connection, we obtain a global topological phase factor which encodes the topological structure of the base space. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. Thus, the understanding of the behavior of topological states of matter is an important problem, whose solution could lead to the discovery of new materials exhibiting new types of global phase factors of electrons depending on variations in the base localization space of the control parameters.

The first fiber bundle theoretic model of Berry’s mechanism leading to the encoding of global invariants in terms of topological phase factors [6,7,8] was formulated by Simon [9] in terms of complex line bundles. Since then, the majority of the mathematical treatments referring to global topological phase phenomena that we are aware of have been formulated exclusively in fiber bundle terms (see, for instance, the collective work [10], as well as the applications in condensed matter physics [4,11,12] and references therein). In this communication, we would like to introduce a paradigm shift from the conventional employment of line bundle models to the utilization of sheaves composed of their sections. Two fundamental rationales that underpin this proposed modification exist: Firstly, the representation of physical information is consistently conveyed functionally within the sections of bundles. Furthermore, the key property lies in the fact that these sections amalgamate harmoniously, exhibiting a sheaf-like structure. This expressive nature is pivotal for elucidating the significant mechanism through which local structural information may be seamlessly integrated, ensuring compatibility with global ones. Secondly, it is noteworthy to highlight that the customary analytic approaches can be effectively formulated and extended to domains beyond differentiable manifolds by leveraging the remarkable tools of sheaf cohomology [13,14]. The latter is considered to be the universal method of figuring out the global invariants bearing physical significance in a way that is invariant under homotopic deformation. Under this state of affairs, the sheaf cohomology groups measure—in terms of an appropriate coefficient sheaf—the global obstructions preventing the extension of locally defined sections to global ones, for instance, the extension of the local solutions of a differential equation to a global solution. The focus of this work is to demonstrate that a sheaf cohomological understanding of the origin of global geometric phase factors in relation to the invariants characterizing the behavior of electron beams on the Brillouin zone is physically significant. For the matter of space, we assume some basic familiarity of the reader with the notion of a sheaf in topology and geometry [13,14,15,16,17,18,19] and some of its broader conceptual implications [20,21,22,23]. The utilization of the methods of sheaf theory and sheaf cohomology to capture the essential aspects of gauge theories has been elaborated in detail by Mallios [24,25], inspired by the approach suggested initially by Selesnick [26]. We employ similar sheaf cohomological concepts targeting the crucial gauge theoretic aspect of electron beams, that is, the local phase invariance of a quantum state vector, in relation to the observation of global gauge-invariant topological phase factors accompanying the periodic evolution over the Brillouin zone, according to the Floquet–Bloch theory. For this purpose, we closely follow the premises and terminology expounded in the review [12].

The proposed fibration model is formulated in terms of a locally free sheaf of module sections (representing the energy states) endowed with a connective structure, which is localized over the base localization space of quasi-momentum parameters identified with the Brillouin zone. The three crucial ingredients of this representation scheme are the local phase invariance of the energy states, the non-trivial topology of the base space of parameters, and the implicit parametrization of the base parameters by a temporal variable giving the means to form closed paths on the Brillouin zone.

2. Integer and Spin Quantum Hall States: Chern Numbers and Time-Reversal Invariance

The integer quantum Hall effect is characterized by a non-trivial $\mathbb{Z}$-valued Chern cohomology class that pertains to the quantization of the Hall conductivity ${\sigma }_{xy}=n{e}^2/h$. In this manner, the latter is expressed by the integral of the first Chern class over the toroidal Brillouin zone ${\mathbb{T}}^2$. The quantum spin Hall effect that was first correlated with spin flow in graphene is characterized by a ${\mathbb{Z}}_2$-invariant tied with the notion of invariance under time-reversal conditions. Both of them can be considered to be different types of topological insulators where, in the first case, the time-reversal symmetry is broken due to the action of the magnetic field, whereas in the second case, it is maintained. In topological band theory, the band structure constituted by the Landau levels is modeled in terms of a complex vector bundle over the Brillouin zone. The electron states are usually described in terms of Bloch functions:

$$\psi \left(\underline{r}\right)=e^{i\underline{k}·\underline{r}}u\left(\underline{r}\right),u(\underline{r}+\underline{s})=u\left(\underline{r}\right),\forall \underline{s}\in {\mathbb{Z}}^2\subset {\mathbb{R}}^2$$

In a quasi-momentum or modular momentum representation, the quasi-periodic eigenstates $u_n\left(\underline{k}\right)$, where $\mathbf{k}$ is the modular parameter, are identified as the solutions to the eigenvalue equation

$$H\left(\underline{k}\right)u_n\left(\underline{k}\right)=E_n\left(\underline{k}\right)u_n\left(\underline{k}\right)$$

where $E_n\left(\underline{k}\right)$ represents the n-th band function. More precisely, $u_n\left(\underline{k}\right)$ are identified as sections of a complex line bundle over the toroidal Brillouin zone, such that the totality of all sections gives rise to a Hilbert space. The top occupied band is called the valence band, whereas the bottom one among the empty ones is called the conduction band, such that the energy gap between these two bands is defined as the band gap. Therefore, the eigenbundle to the nth energy band $E_n$ is identified with a complex line bundle corresponding to the nth occupied band, such that the Whitney sum of all of these eigenbundles is defined as the Bloch bundle. The first Chern number of a complex line bundle of this form characterizes a bulk state in the integer quantum Hall effect. When considering an interface with an insulator, an edge state appears. An edge state is an eigenstate that essentially connects the valence with the conduction band, in the sense that variation of the modular parameter causes the energy of this eigenstate to be evaluated at both the valence and conduction bands. In the simplest scenario, we consider a two-band system separated by a gap, where the systems are characterized by different Chern numbers. Then, the spectral flow of the edge states is correlated with the difference of the first Chern numbers of the corresponding bulk states, establishing the so-called bulk-edge correspondence.

The models referring to the quantum spin Hall effect are formulated in terms of the process of spin-orbit coupling, such that they are time-reversal invariant systems. This means that they are invariant under the time-reversal transformation $\mathcal{T}:t\mapsto -t$, which is represented by an antiunitary operator $\mathrm{\Theta }$, which is realizable via the product of a unitary operator and the complex conjugation operator. In the context of the Brillouin zone, the action of $\mathcal{T}$ changes the sign of the modular parameter $\underline{k}\mapsto -\underline{k}$. The antiunitary operator $\mathrm{\Theta }$ acts by conjugation on $H\left(\underline{k}\right)$, such that $\mathrm{\Theta }H\left(\underline{k}\right){\mathrm{\Theta }}^{-1}=H(-\underline{k})$. In the case of an electron that is a spin-$\frac{1}{2}$ fermion, ${\mathrm{\Theta }}^2=-1$, this gives rise to the phenomenon of Kramers degeneracy, signifying the double degeneracy of all energy levels under time-reversal invariance. The operator $\mathrm{\Theta }$ is also antisymmetric, that is, $\langle \mathrm{\Theta }\psi ,\varphi \rangle =\langle \mathrm{\Theta }\varphi ,\mathrm{\Theta }\mathrm{\Theta }\psi \rangle =-\langle \mathrm{\Theta }\varphi ,\psi \rangle$ for states $\psi$ and $\varphi$, where $\varphi$ and $\mathrm{\Theta }\left(\varphi \right)$ are orthogonal to each other, $\langle \mathrm{\Theta }\varphi ,\varphi \rangle =\langle \mathrm{\Theta }\varphi ,\mathrm{\Theta }\mathrm{\Theta }\varphi \rangle =-\langle \mathrm{\Theta }\varphi ,\varphi \rangle =0$.

An important observation that we would like to point out, and according to our knowledge is new, is that the two levels referring to Kramers degeneracy are folded together at the fixed points of the time-reversal symmetry and can be thought of as the branch points of a ramified double covering. The torus representing the Brillouin zone remains invariant under rotation by $\pi$ with respect to an axis piercing it at four points. This is an involution that changes the sign of the modular parameter, leaving the torus invariant. This $\pi$-involution has four pinched points representing geometrically, in this context, the time-reversal invariance with respect to the modular parameter bearing four fixed points correspondingly. Equivalently, the time-reversal symmetry gives rise to the double branched covering of the sphere $S^2$ by the torus $T^2$ with four branch points. In this ramified covering of the geometric representation of the sphere—identified with the complex projective line—the sphere appears as a pillowcase folded at these four branch points, that is, the fixed points of the time-reversal involution. Equivalently, the pillowcase P is the quotient of the torus by the time-reversal involution. Topologically, it is characterized as a sphere that bears four singular points that are aligned with the four fixed points of this involution. To be precise, we may define P as the quotient of ${\mathbb{R}}^2$ by the group of orientation-preserving isometries, which are generated as follows:

$$(\phi ,\theta )\mapsto (\phi +2\pi ,\theta ),(\phi ,\theta )\mapsto (\phi ,\theta +2\pi ),(\phi ,\theta )\mapsto (-\phi ,-\theta )$$

The above structure may be thought of as a group obtained through the semi-direct product ${\mathbb{Z}}^2⋉{\mathbb{Z}}_2$. The quotient map gives rise to the double branched covering:

$${\mathbb{R}}^2\to P$$

A fundamental domain can be expressed through the rectangle $(\phi ,\theta )\in [0,\pi ]\times [0,2\pi ]$. A point of the pillowcase P can be expressed by the coordinates $(\phi ,\theta )\in {\mathbb{R}}^2$. The points in ${\mathbb{Z}}^2$ are characterized as modular or lattice points, such that the four singular points of the pillowcase P constitute the image of the modular points. The pillowcase P, as well as the complement of the four singular points, bears a well-defined orientation and symplectic structure obtained from $d\phi \wedge d\theta$ on ${\mathbb{R}}^2/{\mathbb{Z}}^2$ through the double branched cover.

If the double branched covering of the sphere by the torus is though of in the context of the three-sphere $S^3$ via the Hopf fibration $S^1\hookrightarrow S^3\rightrightarrows S^2$, then $\mathrm{\Theta }$ may be viewed as the spinorial square root of the involution acting by conjugation as above. Essentially, the four branch points where the two levels of the covering are folded together may be identified by the north, south, west, and east poles of the sphere, as follows: We imagine the rotation of the sphere with respect to the polar north–south axis, and synchronically, the rotation of the sphere with respect to the horizontal west–east axis. The synchronization of these two rotations takes places in a ratio of 2:1, such that a 2$\pi$ rotation around the polar axis is in synchrony with a $\pi$-rotation around the horizontal axis. Thus, the sphere returns to its original position under a $4\pi$ rotation around the polar axis in synchrony with a $2\pi$ rotation around the horizontal axis. The crucial point is that the $\pi$ rotation of the sphere around the horizontal axis is geometrically thought of as an inversion of the sphere that corresponds to the $\pi$-involution. Therefore, the invariance under the $\pi$-involution is in synchrony with the $2\pi$ rotation around the polar axis, which is thought of as the invariance under the flipping of the north with the south pole of the sphere and, conversely, gives rise to the double branched covering $T^2\to S^2$ of the sphere $S^2$ by the torus $T^2$ with the four poles playing the roles of the branch points. In a nutshell, the torus bears the time-reversal symmetry, which is a rotation of order 2, that is, the $\pi$-involution, with four fixed points. Its quotient under the branched cover is a pillowcase, whose underlying space is a sphere with four singular points, where the local group is ${\mathbb{Z}}_2$.

A time-reversal system giving rise to the quantum spin Hall effect may be thought of as comprising two integer quantum Hall systems bearing opposite Chern numbers, such that the total first Chern number is zero. Despite this fact, the ${\mathbb{Z}}_2$-invariance at the four fixed points of the time-reversal operation gives rise to a non-trivial ${\mathbb{Z}}_2$-invariant characterizing the spin flow. If we think of the Bloch bundle of a time-reversal system, it does not split into line bundles according to the energy eigenvalues, but it splits into a direct sum of rank 2 vector bundles. In other words, if the rank of this bundle is $2N$—meaning that there are $2N$ states—due to the double degeneracy, the states form pairs at the singular points, whereas they split into two energy levels everywhere else. In a local region around any of the four fixed points, a local section of the bundle consists of the pair formed by a state and its mirror image under time-reversal. Therefore, the rank 2 vector bundle over the torus is essentially a trivial vector bundle, which is twisted at the four fixed points by means of the involution symmetry. Hence, the transition function of the bundle is antisymmetric at the fixed points of the time-reversal symmetry. The first Chern number of the joint Bloch bundle in this case is zero, which implies that the first Chern number of its determinant line bundle is also zero, since the Brillouin zone torus does not bear any torsion cohomology classes. An immediate consequence is that the determinant line bundle has a global section, and thus, it has a square root, called the Pfaffian line bundle, that is defined precisely over the isotropic Grassmannian, being isomorphic with the complex projective line, that is, $SO\left(4\right)/U\left(2\right)\cong \mathbb{C}{P}^1\cong S^2$. The Pfaffian line bundle involves the selection of either a state or its mirror image as a local section that justifies the square root operation. Its complexity pertains solely to the four fixed points, where the transition function flips over a state to its mirror under the time-reversal symmetry condition. The Pfaffian is a complex line bundle with the structure group $S^1$, which, due to the flipping operation at the fixed points, reduces to a real bundle with the structure group ${\mathbb{Z}}_2$. Henceforth, the characteristic class of the Pfaffian under its reduction by time-reversal symmetry is expressed through its top (second) Stiefel–Whitney cohomology class in $H^2(T^2,{\mathbb{Z}}_2)\cong H^2(S^2,{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2$, which specifies the ${\mathbb{Z}}_2$-invariant in terms of the twofold orientability of the Pfaffian considered as a real line bundle under time-reversal symmetry. In turn, viewing the Pfaffian as a complex line bundle over the complex projective line—according to the square root construction of the determinant line bundle—the ${\mathbb{Z}}_2$-invariant is identified by the $mod2$ first Chern class of the Pfaffian.

3. Sheaf of Fourier Algebras on the Torus as the Argument for the Cohomology of Topological States

Grothendieck’s approach to sheaf theory has emphatically clarified that the natural argument characterizing a cohomology theory is not just a base space, where sheaves are topologically localized, but a pair consisting of a base topological space together with a sheaf of commutative algebras defined over it that gives rise to the pertinent coefficients of the cohomology [27,28]. This has served as the foundation for a generalized approach to differential geometry, conceived as the abstract geometric theory of vector sheaves equipped with a connection. In particular, the approach developed by Mallios, called Abstract Differential Geometry (ADG) [13,14] (see also, [19,29,30]), is based on the appropriate generalization of the analytic methods developed for the case of smooth manifolds by their abstract sheaf cohomological analogues that are applied over general base topological spaces equipped with certain covering systems and complying with fairly simple conditions that allow the formulation of the techniques of homological algebra. This is essentially relevant to all physical systems whose descriptions involve the requirement of local gauge invariance without being subordinate to a smooth manifold base space, in the sense that these systems are categorically amenable to a cohomological process of sheaf-theoretic localization over a general topological space with fairly mild additional qualifying conditions [31]. In this setting, the underlying commutative algebra sheaf of the coefficients, whose spectrum is the base topological space, bears the fundamental role, in the sense that all constructed geometric objects are locally expressed in terms of the sections of this sheaf. In physical terms, we interpret the sheaf of coefficients as a sheaf of observable algebras [21,22,32,33,34]. In this manner, it is worth noting that an algebra sheaf of coefficients can, from its very inception, elude the imperative of being explicitly smooth (which conventionally represents the modeling of an inherent smooth manifold as its spectrum). To exemplify this, it has the potential to metamorphose into a singular sheaf of generalized functions, encompassing distributions in its broad embrace [35,36].

We would like to bring attention to the utmost importance of having an appropriate sheaf of observable algebras, as its fundamental physical significance becomes evident when considering the vector sheaves of states, the group sheaves of automorphic symmetries, and the sheaves of differential forms. It is imperative that these sheaves be finitely generated locally free sheaves of modules over the sheaf of observable algebras. The concept of differentiability of a section of a vector sheaf, known as the covariant derivative, can be expressed in terms of a connection. Locally, a connection on a vector sheaf can be interpreted as a potential 1-form which, in turn, induces the differentiability of a local section. In this particular manner, it can be asserted that a vector sheaf that is appropriately equipped with a connection constitutes the coordinate-free generalization of a differential equation. Furthermore, the integrability properties of this connection, when subjected to prolongation of the differential mechanism, iterating from the local level to the global level, can be adequately described by utilizing the depiction of cohomology combined with an appropriate sheaf of coordinatizing coefficients, instead of relying solely on the conventional analytic methods. Moreover, it should be noted that the curvature of this connection is appropriately measured in a covariant manner by considering the strength of the associated potential relative to the corresponding sheaf of coefficients.

Generally, we may consider soft and fine topological algebra sheaves to play the roles of the sheaves of observable algebras, which are “closed” with respect to complete projective topological tensor products [37,38,39]. In particular, a self-adjoint topological algebra sheaf is suitable for the generalization of the differential geometry pertaining to the case of a non-normed topological algebra sheaf of smooth coefficients. This important generalization of the differential geometric machinery has been inspired by former work of Selesnick on Banach-$\check{S}$ilov algebras bearing the Banach approximation property [26,40]. In this manner, a soft self-adjoint Fréchet-$\check{S}$ilov locally m-convex topological algebra can be assigned as an algebra of observables, such that its Gelfand spectrum is a (Hausdorff) paracompact space [39]. Then, any vector sheaf of rank 1, that is, a line sheaf, which is defined on the Abelian topological spectrum of this algebra of observables can be endowed with a connection, whose curvature gives rise to an integral cohomology Chern class.

More concretely, if G is a compact Abelian group whose group of characters is $\widehat{G}$, its Fourier algebra $\mathcal{A}\left(G\right)$, namely the algebra of the absolutely convergent Fourier series on G with the $l^1$-norm, is a complex unital commutative and self-adjoint Banach-$\check{S}$ilov algebra with the Banach approximation property, whose topological spectrum is (homeomorphic to) G. Thus, in the case where G is a torus, the spectrum of $\mathcal{A}\left(G\right)$ may be canonically identified with the Brillouin zone. Therefore, we model the Brillouin zone as a k-dimensional torus, denoted by

$$G:=T^k\cong {\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^k$$

Each element n of ${\mathbb{Z}}^k$ defines a character on $T^k$ which, for z in $T^k$ and x in ${\mathbb{R}}^k$, takes the form

$${\chi }_n\left(z\right)=exp(2\pi in·x)$$

Thus, we obtain ${\widehat{T}}^k\cong {\mathbb{Z}}^k$. We consider the standard orthonormal basis in ${\mathbb{Z}}^k$ and denote the corresponding characters by ${\chi }_1,{\chi }_2,\dots {\chi }_k$. The important issue is that the differential geometric mechanism referring to electron beams on the Brillouin zone can be efficiently formulated in terms of the observable algebra sheaf $\mathbb{A}\left(T^k\right)$ of the absolutely convergent Fourier series on $T^k$ instead of the sheaf of germs of smooth function ${\mathbb{C}}^{\infty }\left(T^k\right)$. This is a more natural point of view with reference to electron beams when the smoothness assumption is not attainable, like in the lattice formulation of the Floquet–Bloch theory.

Regarding the modeling of global phase factors referring to electron beams on the Brillouin zone, the most appropriate notion is the anholonomy of a connection on a vector sheaf of states defined over the observable algebra sheaf $\mathbb{A}\left(T^k\right)$ of the absolutely convergent Fourier series on $T^k$. Usually, the connection is identified in terms of the standard Chern–Berry connection that is naturally induced by the Hermitian inner product. A Hermitian connection on a vector sheaf of states is employed to express the parallel transport of a state along a path on the base space $T^k$. The transformation induced on a state during its parallel transport along a closed path on the base space—implicitly parameterized by a temporal parameter—gives rise to the anholonomy of the vector sheaf connection. Since our approach is cohomological, the origin of a global phase factor of an electron beam should be identified in terms of an appropriate cohomology class.

In this context, beyond the cohomological aspect of a global phase factor referring to an electron beam on the Brillouin zone, it is important to emphasize the integral symplectic structure of this base localization space that is derived from the identification of the torus $T^2$ with the modular lattice of quasi-momenta ${\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^2$, extending to the integral symplectic isomorphism $T^{2n}\cong {\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^{2n}$ for each positive n. As a point of fact, it is the integrality of the symplectic form on the toroidal Brillouin zone that underlies the pertinent type of quantization observed in the case of the quantum Hall effect. Not only this, but additionally, the symplectic form bears a $\frac{\mathbb{Z}}{2\mathbb{Z}}:={\mathbb{Z}}_2$ type of invariance, which is characteristic of the invariance pertaining to spinorial symplectic variables [41]—apparent and deriving from the primordial case of the symplectic two-dimensional Brillouin zone—that underlies the spin Hall effect. The symplectic structure of the modular lattice in the case of the standard Floquet–Bloch theory from both the perspective of the integrality of the symplectic form it bears and the perspective of its invariance under the antipodal involution in the symplectic variables is not only instrumental for the understanding and modeling of these effects, but is also conceptually indispensable, since the calculations of both a quantum transition probability and a global phase factor bear the symmetry of a symplectic area bounding loop on the base localization space [41,42].

Therefore, from the viewpoint of our analysis, it is equally significant to examine, in detail, first of all, the compatibility of the Floquet–Bloch theory by assigning to the Brillouin zone of quasi-momentum variables the role of the base localization space in the form of the modular lattice $T^{2n}\cong {\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^{2n}$—with the unitary representation theory of the Heisenberg group pertaining to Weyl’s commutation relations. It is characteristic in this manner that the quasi-momentum refers to the modular momentum variable that is adaptable to the periodic structure of the lattice ${\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^{2n}$. In this way, its conjugate is a modular position variable that is adaptable to the periodic structure of the lattice ${\mathbb{Z}}^{2n}$ in ${\mathbb{R}}^{2n}$. Under this state of affairs, it is remarkable that the commutation relations pertain to the structure of the discrete Heisenberg group—to be thought of in conjunction with its continuous counterpart—giving rise to a commutative and modular structure interpreted as a modular symplectic space [42,43], which underlies the assignment of the toroidal Brillouin zone as the base localization space of the fiber bundles of sections modeling the states of quantum electron beams. This provides the best justification for the reason why the commutative Fourier algebra $\mathcal{A}\left(T^{2n}\right)$, whose topological spectrum is homeomorphically identified by the Brillouin zone, where $T^{2n}$ is the torus endowed with its integral and ${\mathbb{Z}}_2$-invariant symplectic structure, is the appropriate algebra for coefficients in terms of which the sections of the sheaves of the quantum states are locally expressed. Therefore, instead of square integrable wave functions, the unitarily equivalent description in terms of theta functions supplies the appropriate sheaf of coefficients to express the topological states of matter, which remarkably emerges naturally from the unitary representation theory of the Heisenberg group on a Hilbert space of states respecting the group-theoretic form of the Weyl commutation relations.

In particular, it is the unitary representation theory of the discrete Heisenberg group, expressed in terms of commutative modular symplectic variables and giving rise to a joint commutative representation space endowed with an integral and ${\mathbb{Z}}_2$-invariant symplectic form that articulates the specific types of topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework. Henceforth, the representation of the states of a quantum electron beam in terms of theta functions [44]—qualified as the sections of a complex line sheaf—incorporate both the integrality condition emanating from the integral symplectic structure of the toroidal Brillouin zone and the pertinent ${\mathbb{Z}}_2$-invariance in terms of cocycle multipliers that are even functions. Thus, the unitary theta representation of the discrete Heisenberg group serves as our conceptual and technical compass for delving deeper into the nature of these topological effects.

4. Central Extensions and the Heisenberg Group: The Emergence of the Modular Commutative and Integral Symplectic Spectrum

In light of the fact that the Fourier–Pontryagin dual of the Abelian group $\mathbb{R}$ is identical to itself, it is feasible to align the one-parameter unitary group, which stems from the continuous group operation of $\mathbb{R}$ with a displacement operator. This association can be established in either the realm of position space or in momentum space. This suggests the consideration of the corresponding one-parameter unitary groups in both conjugate domains synchronically and the derivation of the condition that allows the synchronization of these one-parameter unitary actions over the projective space of rays—identified as a phase space—although the bijectively associated, via Stone’s theorem, self-adjoint operators are not simultaneously observable. It turns out that this attempt is intrinsically tied to the symplectic structure of ${\mathbb{R}}^2:=\mathbb{R}\times \widehat{\mathbb{R}}\cong \mathbb{C}$, which eventually underlies the symmetry of the quantum transition probability with respect to a process evolving around a loop in the symplectic phase space of rays [41,45]. The case involving the symplectic structure on ${\mathbb{R}}^2$ is generalized to ${\mathbb{R}}^{2n}$ in a straightforward manner. For this purpose, let us consider the simplest case of a one-dimensional Hilbert space $\mathcal{H}\cong \mathbb{C}$. For the complex amplitudes $z_1=a_1+b_{1}i$ and $z_2=a_2+b_{2}i$, the Hermitean inner product is defined as follows:

$$\langle z_1|z_2\rangle ={\overline{z}}_1{z}_2=\underset{g\left(z_1,z_2\right)}{\underbrace{\left(a_1{b}_1+a_2{b}_2\right)}}+\underset{\omega \left(z_1,z_2\right)}{\underbrace{\left(a_1{b}_2-a_2{b}_1\right)}}i$$

where $g\left(z_1,z_2\right)$ is the scalar product and $\omega \left(z_1,z_2\right)$ is the symplectic product of $z_1$ and $z_2$. The norm giving rise to the probability assignment is expressed by ${\parallel z\parallel }^2={\left|z\right|}^2$, whence the symplectic conjugation corresponds to the map $z\mapsto z^{★}=iz$, such that $\omega \left(z_1,z_2\right)=g\left(i{z}_1,z_2\right)$. The n-complex dimensional Hilbert space $\mathcal{H}$ may be thought of as a $2n$-real dimensional space. Let the complex coordinates ${\mathrm{q}}_n={\psi }_n$ and ${\mathrm{p}}_n=i{\psi }_n$ denote the positions and momenta, where ${\psi }_n$ are the components of $|\psi \rangle$ in some orthonormal basis of $\mathcal{H}$. Moreover, we consider the one-form $A\equiv i\langle \psi \left|d\right|\psi \rangle ={\mathrm{p}}_{j}d{\mathrm{q}}_j$, and the two-form $\omega \equiv dA=d{\mathrm{p}}_j\wedge d{\mathrm{q}}_j$, where d is the exterior differential operator on the projective space of rays $\mathcal{PH}$. Then, $\omega (\psi ,\varphi )$ becomes expressible through the imaginary part of the Hermitean inner product of $\psi$ and $\varphi$, to be thought of as tangent vectors. Therefore, $\mathcal{H}$ bears the status of a phase space $\left\{\right(\mathrm{q},\mathrm{p}\left)\right\}$ endowed with the natural symplectic structure derived from $\omega$ and thus generated from the Hermitean inner product in $\mathcal{H}$. The section ${\sigma }_{\alpha }$ is considered as a mapping from ${\mathrm{U}}_{\alpha }\subset \mathcal{PH}$ into $\mathcal{H}$, and the antisymmetric two-form ${\omega }^{*}$ on ${\mathrm{U}}_{\alpha }$ constitutes the pull back of $\omega$ with respect to ${\sigma }_{\alpha }$. Since ${\omega }^{*}$ is independent of the choice of section ${\sigma }_{\alpha }$, ${\omega }^{*}$ is non-degenerate, and $d{\omega }^{*}=0$; therefore, ${\omega }^{*}$ is identified as a symplectic two-form on $\mathcal{PH}$. In this sense, $\mathcal{PH}$ is a phase space bearing the symplectic structure of ${\omega }^{*}$. Further, we may consider ${\psi }_k=\alpha +i\beta =rexp\left(i\varphi \right)$, such that $\omega =2d\beta \wedge d\alpha =-d\left(r^2\right)\wedge d\varphi$. In this way, the conjugate symplectic variables can also be expressed as either ${\mathrm{q}}_k=\sqrt{2}\alpha ,{\mathrm{p}}_k=\sqrt{2}\beta$ or ${\mathrm{q}}_k=r^2,{\mathrm{p}}_k=-\varphi$. For any of the above three pairs of $\left({\mathrm{q}}_k,{\mathrm{p}}_k\right)$ evaluated on $\sigma$, which is defined on $\mathrm{U}$ that contains the loop $\mathrm{o}$, we have the following:

$$\epsilon ={\mathit{\oint }}_{\mathrm{o}}{\mathrm{p}}_{k}d{\mathrm{q}}_k=\mathit{\int }{\mathit{\int }}_{\mathrm{\Sigma }}d{\mathrm{p}}_k\wedge d{\mathrm{q}}_k$$

Since $\epsilon$ is independent of the chosen section $\sigma$, the above gives the symplectic area of $\mathrm{\Sigma }$, which is determined by the symplectic two-form ${\omega }^{*}$ of the projective space $\mathcal{PH}$. As such, it remains invariant under symplectic transformations given the specification of $\mathrm{o}$ as a loop. In other words, $\epsilon$ becomes geometrical through its association with a symplectic area-bounding loop in $\mathcal{P}H$. In this setting, the important thing is that the lifting of $\epsilon$ in the exponent, such that it gives rise to an invariant geometric phase factor, requires $\mathit{\int }{\mathit{\int }}_{\mathrm{\Sigma }}d{\mathrm{p}}_k\wedge d{\mathrm{q}}_k$ to be actually dimensionless, evoking its quotient by means of a constant, which bears the units of action and is identified with Planck’s constant ћ.

The importance of the Heisenberg group in the context of the above symplectic interpretation emerges from Weyl’s transcription of the canonical commutation relations in group-theoretic form [46]. In particular, the Lie algebraic commutation relations involving conjugate self-adjoint operators are replaced by the group-theoretic commutation relations between the corresponding bounded one-parameter unitary groups by means of Stone’s theorem and the Lie exponential morphism. This viewpoint is also closely related to the bilinear pairing expressing the Fourier–Pontryagin duality. From this perspective, the uniqueness theorem established by Stone and von Neumann [47,48,49] gains new insights. It reveals that the standard Schrödinger model can be unitarily equivalent to the direct sum of its modulated copies with respect to the continuous group action of $\mathbb{R}\times \widehat{\mathbb{R}}\cong {\mathbb{R}}^2$. It is remarkable that the continuous group action of the group $\mathbb{R}$, which generates a one-parameter unitary group through the exponential morphism, defines a translation operator in both position and momentum representation spaces. This is due to the fact that the dual of $\mathbb{R}$ is identical to itself. The distinction lies in the physical dimensions of the indexing parameter in both cases, which need to be reciprocally related to ensure that the exponents of the unitary groups become dimensionless when divided by Planck’s constant. Given that the complex projective space of rays is regarded as the quantum phase space, possessing a natural symplectic structure, it is crucial to consider translations in both position and momentum representations synchronized to each other. The synchronization involves a modular and commutative compact group structure, whose spectrum functions as a joint base topological localization space of modular position and modular momentum observables. As we shall explain, this involves the representation theory of the discrete Heisenberg group in conjunction with its continuous counterpart, which makes it necessary to re-evaluate Weyl’s approach.

As a point of fact, Weyl’s approach involves the consideration of the continuous group action of the Abelian group ${\mathbb{R}}^2$, which gives rise to the following one-parameter unitary groups:

$$\alpha \mapsto V_{\alpha }:=e^{-(i/ћ)\alpha P}$$

$$\beta \mapsto U_{\beta }:=e^{-(i/ћ)\beta Q}$$

such that the following group-theoretic commutation relation is obtained:

$$U_{\beta }{V}_{\alpha }=e^{-(i/ћ)\beta ·\alpha }{V}_{\alpha }{U}_{\beta }$$

The continuous group action of ${\mathbb{R}}^2$ on the group of unitary operators, denoted as $(\alpha ,\beta )\mapsto V_{\alpha }{U}_{\beta }$, signifies that, while both $V_{\alpha }$ and $U_{\beta }$ give rise to unitary representations of $\mathbb{R}$, their combined action does not generally yield a unitary representation of ${\mathbb{R}}^2$. However, it does induce a projective unitary representation of ${\mathbb{R}}^2$, where a complex multiplicative phase factor ${\mu }_{\alpha \beta }:\mathbb{R}\times \mathbb{R}\to S^1$, known as the multiplier of the obtained projective representation [48,50,51], can freely scale through a Borel map $\beta :\mathbb{R}\to S^1$. This ensures that the continuous group homomorphism described above is preserved. Consequently, the normalized continuous group homomorphism from ${\mathbb{R}}^2$ into the projective unitary group, giving rise to an “Abelian group of unitary ray rotations” in the Hilbert space of state vectors, according to Weyl’s term, can be expressed as

$$W:(\alpha ,\beta )\mapsto e^{(i/2ћ)\alpha ·\beta }{V}_{\alpha }{U}_{\beta },$$

such that:

$$W(\alpha ,\beta )W({\alpha }^{\prime },{\beta }^{\prime })=\mu (\alpha ,\beta :{\alpha }^{\prime },{\beta }^{\prime })W(\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime }),$$

with $\mu (\alpha ,\beta :{\alpha }^{\prime },{\beta }^{\prime }):=e^{\frac{i}{2ћ}({\alpha }^{\prime }·\beta -\alpha ·{\beta }^{\prime })}$, and the antisymmetric bilinear product $({\alpha }^{\prime }·\beta -\alpha ·{\beta }^{\prime })=\omega$ represents the symplectic form on ${\mathbb{R}}^2$. It is important to note that the same reasoning applies to ${\mathbb{R}}^n\times {\mathbb{R}}^n$, where $\omega$ represents the symplectic form on ${\mathbb{R}}^{2n}$.

The significance of this approach lies in the Abelian group and, thus, the commutative specification of the domain for the induced projective representation and the involvement of the symplectic bilinear pairing in the determination of the representation’s multiplier. As a direct consequence of the projective representation, given by

$$W(\alpha ,\beta )W({\alpha }^{\prime },{\beta }^{\prime })=e^{(i/2ћ)\omega }W(\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime }),$$

we conclude that the commutativity condition is attained if and only if the complex exponential phase factor is unity, a condition indicating that the symplectic area should be an integral multiple of Planck’s constant. This implies the existence of a fundamental symplectic area scale that represents the indivisibility of the quantum of action, which is precisely expressed by Planck’s constant in units of action. The requirement of integrality in the specification of the symplectic form, which determines the symplectic Abelian spectrum of the group-theoretic commutation relations, necessitates the consideration of modular observables, that is, observables valued in $\mathbb{R}/\mathbb{Z}$. It is with respect to these modular observables that the commutativity condition can be attained.

For instance, let us consider the one-parameter unitary group of $\alpha$-translations in the position representation, given by $e^{-i\alpha P/ћ}$. In this case, the physically relevant momentum observable is not the $\mathbb{R}$-valued one, but its exponential image valued in $\mathbb{R}/\mathbb{Z}$, which is known as the modular momentum or quasi-momentum. Due to the periodic nature of the complex exponential, we have the relationship $e^{-i\frac{\alpha P}{ћ}}=e^{-i\frac{\alpha }{ћ}·\frac{P\alpha +2k\pi ћ}{\alpha }}$, where $k\in \mathbb{Z}$. The modular momentum is then given by

$$(⥀P)=Pmod\frac{2k\pi ћ}{\alpha }:=Pmodk·p_0$$

where $p_0·\alpha =h$. Similarly, the same argument applies to the one-parameter unitary group of momentum translations in the momentum representation, leading to the corresponding modular position. The synchronized joint action of modular observables becomes possible under the constraint that the product of their moduli is proportional to an integer multiple of Planck’s constant. Specifically, the product of the conjugate variables $\alpha ·\beta$ should be the constant modulo $\mathbb{Z}$, and this constant can be expressed in terms of the quantum of action $\pi ћ$.

It is worth noting that the real-valued indexing parameters of the corresponding one-parameter unitary groups are treated as symplectic spinorial variables in our set-up. They enable Archimedes’ measure-preserving bijection from the quantum phase space to the symplectic Abelian spectrum of the group-theoretic commutation relations [41,52]. The constancy of the product of these variables can be geometrically expressed through a rectangular hyperbola, which encodes their intrinsic ${\mathbb{Z}}_2$ invariance. In particular, since $\alpha ·\beta =\mu ·h$, where $\mu$ is a positive integer and h represents Planck’s constant, if we extend horizontally by a positive real value $\lambda$ and then contract vertically by the same factor $\lambda$, a symplectic action equivalent to multiplying $\alpha$ by $\lambda$ and $\beta$ by $1/\lambda$, the rectangular hyperbola remains invariant due to $\frac{\alpha }{\lambda }·\beta \lambda =\mu ·h$. As the translation-indexing variables appear in the exponents of the corresponding one-parameter unitaries, which are defined as the modulo $2\pi \mathbb{Z}$, the following relation expressing the minimal invariant symplectic area holds:

$$\frac{\alpha }{\lambda }·\beta \lambda ={\left(\frac{\sqrt{h}}{2}\right)}^2$$

Furthermore, the symplectic area is preserved if we change the signs of the symplectic variables $\alpha$ and $\beta$ in the Abelian spectrum, i.e., $(\alpha ,\beta )\to (-\alpha ,-\beta )$. This property justifies their characterization as symplectic spinorial variables and indicates their $Z_2$-double covering attribute. Moreover, any positive real-number factor $\lambda$ in ${\mathbb{R}}^{+}$ defines an area-preserving symplectic transformation with respect to $\alpha$ and $\beta$. Thus, the modularity can still be expressed in terms of $2\pi \mathbb{Z}$, as in the initial argument, rather than $4\pi \mathbb{Z}$, if we redefine the invariant minimal area as follows:

$$\frac{\alpha }{\lambda }·\beta \lambda ={\left(\sqrt{\frac{h}{2}}\right)}^2$$

where the relevant $Z_2$-double covering metaplectic attribute is taken into account. Hence, $\sqrt{\frac{h}{2}}$ represents the geometric mean of $\frac{\alpha }{\lambda }$ and $\beta \lambda$, and its square, $\frac{h}{2}$, specifies the minimal invariant symplectic area in the modular Abelian spectrum, thus giving rise to the fundamental area scale pertaining to the integrality condition. It is important to note that the number of degrees of freedom does not affect the argument pertaining to the integral Abelian symplectic spectrum of ${\mathbb{R}}^2$ via the consideration of modular observables, since the $2n$-dimensional case constitutes a straightforward generalization.

The existence of a minimal invariant area in the Abelian symplectic spectrum of modular observables is intrinsically tied to Gromov’s non-squeezing theorem in symplectic geometry [53]. According to this theorem, there is no symplectic transformation in a $2n$-dimensional symplectic phase space that can squeeze a symplectic ball of radius $\rho$ through a circle in the symplectic plane with a radius smaller than $\rho$. As a consequence of the non-squeezing theorem, an invariant minimal symplectic area is associated with each pair of symplectic variables. Consider the two-dimensional real, or equivalently, the one-dimensional complex Abelian spectrum of the symplectic ball, obtained by the orthogonal projection onto the corresponding symplectic plane. The non-squeezing property implies that the ball of radius $\rho$ cannot be mapped under the action of any symplectic transformation to the cylinder ${\alpha }^2+{\beta }^2<{\kappa }^2$ in any $(\alpha ,\beta )$ symplectic plane if $\kappa <\rho$. From a physical standpoint, this implies that the symplectic rigidity is intrinsic to the two-dimensional symplectic plane of the Abelian spectrum where the circular base of this cylinder lies. It is precisely this symplectic rigidity that underlies the irreducibility of the two-dimensional integral symplectic flow in a topological state of matter. Further, since a minimal irreducible rigid invariant area $\frac{h}{2}=\pi ћ$ is associated with each pair of symplectic variables, the quantity $\sqrt{ћ}$ is naturally identified with the radius $\rho$ of the symplectic ball, such that $\rho :=\sqrt{ћ}$. Therefore, the commutative modular spectrum possesses an area of at least $A=\frac{h}{2}=\pi ћ$, which is preserved under symplectic transformations. Recalling that the quantum modular variables are defined as the modulo $\mathbb{Z}$, due to the integrality condition, all $\mathbb{R}$-valued symplectic areas of regions in ${\mathbb{R}}^2$ that differ by an integer yield the same $S^1$-valued geometric phase. Therefore, we conclude that the observability of the global irreducible geometric phase factor emerges from the non-squeezable area $\pi ћ$ of the two-dimensional symplectic Abelian spectrum. It has to be emphasized that, although modular observables commute under the condition of integrality of the symplectic form, the cells of the modular lattice $\frac{{\mathbb{R}}^2}{{\mathbb{Z}}^2}$ cannot be distinguished experimentally in a deterministic manner. The reason for this lies in the fact that their equivalence classes are characterized by a certain winding number, which is only specifiable with respect to an intertwining homological boundary corresponding to the neutral condition of zero total winding. A homological boundary or coboundary of this form, although trivial from a topological viewpoint, is non-trivial from a geometric symplectic perspective, since it bounds the minimal non-squeezable symplectic area derived from the irreducibility of the quantum of action. In the case of the Abelian symplectic spectrum ${\mathbb{R}}^2$, the integrality of the symplectic area implies that the symplectic flow associated with the quantum of action is topologically toroidal. It is universally covered by ${\mathbb{R}}^2$, where ${\mathbb{Z}}^2$ plays the role of the free Abelian fundamental group in two generators of the torus. This process arises from the Abelianization of the free group in two non-commuting generators ${\mathrm{\Theta }}_2$ and remarkably takes place universally through the intervention of the discrete Heisenberg group, whose unitary representation theory underlies our approach.

The emergence of the Heisenberg group in its continuous guise takes place in the context of the group-theoretic quantum commutations relations if we shift our focus to the projective unitary representation of the Abelian group ${\mathbb{R}}^2$ with the multiplier $\mu (\alpha ,\beta :{\alpha }^{\prime },{\beta }^{\prime })$, expressed in terms of the symplectic form $\omega$ on ${\mathbb{R}}^2$, through the exponential phase factor $e^{(i/2ћ)\omega }$:

$$W(\alpha ,\beta )W({\alpha }^{\prime },{\beta }^{\prime })=\mu (\alpha ,\beta :{\alpha }^{\prime },{\beta }^{\prime })W(\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime })=e^{(i/2ћ)\omega }W(\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime }).$$

The main concern lies in reformulating the above expression in terms of a genuine unitary representation of a new group that encodes and extends the information of $\omega$ on ${\mathbb{R}}^2$ in a novel way. This new group is precisely the Heisenberg group, which is constructed by the central extension of the additive group ${\mathbb{R}}^2$ by the multiplicative group $S^1$:

$$1\to S^1\hookrightarrow \mathbb{H}\twoheadrightarrow {\mathbb{R}}^2\to 0,$$

Here, the kernel $\mathbb{H}\twoheadrightarrow {\mathbb{R}}^2$ gives rise to a central subgroup of $\mathbb{H}$ that is isomorphic to $S^1$. According to this central extension, the underlying set of the Heisenberg group, denoted as $\mathbb{H}:={\mathbb{R}}^2\times S^1$, is equipped with the composition operation:

$$(v,\tau )(v^{\prime },{\tau }^{\prime }):=(v+v^{\prime },(\tau +{\tau }^{\prime })·\mu (v:v^{\prime })),$$

where $v=(\alpha ,\beta )$ in ${\mathbb{R}}^2$. The key characteristic of $\mathbb{H}$, defined through the central extension of ${\mathbb{R}}^2$ by $S^1$, is that it is a non-Abelian nilpotent group of order two, meaning that all three-fold and higher group commutators vanish. Consequently, the non-Abelian nature of $\mathbb{H}$ is captured by the commutator, which is reduced to a central element or, equivalently, a complex multiplicative phase in $S^1$. If we consider A and $\widehat{A}$ as locally compact Abelian Lie groups, then $\mathbb{H}$ becomes a non-Abelian nilpotent Lie group. It is important to note that the central extension mentioned above relates to the $S^1$-valued bilinear pairing of two locally compact groups, A and $\widehat{A}$, which are Fourier–Pontryagin duals of each other, both identified with $\mathbb{R}$ in the current setting. The $S^1$-central extension of ${\mathbb{R}}^2$ leads to a unitary representation $W_{\mu }$ of the Heisenberg group $\mathbb{H}$ in the Hilbert space of quantum states.

The unitary representation $W_{\mu }$ of the Heisenberg group $\mathbb{H}$ in the Hilbert space can be defined as follows:

$$W_{\mu }:(v,\tau )\mapsto \tau ·W\left(v\right),$$

where $W_{\mu }(1,\tau )=\tau$ and $W_{\mu }(v,1)=v$. Thus, we establish a bijective correspondence between the set of projective unitary representations of $\mathbb{R}\times \mathbb{R}$ with the multiplier $\mu (v:v^{\prime })$ and the set of unitary representations of the Heisenberg group $\mathbb{H}$. The latter group preserves both the complex-valued Hermitian inner product and the unit sphere of normalized state vectors in the Hilbert space. As $\mathbb{H}$ is not simply connected, we construct its simply-connected universal covering Lie group, denoted as $\overline{\mathbb{H}}$. This corresponds to the universal covering of $S^1$ by $\mathbb{R}$ with the kernel $\mathbb{Z}$ achieved through exponentiation. Therefore, we have $\overline{\mathbb{H}}={\mathbb{R}}^2\times \mathbb{R}$, and its group composition is given by

$$(\alpha ,\beta ,w)({\alpha }^{\prime },{\beta }^{\prime },w^{\prime }):=(\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime },w+w^{\prime }+\frac{1}{2}\omega ).$$

It is evident that the universal covering Heisenberg group $\overline{\mathbb{H}}$ is also non-Abelian and nilpotent of order two. Consequently, the Baker–Campbell–Hausdorff formula simplifies to $e^X{e}^Y=e^{X+Y+\frac{1}{2}[X,Y]}$.

Regarding the integrality condition of the symplectic form that gives rise to the notion of modular observables, it is crucial to consider the discrete Heisenberg group ${\mathbb{H}}_{\mathbb{Z}}$ obtained by restricting the variables $(\alpha ,\beta ,w)$ to integer values. Thus, ${\mathbb{H}}_{\mathbb{Z}}$ is characterized by the following exact sequence of groups:

$$0\to \mathbb{Z}\hookrightarrow {\mathbb{H}}_{\mathbb{Z}}\twoheadrightarrow {\mathbb{Z}}^2\to 0,$$

This means that the image of $\mathbb{Z}$ is central to ${\mathbb{H}}_{\mathbb{Z}}$, making ${\mathbb{H}}_{\mathbb{Z}}$ the uniquely defined central extension of ${\mathbb{Z}}^2$ by $\mathbb{Z}$. The process of spectral Abelianization can be effectively understood through ${\mathbb{H}}_{\mathbb{Z}}$.

To explore the Abelianization process, we start with the non-Abelian free group ${\mathrm{\Theta }}_2$ in two generators, denoted as ${\mathrm{\Theta }}_2=\langle a,b\rangle$. This group exhibits genuine non-commutativity through its irreducible commutators, which encode non-trivial linking properties [54]. Most importantly, ${\mathrm{\Theta }}_2$ admits a unitary representation in the Hilbert space of quantum state vectors in terms of conjugate one-parameter unitary groups [45]. We can express the first commutator subgroup of ${\mathrm{\Theta }}_2$, denoted as $[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]$, and maintain this notation for all higher commutators, such as ${\mathrm{\Theta }}_2\supset [{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]\supset [[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2],{\mathrm{\Theta }}_2]$, and so on. Then, the Abelianization of ${\mathrm{\Theta }}_2$ is isomorphic to the quotient ${\mathrm{\Theta }}_2/[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]$. In the case of the order two nilpotency of ${\mathbb{H}}_{\mathbb{Z}}$, it can be identified up to the isomorphism by the quotient ${\mathrm{\Theta }}_2/[[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2],{\mathrm{\Theta }}_2]$. The significant observation is that the Abelianization group homomorphism of the free group ${\mathrm{\Theta }}_2\to {\mathrm{\Theta }}_2/[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]\cong {\mathbb{Z}}^2$ induces a surjective group homomorphism:

$${\mathbb{H}}_{\mathbb{Z}}={\mathrm{\Theta }}_2/[[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2],{\mathrm{\Theta }}_2]\twoheadrightarrow {\mathbb{Z}}^2={\mathrm{\Theta }}_2/[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]$$

The kernel of the above homomorphism is the normal subgroup of ${\mathbb{H}}_{\mathbb{Z}}$ generated by the equivalence class of the commutators $c=[a,b^{-1}]$, which are central to $\left[{\mathbb{H}}_{\mathbb{Z}}\right]$. Moreover, due to the inclusion relation between commutator subgroups, a homomorphism ${\mathrm{\Theta }}_2\to \mathbb{Z}$ always factors through $\mathbb{H}\mathbb{Z}$. Let $\overline{\alpha }:{\mathrm{\Theta }}_2\to \mathbb{Z}$ and $\overline{\beta }:{\mathrm{\Theta }}_2\to \mathbb{Z}$ be group homomorphisms, such that $\overline{\alpha }\overline{\beta }:{\mathrm{\Theta }}_2\to {\mathbb{Z}}^2$ is the spectral Abelianization group homomorphism, whose kernel is generated by the equivalence class of the commutators that are central to $\left[{\mathbb{H}}_{\mathbb{Z}}\right]$. We define $\overline{\alpha }\left(\varrho \right)=\alpha$ and $\overline{\beta }\left(\varrho \right)=\beta$ as the $\mathbb{Z}$-images of these group homomorphisms for an element $\varrho$ in ${\mathbb{H}}_{\mathbb{Z}}$. The group $\left[{\mathbb{H}}_{\mathbb{Z}}\right]$ consists of upper triangular matrices with integer entries, that is,

$$\left[\begin{array}{ccc}1& \alpha & w\\ 0& 1& \beta \\ 0& 0& 1\end{array}\right]=\mathrm{\Pi }(\alpha ,\beta ,w)$$

identified with a matrix representation of the discrete Heisenberg group ${\mathbb{H}}_{\mathbb{Z}}$. We map a and $b^{-1}$ in ${\mathrm{\Theta }}_2$ to $\mathrm{\Pi }(1,0,0)$ and $\mathrm{\Pi }(0,1,0)$, respectively, in $\left[{\mathbb{H}}_{\mathbb{Z}}\right]$. The commutator $c=[a,b^{-1}]$ is central to $\left[{\mathbb{H}}_{\mathbb{Z}}\right]$ and is mapped to $\mathrm{\Pi }(0,0,1)$. Note that $c^k$ is mapped to $\mathrm{\Pi }(0,0,k)$, which is non-trivial for $k\ne 0$.

The spectral Abelianization process conducted in terms of modular observables and pertaining to the integrality of the symplectic form is elucidated if we express the group-theoretic structure of the Heisenberg group in each one of its three guises in terms of a principal fiber bundle equipped with a connection, which geometrizes the information encoded in the corresponding group central extension. In the case of $\mathbb{H}$, the base space of the principal bundle is ${\mathbb{R}}^2$, and the structure group consists of the central elements in $S^1$. A connection is defined by a certain splitting into vertical and horizontal subspaces at a point $\zeta =(v,\tau )$ in $\mathbb{H}$. The vertical subspace $V_{\zeta }$ is identified by the one-dimensional subspace comprising the tangent vectors in the tangent space $T_{\zeta }\mathbb{H}$ that are tangent to $S^1$ via $\zeta$. On the other hand, the horizontal subspace $H_{\zeta }$ is identified by the orthogonal complement of $V_{\zeta }$ in $T_{\zeta }\mathbb{H}$. The distribution of two-dimensional horizontal subspaces is transverse to the $S^1$-fibers and invariant under the left action of group multiplication. This splitting gives rise to a connection on the total space of $\mathbb{H}$, where the projection $\pi :\mathbb{H}\to {\mathbb{R}}^2$ makes $H_{\zeta }$ and $T_v{\mathbb{R}}^2$ isomorphic, i.e., $H_{\zeta }\cong T_v{\mathbb{R}}^2$ for any $\zeta$ in $\mathbb{H}$ corresponding to a vector v in $T_v{\mathbb{R}}^2$.

The connection can also be expressed equivalently in terms of a differential one-form $\sigma$ taking values in the Heisenberg Lie algebra $\mathbf{h}$. The Heisenberg Lie algebra $\mathbf{h}$ is a non-Abelian and nilpotent Lie algebra of order two, obtained from the central extension of the Abelian Lie algebra ${\mathbb{R}}^2$ by $\mathbb{R}$. It can be diffeomorphically identified with the simply-connected and nilpotent universal covering Lie group of the Heisenberg group $\overline{\mathbb{H}}$ through the exponential Lie morphism. The Heisenberg Lie algebra $\mathbf{h}$ consists of all left-invariant vector fields on the Heisenberg group, and the Lie bracket operation is realized by their Lie algebra commutator, which follows the canonical commutation relations. The splitting of $T_{\zeta }$ for each $\zeta$ in $\mathbb{H}$ induces a connection one-form $\sigma$ with values in $\mathbf{h}$. At each point $\zeta$, the horizontal subspace $H_{\zeta }$ is the subspace where the connection one-form vanishes, i.e., $\sigma \left(\xi \right)=0$ for any tangent vector $\xi$ at $\zeta$. The Lie commutator $[z,z^{\prime }]$ for $z,z^{\prime }\in \mathbf{h}$ gives rise to a symplectic form $\omega (z,z^{\prime })=[z,z^{\prime }]$ in $\mathbf{h}$, valued in the center of $\mathbf{h}$, which is identified by $\mathbb{R}$. This symplectic structure $\omega$ is translated by group multiplication on the left to any point $\zeta$ in $\mathbb{H}$. Consequently, the bundle projection $\pi :\mathbb{H}\to {\mathbb{R}}^2$ restricts the symplectic form $\omega$ to ${\mathbb{R}}^2$, denoted by the same symbol.

The defined connection on $\mathbb{H}$ allows for the lifting of paths from ${\mathbb{R}}^2$ to $\mathbb{H}$ using the specified horizontal distribution in $\mathbb{H}$. Specifically, for a path $\mathrm{\gamma }:[0,1]\to {\mathbb{R}}^2$ based at a point $\alpha \in {\pi }^{-1}\left(\mathrm{\gamma }\left(0\right)\right)$, the unique horizontal lift $\tilde{\mathrm{\gamma }}:[0,1]\to \mathbb{H}$ is defined such that $\tilde{\mathrm{\gamma }}\left(0\right)=\alpha$ and $\pi \circ \tilde{\mathrm{\gamma }}\left(s\right)=\mathrm{\gamma }\left(s\right)$ for all $s\in [0,1]$. However, if $\mathrm{\gamma }$ is a loop in ${\mathbb{R}}^2$, in general, $\tilde{\mathrm{\gamma }}$ is not a loop in $\mathbb{H}$. The vertical interval between $\tilde{\mathrm{\gamma }}\left(0\right)$ and $\tilde{\mathrm{\gamma }}\left(1\right)$ in the vertical subspace of the fiber ${\pi }^{-1}\left(\mathrm{\gamma }\left(1\right)\right)$ corresponds to the absolute value of the symplectic area enclosed by the loop $\mathrm{\gamma }$ in ${\mathbb{R}}^2$. Thus, the sign of the symplectic area determines which of $\tilde{\mathrm{\gamma }}\left(0\right)$ and $\tilde{\mathrm{\gamma }}\left(1\right)$ lies above the other vertically. Consequently, the distance between $\tilde{\mathrm{\gamma }}\left(0\right)$ and $\tilde{\mathrm{\gamma }}\left(1\right)$ is measured in terms of a real number equal to the signed symplectic area enclosed by the loop $\mathrm{\gamma }$ in the base space ${\mathbb{R}}^2$. This real number, which has units of area, corresponds to the center of $\mathbf{h}$ or, equivalently, the vertical subspace of the fiber ${\pi }^{-1}\left(\mathrm{\gamma }\left(1\right)\right)$ in $\mathbb{H}$. It can be identified through the exponential Lie diffeomorphism with the corresponding central element in the fiber ${\overline{\pi }}^{-1}\left(\mathrm{\gamma }\left(1\right)\right)$ of the simply-connected $\overline{\mathbb{H}}$, which is isomorphic to $\mathbb{R}$. This has particular physical significance when interpreting ${\mathbb{R}}^2$ as the Abelian spectrum spanned by the symplectic spinorial variables. This means that the area is preserved under simultaneous changes in the sign of the symplectic spinors $\alpha$ and $\beta$ of the Abelian spectrum, $(\alpha ,\beta )\to (-\alpha ,-\beta )$. Equivalently, the symplectic area is ${\mathbb{Z}}_2$-invariant with respect to the integral symplectic spinors of the Abelian spectrum of modular observables and is expressed in terms of a $\pi$-rotation. In particular, the non-squeezable minimal symplectic area expressed in terms of Planck’s constant is ${\mathbb{Z}}_2$-invariant, meaning that the symplectic spinorial flow spontaneously synchronizes symplectic antipodes bearing a $\pi$-angular difference to each other for each winding in $\mathbb{Z}$. Equivalently, the symplectic planes of the Abelian spectrum corresponding to opposing integer windings are amalgamated to each other, such that the ${\mathbb{Z}}_2$-invariant Abelian spectrum is characterized uniquely by a positive integer resonance frequency. At resonance frequencies, where the total number of windings is zero, the composite loop becomes a boundary of homological synchronization, containing at least the non-squeezable area $\pi ћ$ that physically constitutes the fundamental irreducible symplectic area scale pertaining to the symplectic integrality condition.

Consider a path $\upsilon$ in the three-dimensional $\overline{\mathbb{H}}$ described by $\upsilon =({\upsilon }_1,{\upsilon }_2,{\upsilon }_3)$. We say that $\upsilon$ is horizontal if all of its tangent vectors lie in the horizontal distribution of two-dimensional subspaces, which can be expressed as $\acute{{\upsilon }_3}=\frac{1}{2}({\upsilon }_1\acute{{\upsilon }_2}-{\upsilon }_2\acute{{\upsilon }_1})$. For any path in ${\mathbb{R}}^2$ with coordinates $({\upsilon }_1,{\upsilon }_2)$ from $(0,0)$ to $(\alpha ,\beta )$, a unique horizontal lift $({\upsilon }_1,{\upsilon }_2,{\upsilon }_3)$ exists in $\overline{\mathbb{H}}$ (or equivalently, in $\mathbf{h}$). This horizontal lift connects $(0,0,0)$ to $(\alpha ,\beta ,\epsilon )$, where $\epsilon$ represents the signed symplectic area of the region $\mathrm{\Sigma }$ in ${\mathbb{R}}^2$ bounded by the loop $\partial \mathrm{\Sigma }$, composed of $({\upsilon }_1,{\upsilon }_2)$ and a straight-line path from $(0,0)$ to $(\alpha ,\beta )$ (which contributes zero to $\epsilon$).

The signed symplectic area of the region $\mathrm{\Sigma }$ in ${\mathbb{R}}^2$ can be calculated using Stokes’ theorem, which gives

$$\epsilon =\frac{1}{2}{\mathit{\oint }}_{\partial \mathrm{\Sigma }}({\upsilon }_1\acute{{\upsilon }_2}-{\upsilon }_2\acute{{\upsilon }_1})=\mathit{\int }{\mathit{\int }}_{\mathrm{\Sigma }}d\alpha \wedge d\beta$$

This leads to the action $S=\epsilon ={\mathit{\oint }}_{\partial \mathrm{\Sigma }}\sigma$ of the contact connection one-form $\sigma =dw-\frac{1}{2}(\alpha d\beta -\beta d\alpha )$, where its zero kernel at any point $\zeta$ lies in the horizontal subspace along $\partial \mathrm{\Sigma }$ that bounds the planar region $\mathrm{\Sigma }$. The closed symplectic two-form $\kappa =d\sigma =d\alpha \wedge d\beta$ plays the role of the curvature form of the connection $\sigma$. Therefore, the symplectic two-form $\kappa$ represents the measure of non-integrability of the associated horizontal distribution induced by the connection $\sigma$. Similarly, the connection $\sigma$, through the horizontal distribution of two-dimensional subspaces, encompasses an Abelian subgroup of the Heisenberg group that relates to the symplectic Abelian spectrum under the integrality condition. It is worth noting that any two points in $\overline{\mathbb{H}}$ (or equivalently, in $\mathbf{h}$) can be connected by a horizontal path, which corresponds to the unique horizontal lift of the corresponding planar path in the Abelian spectrum. Additionally, considering that the fundamental group of $\mathbb{H}$ is $\mathbb{Z}$, the universal covering $\overline{\mathbb{H}}$ can be seen as a flat principal bundle over $\mathbb{H}$ with $\mathbb{Z}$ being its discrete structure group. All points located on an $\mathbb{R}$-fiber of $\overline{\mathbb{H}}$, which differ by an integer, project down to the same point on the corresponding $S^1$-fiber of $\mathbb{H}$. Consequently, $\mathbb{R}$-valued signed symplectic areas of regions $\mathrm{\Sigma }$ in ${\mathbb{R}}^2$ that differ by an integer give rise to the same $S^1$-valued geometric phase factor through the exponential universal morphism, $exp:\epsilon mod\mathbb{Z}\mapsto exp\left(i\epsilon \right)$. The complex exponential phase factor $exp\left(i\epsilon \right)$ captures the anholonomy of the connection, which arises from the non-integrability of the horizontal distribution. As the anholonomy is physically interpreted as an invariant geometric phase factor with a dimensionless exponent, the signed symplectic area $\epsilon$, which has units of area, is made dimensionless by taking its quotient with respect to a constant with units of action, represented by Planck’s constant ћ. Therefore, a bounded symplectic area in ${\mathbb{R}}^2$ becomes proportional to the exponent of the geometric phase factor only through the minimal area $\frac{h}{2}=\pi ћ$ of the symplectic spinorial Abelian spectrum modulo $\mathbb{Z}$, which corresponds to the interpretation of the symplectic integrality condition in terms of the Heisenberg group in its three separate guises.

Considering the integrality condition, we focus our attention on the modular lattice $\frac{{\mathbb{R}}^2}{{\mathbb{Z}}^2}\cong S^1\times S^1$. Equivalently, due to Fourier–Pontryagin duality, we may focus on the discrete lattice ${\mathbb{Z}}^2$ within the plane ${\mathbb{R}}^2$. We consider the composition string ${\chi }_1·{\chi }_2·\dots ·{\chi }_{\nu }$ in a, b, $a^{-1}$, $b^{-1}$ of the free group ${\mathrm{\Theta }}_2$, whose image in the discrete Heisenberg group ${\mathbb{H}}_{\mathbb{Z}}={\mathrm{\Theta }}_2/[[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2],{\mathrm{\Theta }}_2]$ is denoted by $\varrho$. Recall that the Abelianization homomorphism ${\mathrm{\Theta }}_2\to {\mathbb{Z}}^2$, where ${\mathbb{Z}}^2\cong {\mathrm{\Theta }}_2/[{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_2]$, induces a surjective homomorphism: ${\mathbb{H}}_{\mathbb{Z}}\twoheadrightarrow {\mathbb{Z}}^2$, whose kernel is the normal subgroup of ${\mathbb{H}}_{\mathbb{Z}}$ generated by the equivalence class of the commutators $c=[a,b^{-1}]$. Therefore, each composition string of the given form leads to a path on the symplectic plane ${\mathbb{R}}^2$. Specifically, consider the path from the origin $(0,0)$ to one of the points $(1,0)$, $(0,1)$, $(-1,0)$, $(-1,-1)$ depending on whether ${\chi }_1$ is one of the generators a, b, or their inverses $a^{-1}$, $b^{-1}$, respectively. Any path of this form can be extended by joining it to the paths corresponding to ${\chi }_2,\dots ,{\chi }_{\nu -1}$ in irreversible succession. This gives rise to a composite path from the origin $(0,0)$ to a point $(l,m)$ on the discrete lattice ${\mathbb{Z}}^2$ in ${\mathbb{R}}^2$. Finally, at $\nu$, from $(l,m)$, the path is extended, such that it terminates at $(l,m)+(1,0)$, or $(l,m)+(0,1)$, or $(l,m)+(-1,0)$, or $(l,m)+(-1,-1)$, depending on which one of a, b, $a^{-1}$, or $b^{-1}$, ${\chi }_{\nu }$ refers to. If we make the gauge choices $\overline{\alpha }\left(\varrho \right):=\alpha =0$, and $\overline{\beta }\left(\varrho \right):=\beta =0$ for $\varrho$ in ${\mathbb{H}}_{\mathbb{Z}}$, the considered path is actually a loop bounding a region in the symplectic plane ${\mathbb{R}}^2$. Due to the fact that the total winding number of this loop is zero, it qualifies as a homological boundary. The important thing in this context is that, geometrically, the area bounded by this loop cannot be contracted beyond the minimal area $\pi ћ$—expressed through the quantum of action—by means of any symplectic transformation. In general, the commutativity condition in the image of the Abelianization homomorphism ${\mathrm{\Theta }}_2\to {\mathbb{Z}}^2$ factorizing through the discrete Heisenberg group, or equivalently, the relation $[a,b^{-1}]=a{b}^{-1}{a}^{-1}b=1$, implies in homology-theoretic terms that the image of the commutator is a cycle that corresponds to a boundary, since its total winding number vanishes. This boundary loop encloses an area on the symplectic Abelian spectrum, which cannot be squeezed beyond the minimal area $\pi ћ$ under the action of any symplectic transformation. Therefore, the quantum of action expresses—through the integrality condition—the symplectic rigidity of the area bounded by this type of zero winding loop or, equivalently, the spectral observability of the associated bounded symplectic spinorial flow. In other words, the symplectic rigidity provides the physical measure of synchronization within the enclosed symplectic area of a boundary loop with zero winding on the Abelian spectrum. This measure is encoded in the discrete Heisenberg group through the horizontal lift of the loop bounding the symplectic area, providing the center of the Heisenberg algebra with a logarithmic scale, which is based precisely on this synchronization measure and is also time-reversal invariant due to its ${\mathbb{Z}}_2$ symplectic spinorial symmetry. Of course, the time-reversal symmetry pertaining to the spinorial flow can be broken under the intervention of an external agent, like a magnetic field that imposes a certain orientability.

We recall the continuous group homomorphism from ${\mathbb{R}}^2$ to the projective unitary group of the Hilbert space of quantum state vectors, which extends to a faithful unitary representation of the Heisenberg group. The symplectic and spinorial Abelian spectrum associated with this representation possesses an integral symplectic structure. Therefore, based on the faithful representation of the Heisenberg group on the Hilbert space and the integrality of the symplectic form, the commutative quantum modular observables span the horizontal distribution of the Heisenberg group, which, upon Abelianization, corresponds to the cells of the modular lattice $\frac{{\mathbb{R}}^2}{{\mathbb{Z}}^2}$. Through a compatible composition starting from an initial projector, the corresponding one-parameter unitary groups generated by conjugate observables realize the minimal area bounding boundary loops in the quantum projective phase space. In other words, the modular observables generate boundaries in the quantum phase space that enclose areas where the symplectic flow synchronizes in integer multiples of $\pi ћ$. The unitary representation of the Heisenberg group $\mathbb{H}$ on the Hilbert space $\mathcal{H}$ exhibits the property of preserving both the complex-valued Hermitian inner product and the unit sphere $\mathbf{S}\mathcal{H}$, which consists of the normalized state vectors. Remarkably, the action of the central subgroup $S^1$, which emanates from the Heisenberg group $\mathbb{H}$, manifests the property of preserving the integrity of the fibers within the principal fiber bundle $pr:\mathbf{S}\mathcal{H}\to \mathbf{P}\mathcal{H}$. This allows for the natural identification of $S^1$ as the structure group of this principal bundle.

Due to the identification of the base Abelian spectrum with the modular lattice $\frac{{\mathbb{R}}^2}{{\mathbb{Z}}^2}$, which possesses an integral symplectic structure, the topological structure of the Abelian spectrum of modular observables is toroidal. Consequently, the corresponding Hilbert state of the quantum state vectors is expressed through the sheaf of sections of the associated complex line bundle over the torus. This complex line bundle inherits a natural connection induced by the Hermitian inner product. In this context, the curvature two-form of the complex line bundle is integral in terms of the units of the quantum of action, as it is proportional to the integral symplectic form of the toroidal Abelian spectrum, which serves as the base symplectic space of the complex line bundle. In general, for a complex line bundle equipped with a connection over a base symplectic space X, if the closed two-form representing the curvature R is proportional to the symplectic form of the base space, then its two-dimensional de Rham cohomology class is integral, meaning that $\left[R\right]\in H^2(X,\mathbb{Z})\hookrightarrow H^2(X,\mathbb{R})$. In our case, the base space is $X=T^2\cong S^1\times S^1\cong {\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^2$, which is a torus. It is worth noting that the Abelianization process occurs universally through the $\mathbb{Z}$-central extension, which forms the discrete and nilpotent Heisenberg group ${\mathbb{H}}_{\mathbb{Z}}$ in conjunction with the $S^1$-central extension of its continuous counterpart, $\mathbb{H}$. Equivalently, the complex line bundle of quantum states over $T^2$, which is associated with the principal circle bundle over $T^2$ that is identified by the Heisenberg nilmanifold $\overline{\mathbb{H}}/{\mathbb{H}}_{\mathbb{Z}}$, classifies the discrete Heisenberg group in terms of the integer Chern classes in $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$.

In this framework, we consider the compact Abelian group structure of the torus $T^2$ that characterizes the Abelian symplectic spectrum through universal factorization via ${\mathbb{H}}_{\mathbb{Z}}$. Treating the torus $T^2$ as a compact Abelian group, we define its Fourier algebra $\mathcal{A}:=\mathcal{A}\left(T^2\right)$ as the algebra of observables associated with the Abelian symplectic spectrum. We further define the corresponding complex line bundle of states over $T^2$ in this context. Note that the Fourier algebra $\mathcal{A}$ is isomorphic with the algebra of absolutely convergent Fourier series on $T^2$ with the $l^1$-norm, characterized, in this way, as a commutative self-adjoint unital Banach-$\check{S}$ilov algebra over the complexes, such that its topological spectrum is (homeomorphic to) X. The differential geometric aspects of a complex line bundle over the torus in association with the harmonic analysis pertaining to its compact Abelian group structure have been elaborated by Selesnick [26,40,55] and generalized by Mallios in the context of ADG. It is the symplectic interpretation of this framework that bears the major significance for our purpose of focusing on the irreducible unitary representation of the Weyl–Heisenberg group obtained by means of the Abelian symplectic spectrum. In particular, the modular symplectic variables of the Abelian spectrum admit a sheaf cohomological instantiation, which generates a unitary representation in terms of theta functions, as we shall explain shortly.

In our setting, the pertinent complex line bundle of quantum state vectors is thought of in terms of the line sheaf of its sections. From a physical viewpoint, the construction of a sheaf constitutes the natural outcome of a complete localization process. In particular, we consider the localization process of the set (Hilbert space basis of vectors) of Hamiltonian eigenstates with respect to the topological structure of the toroidal Brillouin zone $T^k:=X$. Let $\mathbb{A}\left(T^k\right):=\mathbb{A}\left(X\right):=\mathbb{A}$ be the soft and fine observable algebra sheaf of the absolutely convergent Fourier series localized over X. According to our general perspective, we consider the pair $(X,\mathbb{A})$ to be the Gelfand spectrum of the algebra of observables $\mathcal{A}$. Together with the $\mathbb{C}$-algebra sheaf $\mathbb{A}$, we also consider the Abelian group sheaf of invertible elements of $\mathbb{A}$, denoted by $\tilde{\mathbb{A}}$. An $\mathbb{A}$-module $\mathbb{E}$ is called a locally free $\mathbb{A}$-module of states of finite rank m or simply a vector sheaf of states, if for any point $\alpha \in X$ an open set U of X exists such that

$$\mathbb{E}{\mid }_U\cong \stackrel{m}{⨁}\left(\mathbb{A}{\mid }_U\right):={\left(\mathbb{A}{\mid }_U\right)}^m$$

where ${\left(\mathbb{A}{\mid }_U\right)}^m$ denotes the m-terms’ direct sum of the sheaf of $\mathbb{C}$-observable algebras $\mathbb{A}$ restricted to U, for some $m\in \mathbb{N}$. We call ${\left(\mathbb{A}{\mid }_U\right)}^m$ the local sectional frame of states of $\mathbb{E}$ associated via the open covering $\mathcal{U}=\left\{U\right\}$ of X. If the rank is 1, the corresponding vector sheaf is called a line sheaf of states, that is, locally for any point $\alpha \in X$, an open set U of X exists such that $\mathbb{E}{\mid }_U\cong \mathbb{A}{\mid }_U$.

If the Hamiltonian of a quantum system is parameterized in terms of a set of control parameters from a base topological space, the structure of a vector space of states is generalized to the structure of a vector sheaf of states. In the current setting, the base space is identified by the toroidal Brillouin zone. Every section of a vector sheaf can be locally expressed as a finite linear combination of a basis of sections with coefficients from the observable algebra sheaf. Moreover, the set of sections of a vector sheaf can be equipped locally with the structure of a Hilbert space. Therefore, a vector sheaf of states $\mathbb{E}$ bijectively specifies a Čech 1-cocycle in $Z^1(\mathcal{U},GL(m,\mathbb{A}))$ with respect to a local covering $\mathcal{U}=\left\{U\right\}$ of X, according to the following local isomorphisms:

$${\eta }_{\alpha }:\mathbb{E}{\mid }_{U_{\alpha }}\cong {\left(\mathbb{A}{\mid }_{U_{\alpha }}\right)}^m;{\eta }_{\beta }:\mathbb{E}{\mid }_{U_{\beta }}\cong {\left(\mathbb{A}{\mid }_{U_{\beta }}\right)}^m$$

Hence, for every x in $U_{\alpha }\bigcap U_{\beta }$, the isomorphism $g_{\alpha \beta }\left(\alpha \right)={{\eta }_{\alpha }\left(\alpha \right)}^{-1}\circ {\eta }_{\beta }\left(\alpha \right)$ specifies the transition function from $U_{\alpha }$ to $U_{\beta }$. The cocycle $g_{\alpha \beta }$ is an invertible matrix section in the sheaf of germs $GL(m,\mathbb{A})$, which takes values in the general linear group $GL(m,\mathbb{C})$. The transition function $g_{\alpha \beta }$ satisfies the cocycle conditions $g_{\alpha \beta }\circ g_{\beta \mathrm{\gamma }}=g_{\alpha \mathrm{\gamma }}$ on triple intersections whenever this is the case.

Therefore, from the above information, we can construct a complex vector bundle whose fiber is ${\mathbb{C}}^n$, structure group is $GL(m,\mathbb{C})$, and module of sections gives rise to the vector sheaf of states. In particular, for $m=1$, we obtain a line bundle L with the fiber $\mathbb{C}$ and structure group $GL(1,\mathbb{C})\cong \tilde{\mathbb{C}}$ (the non-zero complex numbers), whose sections form a line sheaf of states $\mathbb{L}$. The structure group of this line bundle may be reduced from $\tilde{\mathbb{C}}$ to $U\left(1\right):=S^1$. Hence, a line sheaf of states $\mathbb{L}$ corresponds bijectively with a Čech 1-cocycle $\left(g_{\alpha \beta }\right)$ in $Z^1(\mathcal{U},\tilde{\mathbb{A}})$, where $\tilde{\mathbb{A}}\cong GL(1,\mathbb{A})$ is the group sheaf of invertible elements of $\mathbb{A}$ with values in $\tilde{\mathbb{C}}$, and $Z^1(\mathcal{U},\tilde{\mathbb{A}})$ is the set of Čech 1-cocycles. Every 1-cocycle may be conjugated with a 0-cochain $\left(t_{\alpha }\right)$ in the set $C^0(\mathcal{U},\tilde{\mathbb{A}})$ of 0-cochains, such that, another equivalent 1-cocycle arises:

$$g_{\alpha \beta }^{\prime }=t_{\alpha }·g_{\alpha \beta }·t_{\beta }^{-1}$$

In particular, by defining the coboundary operator ${\Delta }^0:C^0(\mathcal{U},\tilde{\mathbb{A}})\to C^1(\mathcal{U},\tilde{\mathbb{A}})$, we identify the set of one-coboundaries of the form ${\Delta }^0\left(t_{\alpha }^{-1}\right)$ in $B^1(\mathcal{U},\tilde{\mathbb{A}})$:

$${\Delta }^0\left(t_{\alpha }^{-1}\right):=t_{\alpha }·t_{\beta }^{-1}$$

Henceforth, the one-cocycle $g_{\alpha \beta }^{\prime }$ is equivalent to the one-cocycle $g_{\alpha \beta }$ if and only if there exists a zero-cochain $\left(t_{\alpha }\right)$, such that

$$g_{\alpha \beta }^{\prime }·g_{\alpha \beta }^{-1}={\Delta }^0\left(t_{\alpha }^{-1}\right).$$

Therefore, the Abelian group of isomorphism classes of line sheaves of states over X, $Iso\left(\mathbb{L}\right)\left(X\right)$, corresponds bijectively to the Abelian group of cohomology classes $H^1(X,\tilde{\mathbb{A}})$, identified with the Picard group $\mathrm{Pic}\left(X\right)$ of X, as follows:

$$Iso\left(\mathbb{L}\right)\left(X\right)\cong H^1(X,\tilde{\mathbb{A}})\cong \mathrm{Pic}\left(X\right).$$

The above isomorphism is the cohomological expression of the principle of local gauge (phase) invariance. We note that each equivalence class in the above Abelian group has an inverse, ${\mathbb{L}}^{-1}:={Hom}_{\mathbb{A}}\left(\mathbb{L},\mathbb{A}\right)$, where ${Hom}_{\mathbb{A}}\left(\mathbb{L},\mathbb{A}\right):={\mathbb{L}}^{*}$ is the dual line sheaf of $\mathbb{L}$. This is a consequence of the fact that the multiplication in this Abelian group is expressed by the tensor product of two equivalence classes of line sheaves, such that

$$\mathbb{L}{\otimes }_{\mathbb{A}}{\mathbb{L}}^{*}\cong {Hom}_{\mathbb{A}}\left(\mathbb{L},\mathbb{L}\right)\equiv {End}_{\mathbb{A}}\mathbb{L}\cong \mathbb{A}$$

5. Bijective Correspondence between Complex Line Bundles on the Brillouin Zone and Integral Symplectic Forms

If a complex line bundle of states is considered bijectively in terms of the line sheaf of its sections, then exponentiation can be locally expressed through the following short exact sequence of Abelian group sheaves, where Planck’s constant appears explicitly:

$$0\to \mathbb{Z}\stackrel{\iota }{\to }\mathbb{A}\stackrel{exp(\frac{i}{ћ}\left(\right))}{\to }\tilde{\mathbb{A}}\to 1.$$

In the above sequence, $\mathbb{Z}$ denotes the constant Abelian group sheaf of the integers, which consists of all the locally constant sections with values in the group of the integers, such that $Ker\left(e\right)=Im\left(\iota \right)\cong \mathbb{Z}$. The above short exact sequence gives rise to the following short exact sequences of Abelian group sheaves by restriction to locally constant coefficients:

$$0\to \mathbb{Z}\stackrel{\iota }{\to }\mathbb{C}\stackrel{exp(\frac{i}{ћ}\left(\right))}{\to }\tilde{\mathbb{C}}\to 1,$$

$$0\to 2\pi ћ\mathbb{Z}\stackrel{\iota }{\to }\mathbb{R}\stackrel{exp(\frac{i}{ћ}\left(\right))}{\to }\mathbb{U}\left(1\right)\to 1.$$

In our sheaf-theoretic set-up, the pair $(T^2,\mathbb{A})$ is identified with the commutative Gelfand spectrum of the Fourier algebra of observables $\mathcal{A}$ over the toroidal Brillouin zone $T^2$, where $\mathbb{A}$ is the localized $\mathbb{C}$-algebra sheaf of germs associated with the algebra $\mathcal{A}$. The case of the base space $T^{2n}$ follows on from the symplectic topological treatment of $T^2$ in a straightforward way. The invertible elements of $\mathbb{A}$ give rise to the Abelian group sheaf $\tilde{\mathbb{A}}$. The locally free module of sections of a complex line bundle L over $T^2$ defines a line sheaf of states $\mathbb{L}$ over $\mathbb{A}$, which is locally equipped with the structure of a Hilbert space. We consider that $\mathbb{L}$ is also equipped with the Chern–Berry Hermitian connection that is induced by the associated Hermitian inner product structure on its states. An equivalence class of line sheaves endowed locally with the structure of a Hilbert space is identified as the carrier state space of a quantum beam. The central idea is to model a quantum beam in terms of the commutative observable algebra sheaf $\mathbb{A}=\mathbb{A}\left(T^2\right)$ that pertains to the Abelian Gelfand spectrum, expressed in terms of absolutely convergent Fourier series on $T^2$.

From the perspective of harmonic analysis on $T^2$, each element $\nu$ of ${\mathbb{Z}}^2$ gives rise to a character on $T^2$. Specifically, for $\mathrm{\Lambda }$ in $T^2$ and $\upsilon$ in ${\mathbb{R}}^2$, a character is defined as follows:

$${\chi }_{\nu }\left(\mathrm{\Lambda }\right)=exp(\frac{i}{ћ}\nu ·\upsilon ),$$

where the associated group of characters is ${\widehat{T}}^2\cong {\mathbb{Z}}^2$. We denote by ${\chi }_1$ and ${\chi }_2$ the characters corresponding to the standard orthonormal basis of ${\mathbb{Z}}^2$. Since $T^2$ is a connected compact Abelian group, every non-zero element of $\mathbb{A}$ can be expressed as the product of an element of $exp\left(\mathbb{A}\right)$ with a unique character $\chi$, following the relation:

$$H^1(T^2,\mathbb{Z})\cong \frac{\tilde{\mathbb{A}}}{exp\frac{i}{ћ}\mathbb{A}}\cong {\widehat{T}}^2\cong {\mathbb{Z}}^2.$$

The Čech cohomology algebra of $T^2$ with $\mathbb{Z}$-valued coefficients is isomorphic to the exterior algebra, generated by the elements of the first degree, as described below:

$${\wedge }^p{H}^1(T^2,\mathbb{Z})\cong H^p(T^2,\mathbb{Z})$$

$$H^p(T^2,\mathbb{Z})\cong {\wedge }^p{\widehat{T}}^2\cong {\wedge }^p{\mathbb{Z}}^2$$

where $p=1,2$. Note that ${\wedge }^p{\widehat{T}}^2$ is torsion-free, since ${\widehat{T}}^2$ is torsion-free, which informs us that $H^p(T^2,\mathbb{Z})$ is torsion-free. Therefore the natural injection $\mathbb{Z}\hookrightarrow \mathbb{R}$ induces the injective morphism of cohomology groups:

$$H^p(T^2,\mathbb{Z})\to H^p(T^2,\mathbb{R})\cong H^p(T^2,\mathbb{Z}){\otimes }_{\mathbb{Z}}\mathbb{R}$$

The short exact exponential sheaf sequence of Abelian groups gives rise to a long exact sequence in cohomology, which, due to the fact that $T^2$ is a paracompact topological space, is expressed in terms of Čech cohomology groups, as follows:

$$\dots \to H^1(T^2,\mathbb{Z})\to H^1(T^2,\mathbb{A})\to H^1(T^2,\tilde{\mathbb{A}})\to H^2(T^2,\mathbb{Z})\to H^2(T^2,\mathbb{A})\to H^2(T^2,\tilde{\mathbb{A}})\to \dots$$

Further, since $\mathbb{A}$ is a fine and soft sheaf, $H^1(T^2,\mathbb{A})=H^2(T^2,\mathbb{A})=0$, such that

$$0\to H^1(T^2,\tilde{\mathbb{A}})\stackrel{{\delta }_c}{\to }{H}^2(T^2,\mathbb{Z})\to 0.$$

In this manner, we obtain the following isomorphism of Abelian groups, called the Chern isomorphism:

$${\delta }_c:H^1(T^2,\tilde{\mathbb{A}})\stackrel{\cong }{\to }{H}^2(T^2,\mathbb{Z}).$$

Since the group of isomorphism classes of line sheaves of states over $T^2$, that is, $Iso\left(\mathbb{L}\right)\left(T^2\right):=\mathrm{{\rm Y}}$, where $\mathrm{Pic}\left(T^2\right)$ denotes the Picard group of $T^2$, is in bijective correspondence with the Abelian group of the cohomology classes $H^1(T^2,\tilde{\mathbb{A}})$, we conclude that

$$\mathrm{{\rm Y}}\cong H^2(T^2,\mathbb{Z})\cong \mathrm{Pic}\left(T^2\right),$$

where $\mathrm{Pic}\left(T^2\right)$ denotes the Picard group of $T^2$. Therefore, each equivalence class of line sheaves of states in the Picard group $\mathrm{Pic}\left(T^2\right)$—identified as the carrier state space of a quantum beam—is in bijective correspondence with a cohomology class in the integral two-dimensional cohomology group of $T^2$.

Henceforth, a generator $\chi \wedge \zeta$ of ${\wedge }^2{\widehat{T}}^2\cong \mathbb{Z}$ gives rise to a line sheaf of states ${\mathbb{L}}_{\chi \wedge \zeta }$, such that each line sheaf of states $\mathbb{L}$ on $T^2$ of a quantum beam is equivalent to a finite tensor product ${\mathbb{L}}_{\chi \wedge \zeta }$ over $\mathbb{A}$:

$$\mathbb{L}\cong \underset{i,j=1}{\overset{2}{⨂}}{\mathbb{L}}_{{\chi }_i\wedge {\zeta }_j}$$

In the present setting, the Chern isomorphism ${\delta }_c$, that is, $\mathrm{{\rm Y}}\cong H^2(T^2,\mathbb{Z})$, is expressed symplectically by the assignment ${\delta }_c\left(\mathbb{L}\right)={\sum }_{i,j=1}^2{\chi }_i\wedge {\zeta }_j$. Furthermore, due to the injective mapping of cohomology groups $H^2(T^2,\mathbb{Z})\hookrightarrow H^2(T^2,\mathbb{R})$, the integral Chern cohomology class ${\delta }_c\left(\mathbb{L}\right)$ is expressible through its injective image in $H^2(T^2,\mathbb{R})$, that is, via the real Chern class ${\wedge }^2{\mathbb{Z}}^2\hookrightarrow {\wedge }^2{\mathbb{R}}^2$. The orthonormal basis in ${\mathbb{Z}}^2$ with the associated characters ${\chi }_1$ and ${\chi }_2$ gives rise to a basis in ${\wedge }^2{\mathbb{Z}}^2\cong \mathbb{Z}$, such that the Abelian group ${\wedge }^2{\mathbb{Z}}^2\cong \mathbb{Z}$ becomes isomorphic with the group of integer-valued antisymmetric bilinear forms on ${\mathbb{Z}}^2$.

Note that the $\mathbb{R}$-vector space of real-valued antisymmetric bilinear forms, which is isomorphic to ${\wedge }^2{\mathbb{R}}^2\cong \mathbb{R}$, is generated via the injection ${\mathbb{Z}}^2\to {\mathbb{R}}^2$, which is realized simply through the basis extension of the coefficients. Therefore, the real Chern class that classifies a line sheaf of a quantum beam on $T^2$ can be represented by a real-valued antisymmetric bilinear form on ${\mathbb{R}}^2$, which is integral. This form can be identified by the non-degenerate symplectic form of the Abelian spectrum. Conversely, any real-valued antisymmetric bilinear form on ${\mathbb{R}}^2$ that is integral represents a real Chern class of this line sheaf on $T^2$. Since a line sheaf of a quantum beam is a Hermitian differential sheaf, called a unitary ray, we can apply the Chern–Weil integrality theorem.

According to this theorem, a global and closed two-form is identified with the curvature R of a unitary ray if and only if the two-dimensional de Rham cohomology class $\left[R\right]$ of this form satisfies the integrality condition $\left[R\right]\in H^2(T^2,\mathbb{Z})\hookrightarrow H^2(T^2,\mathbb{R})$. Therefore, any real-valued antisymmetric bilinear form on ${\mathbb{R}}^2$ that is integral and qualifies as a symplectic form of the Abelian spectrum represents the curvature two-dimensional cohomology class of a gauge class of unitary rays constituting a quantum beam on $T^2$, and vice versa. In fact, we can directly obtain a unitary ray on $T^2$ from such a real-valued antisymmetric bilinear form on ${\mathbb{R}}^2$ that is integral and serves as a symplectic form of the Abelian spectrum.

If we consider the lattice ${\mathbb{Z}}^2$ in ${\mathbb{R}}^2$ for $\upsilon$ in ${\mathbb{R}}^2$ and $\nu$ in ${\mathbb{Z}}^2$, the projective module of continuous sections of the unitary ray on $T^2$ can be determined by the projective module of continuous sections of the corresponding unitary ray on ${\mathbb{R}}^2$, as follows:

$$\psi (\upsilon +\nu )=u_{\nu }\left(\upsilon \right)\psi \left(\upsilon \right)$$

where $u_{\nu }\left(\upsilon \right)$ is a unitary phase. Thus, if we make use of the relation

$$\psi (\upsilon +\nu +{\nu }^{\prime })=u_{\nu +{\nu }^{\prime }}\left(\upsilon \right)\psi \left(\upsilon \right)=u_{{\nu }^{\prime }}(\upsilon +\nu )u_{\nu }\left(\upsilon \right)\psi \left(\upsilon \right)$$

we can derive the cocycle condition:

$$u_{\nu +{\nu }^{\prime }}\left(\upsilon \right)=u_{{\nu }^{\prime }}(\upsilon +\nu )u_{\nu }\left(\upsilon \right)$$

We conclude that a cocycle $u_{\nu }\left(\upsilon \right)$ defined stalk-wise gives rise to a unitary ray associated with a complex line bundle on the torus $T^2\cong S^1\times S^1\cong {\left(\frac{\mathbb{R}}{\mathbb{Z}}\right)}^2$ under the condition that, for each $\nu$∈${\mathbb{Z}}^2$, the action on $\mathbb{C}\times {\mathbb{R}}^2$ is realized as follows:

$$(z,\upsilon )\mapsto (u_{\nu }\left(\upsilon \right)·z,\upsilon +\nu )$$

It is clear that by taking the quotient of the trivial complex line bundle on ${\mathbb{R}}^2$ by the action of ${\mathbb{Z}}^2$ defined above, we can generate a unitary ray on $T^2$, and further, all unitary rays on $T^2$ can be realized in this way.

At the next stage, we identify an integral antisymmetric bilinear form on ${\mathbb{R}}^2$—being representative of the curvature two-dimensional cohomology class of a gauge class of unitary rays making a quantum beam on $T^2$—with the symplectic form $\omega$ of the Abelian spectrum. We also consider a function $\varsigma :{\mathbb{Z}}^2\to \mathbb{R}$, which satisfies, for $\mu$, $\nu$∈${\mathbb{Z}}^2$, the following condition:

$${\varsigma }_{\mu \nu }:=\varsigma (\mu +\nu )={\varsigma }_{\mu }+{\varsigma }_{\nu }+\omega (\mu ,\nu )mod2.$$

Note the $mod2$-dependence in the above condition, which is indicative of the ${\mathbb{Z}}_2$-invariance of the integral symplectic form of the Abelian spectrum. Then, the complex line bundle associated with a unitary ray on $T^2$ corresponds to the cocycle $u_{\nu }\left(\upsilon \right)$, such that

$$u_{\nu }\left(\upsilon \right)=exp\frac{i}{2ћ}({\varsigma }_n+\omega (\mu ,\nu ))$$

Therefore, the representative unitary ray is independent of $\varsigma$ up to the isomorphism, and its real Chern class is determined by the symplectic form $\omega$ of the Abelian spectrum. In this context, $\omega$ is a real symplectic form in the Abelian group $H^2(T^2,\mathbb{R})\cong {\wedge }^2{\mathbb{R}}^2\cong \mathbb{R}$, which constitutes the image of the integral symplectic form $\omega$ in the Abelian group $H^2(T^2,\mathbb{Z})\cong {\wedge }^2{\mathbb{Z}}^2\cong \mathbb{Z}$, which is denoted in the same way due to injectivity.

Therefore, the locally free module (vector sheaf) of continuous sections of the representative unitary ray of a quantum beam on $T^2$, which is classified by the Chern class $\omega$, identified by the integral symplectic form of the Abelian spectrum, is determinable isomorphically by the locally free module of continuous sections of the corresponding unitary ray on ${\mathbb{R}}^2$ consisting of functions $\rho :{\mathbb{R}}^2\to \mathbb{C}$, such that

$$\rho (\upsilon +\nu )=\rho \left(\upsilon \right)exp\frac{i}{2ћ}({\varsigma }_{\nu }+\omega (\mu ,\nu )),$$

and thus, quantum states should be modularly identified in terms of the doubly quasi-periodic theta functions [44,56,57]. At the final stage, if we recall that the carrier state space $Iso\left(\mathbb{L}\right)\left(T^2\right):=\mathrm{{\rm Y}}\equiv \mathrm{Pic}\left(T^2\right)$ of a quantum beam is in bijective correspondence with a cohomology class in the integral two-dimensional cohomology group of $T^2$, that is, $H^2(T^2,\mathbb{Z})\cong {\wedge }^2{\mathbb{Z}}^2\cong \mathbb{Z}$, we can conclude that there is a single generator $\omega$ of the carrier space of a quantum beam, which is concomitantly identified with the integral and the ${\mathbb{Z}}_2$-invariant symplectic form $\omega$ of its Abelian spectrum. Henceforth, due to the Stone–von Neumann theorem, for each positive integer frequency $\kappa$, the Hilbert space of sections ${H{}_{\omega }}^{\kappa }$ has an irreducible unitary representation of the Heisenberg group, characterized by the multiplier $exp\pi i\kappa \frac{\omega }{h}=exp\frac{i}{2ћ}·\kappa \omega$. In particular, the Jacobi theta functions constitute an irreducible unitary representation of the discrete Heisenberg group.

6. Group Cohomological Treatment of the Integer and Spin Quantum Hall Effects

We consider two unitary rays $(\mathbb{L},\nabla )$ and $({\mathbb{L}}^{\prime },{\nabla }^{\prime })$, along with an isomorphism $h:\mathbb{L}\stackrel{\cong }{\to }{\mathbb{L}}^{\prime }$ between their carrier state spaces. We say that ∇ is gauge equivalent to ${\nabla }^{\prime }$ if they are conjugate connections under the action of h, that is, ${\nabla }^{\prime }=h·\nabla ·h^{-1}$. The set of equivalence classes of pairs of the form $(\mathbb{L},\nabla )$ induced by an isomorphism h of their carrier spaces gives rise to a quantum beam denoted by $Iso(\mathbb{L},\nabla )$ [32,58]. Remarkably, this set not only has an Abelian group structure but also constitutes an Abelian subgroup of the Abelian group $Iso\left(\mathbb{L}\right)$.

In the case of Hermitean connections, a constant ${\mathbb{S}}^1$-valued one-cocycle gives rise to a linear local system of rank one, which serves as the carrier state space for a particular type of integrable unitary ray. In fact, there exists a bijective correspondence between integrable unitary rays and linear local systems. Let $(\mathbb{L},\nabla )$ be an integrable unitary ray, meaning that its curvature is zero. Then, the set of sections belonging to the kernel of the connection ∇, denoted as $Ker(\nabla ):=s\in \mathbb{L}:\nabla \left(s\right)=0$, gives rise to a linear local system whose sections are covariantly constant with respect to ∇. Conversely, if we have a linear local system $\mathrm{\Lambda }$, it gives rise to a unitary ray $(\mathbb{L},\nabla )$ in the following way: $\mathbb{L}:=\mathbb{A}{\otimes }_{\mathbb{C}}\mathrm{\Lambda }$. For every pair of local sections $a\in \mathbb{A}\left(U\right)$ and $s\in \mathrm{\Lambda }\left(U\right)$, we define $\breve{\nabla }(a\otimes s):=da\otimes s$. Since this connection $\breve{\nabla }$ is integrable, we establish a bijective correspondence between integrable unitary rays $(\mathbb{L},\breve{\nabla })$ and linear local systems, where the latter can always be identified by the system of covariantly constant sections of an integrable unitary ray.

Applying the above considerations to the case of the toroidal Abelian spectrum, we obtain the following result:

$$Hom({\pi }_1\left(T^2\right),{\mathbb{S}}^1)\cong H^1(T^2,{\mathbb{S}}^1)\cong \left[(\mathbb{L},\breve{\nabla })\right]\cong \left[Ker{\left(\breve{\nabla }\right)}_{\mathbb{L}}\right]\cong \left[\mathrm{\Lambda }\right],$$

where the first term denotes the set of representations of the fundamental group of the torus $T^2$ to ${\mathbb{S}}^1$. Since $Hom({\pi }_1\left(T^2\right),{\mathbb{S}}^1)\cong H^1(T^2,{\mathbb{S}}^1)\cong {\mathbb{S}}^1\times {\mathbb{S}}^1$, we conclude that the group structure of the Abelian spectrum itself cohomologically classifies the integrable unitary rays, or equivalently, the linear local systems of their covariantly constant sections. In other words, the Abelian spectrum, beyond its topological role as the base space of the considered unitary rays, also bears a cohomological role—pertaining to its Abelian group structure—that classifies the integrable unitary rays. The latter cohomological functioning of the Abelian spectrum is intricately weaved together with the integral symplectic structure that it bears.

Let us recall our previous identification of an integral antisymmetric bilinear form on ${\mathbb{R}}^2$, which represents the curvature two-dimensional cohomology class of a gauge class of unitary rays forming a quantum beam on $T^2$ with the integral symplectic form $\omega$ of the toroidal Abelian spectrum. This is realized through integral Chern classes in $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$, where we have

$$c_1=-\frac{1}{2\pi i}\left[\omega \right]\in H^2(T^2,\mathbb{Z}){\overline{c}}_1=c_{-1}=\frac{1}{2\pi i}\left[\omega \right]\in H^2(T^2,\mathbb{Z}),$$

such that the curvature class $\left[R\right]\equiv \left[\omega \right]$, and $H^2(T^2,\mathbb{Z})\to H^2(T^2,\mathbb{R})$. Considering the Bockstein morphism ${\delta }_c:H^1(T^2,{\mathbb{S}}^1)\to H^2(T^2,\mathbb{Z})$, we find that the inverse image of any integral symplectic Chern class in $H^2(T^2,\mathbb{Z})$ corresponds to the neutral element of the Abelian group $H^1(T^2,{\mathbb{S}}^1)$, which is cohomologically classified as a one-coboundary. Since $H^1(T^2,{\mathbb{S}}^1)\cong H_1(T^2,{\mathbb{S}}^1)\cong {\mathbb{S}}^1\times {\mathbb{S}}^1$, we obtain a dual one-boundary, which, although trivial homologically, is non-trivial symplectically. Specifically, it is a zero-winding closed chain that encloses the non-squeezable symplectic area, serving as the unit in the integrality condition and physically identified with the irreducible symplectic flow of the quantum of action.

This zero-winding boundary captures the essence of the Abelian spectrum from a group homological perspective, along with the non-squeezable symplectic area it bounds from a geometric standpoint. These aspects are inseparable and fundamentally stem from the role of the discrete Heisenberg group in the process of Abelianization, as well as its irreducible unitary representation in terms of theta functions. These zero-winding (co)-boundaries in $H^1(T^2,{\mathbb{S}}^1)\cong H_1(T^2,{\mathbb{S}}^1)\cong {\mathbb{S}}^1\times {\mathbb{S}}^1$ are instrumental to the constitution of the homological Abelian spectrum and are subordinated to its symplectic qualification under the quantum of action.

Consider a gauge equivalence class of unitary rays forming a quantum beam, denoted by $Iso(\mathbb{L},\nabla ):={\mathrm{{\rm Y}}}^{\nabla }$. The Abelian group ${\mathrm{{\rm Y}}}^{\nabla }$ can be partitioned into orbits over the projective module of global closed two-forms ${\mathrm{\Omega }}_{c,\mathbb{Z}}^2$, where each orbit is labeled by an integral global closed two-form R from ${\mathrm{\Omega }}_{c,\mathbb{Z}}^2$, which is identified with the integral symplectic form $\omega$ of the underlying Abelian spectrum. In other words, we have the decomposition:

$${\mathrm{{\rm Y}}}^{\nabla }=\sum _{R\in {\mathrm{\Omega }}_{c,\mathbb{Z}}^2}{\mathrm{{\rm Y}}}_R^{\nabla }$$

Each orbit corresponds to a spectral $\left[R\right]$-beam, which, due to the integrality condition and the fact that $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$, is characterized by its specific positive integer frequency. All unitary rays within a beam indexed by a certain frequency are indistinguishable from each other in terms of their common differential and symplectic invariants.

However, there is free group action of the Abelian group of integrable unitary rays $H^1(T^2,{\mathbb{S}}^1)\cong {\mathbb{S}}^1\times {\mathbb{S}}^1$ on the Abelian group ${\mathrm{{\rm Y}}}^{\nabla }$ [58]. This action is restricted to free group action on each spectral $\left[R\right]$-beam of a positive integer frequency. The existence of this free group action is fundamentally induced by the group cohomological role of the Abelian spectrum itself.

$$\xi ·\left[(\mathbb{L},\nabla )\right]:=\left[(\xi ·\mathbb{L},\nabla )\right]\equiv \left[({\mathbb{L}}^{\prime },\nabla )\right]$$

such that $\left[\left({\xi }_{\alpha \beta }\right)\right]=1$ in $H^1(T^2,{\mathbb{S}}^1)$. In this manner, a cohomology class in $H^1(T^2,{\mathbb{S}}^1)$ indicates a polarization phase germ of a spectral $\left[R\right]$-beam of a certain frequency. In particular, its evaluation in a homology cycle $\mathrm{\gamma }$∈$H_1\left(T^2\right)$ by means of the standard bilinear pairing $H_1\left(T^2\right)\times H^1(T^2,{\mathbb{S}}^1)\to {\mathbb{S}}^1$ gives rise to a global gauge-invariant phase factor in ${\mathbb{S}}^1$. Thus, the free group action of the Abelian spectrum has the capacity to classify the unitary rays of a spectral $\left[R\right]$-beam of a certain positive integer frequency by means of their global polarization phase in ${\mathbb{S}}^1$, if and only this action is also transitive. The transitivity of this free group action can be attained only by restriction to the logarithmically exact closed one-forms [58]. This is the case because, under this restriction, the following sequence of Abelian group sheaves is exact:

$$1\to \tilde{\mathbb{C}}\stackrel{ϵ}{\to }\tilde{\mathbb{A}}\stackrel{{\tilde{d}}^0}{\to }{\mathrm{\Omega }}^1\stackrel{d^1}{\to }{d}^1{\mathrm{\Omega }}^1\to 0$$

Since the cohomological free group action of the Abelian spectrum $H^1(T^2,{\mathbb{S}}^1)\cong {\mathbb{S}}^1\times {\mathbb{S}}^1$ on a spectral $\left[R\right]$-beam of a certain frequency is qualified as transitive under restriction to the logarithmically exact closed one-forms, a spectral $\left[R\right]$-beam of a certain frequency becomes an affine space with the structure group being the Abelian spectrum itself. Therefore, we are finally led to the following short exact sequence of Abelian groups:

$$1\to H^1(T^2,{\mathbb{S}}^1)\stackrel{\sigma }{\to }{\mathrm{{\rm Y}}}^{\nabla }\stackrel{\kappa }{\to }{\mathrm{{\rm Y}}}_R^{\nabla }\to 0$$

which, due to the fact that $\frac{1}{2\pi i}\left[\omega \right]$ is integral in $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$, where $\left[R\right]\equiv \left[\omega \right]$, can be equivalently written as follows:

$$1\to H^1(T^2,{\mathbb{S}}^1)\stackrel{\sigma }{\to }{\mathrm{{\rm Y}}}^{\nabla }\stackrel{2\pi i\kappa }{\to }{H}^2(T^2,\mathbb{Z})\to 0$$

$$1\to {\mathbb{S}}^1\times {\mathbb{S}}^1\stackrel{\sigma }{\to }{\mathrm{{\rm Y}}}^{\nabla }\stackrel{2\pi i\kappa }{\to }\mathbb{Z}\to 0$$

In conclusion, a quantum beam ${\mathrm{{\rm Y}}}^{\nabla }$ can be understood as the Abelian central extension of a spectral $\left[R\right]$-beam with a positive integer frequency ${\mathrm{{\rm Y}}}_R^{\nabla }$, where $\left[R\right]\equiv \left[\omega \right]$, or equivalently, the Abelian group $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$ by the Abelian spectrum ${\mathbb{S}}^1\times {\mathbb{S}}^1$. The carrier space of a spectral beam over the symplectic toroidal Abelian spectrum is classified by the cohomology class of an integral symplectic form in $H^2(T^2,\mathbb{Z})\cong \mathbb{Z}$, and this classification itself relates to the discrete Heisenberg group ${\mathbb{H}}_{\mathbb{Z}}$ in its association with the quotient $\overline{\mathbb{H}}/{\mathbb{H}}_{\mathbb{Z}}$. Therefore, we can conclude that any spectral beam is an Abelian central extension of $\mathbb{Z}$ by ${\mathbb{S}}^1\times {\mathbb{S}}^1$ with respect to the group action of the integrable rays of the Abelian spectrum.

In summary, the Abelian spectrum not only determines a quantum beam through its integral symplectic form as a theta-representation of the Heisenberg group, based on the non-squeezable symplectic area of the quantum of action, but also specifies the global anholonomic observable phase factors cohomologically through the Abelian central extension of $\mathbb{Z}$ by ${\mathbb{S}}^1\times {\mathbb{S}}^1$ as well as the ${\mathbb{Z}}_2$-invariant symplectic spinorial flow. Therefore, it is the integrality of the symplectic form on the toroidal Brillouin zone that characterizes the quantum Hall effect. Additionally, the symplectic form bears a ${\mathbb{Z}}_2$-type of invariance—characteristic of the invariance pertaining to spinorial symplectic variables—that underlies the spin Hall effect, as it is expressed by the $mod2$ first Chern class of the Pfaffian.