Manifestation of the topological index formula in quantum waves and geophysical waves

Using semi-classical analysis in $\mathbb{R}^{n}$ we present a quite general model for which the topological index formula of Atiyah-Singer predicts a spectral flow with the transition of a finite number of eigenvalues between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [faure_zhilinskii_2000,2001], also in geophysics for the oceanic or atmospheric equatorial waves [Matsuno_1966, Delplace_Marston_Venaille_2017] and expected to be observed in plasma physics [Qin, Fu 2022].

Remark 0.1. On this pdf file, you can click on the colored words, they contain an hyper-link to wikipedia or other multimedia contents.

Introduction
The famous index theorem of Atiyah Singer obtained in the 60' relates two different domains of mathematics: spectral theory of pseudo-differential operators and differential topology [5] [21]. This theorem has a strong importance in mathematics with many applications (e.g. the Riemann-Roch-Hirzebruch index formula that is used in geometric quantization [11]) but also in physics: in quantum field theory with anomalies [32, chap.19], in molecular physics with energy spectrum [17,18,20]. Recently P. Delplace, J.B. Marston and A. Venaille [8] have discovered that a famous model of oceanic equatorial waves established by Matsuno in 1966 [30] has remarkable topological properties, namely that the existence of N = +2 equatorial modes in the Matsuno's model is related to the fact that the dispersion equation of this model defines a vector bundle over 2 S 2 whose topology is characterized by a Chern index with value C = +2. In the similar context of waves but in plasma physics, Hong Qin, Yichen Fu [34] have recently predicted a manifestation of the index formula.
In this paper we propose a general mathematical model that contains as particular cases the normal form used for molecular physics in [17,18] and the model of Matsuno [30][8] of equatorial waves. For this general model we have on one side a spectral index N ∈ Z that counts the number of eigenvalues that move upwards as a parameter µ increases and on the other side a topological Chern index C ∈ Z associated to a vector bundle that characterizes the equivalence class of the model. We establish the index formula N = C.
The paper is organized as follows. In Section 2 we present the general model and the main result of this paper, Theorem 2.7. In Section 2.5 we give the proof of Theorem 2.7. The proof relies on the index theorem on Euclidean space of Fedosov-Hörmander given in [25, thm 7.3 p. 422],[5, Thm 1, page 252] and explained in the appendix.
Sections 3 and 4 are applications of this general model in physics. In Section 3 we present a simple model used in [17,18] to show the manifestation of the index formula in experimental molecular spectra of quantum waves. In Section 4 we present the model of equatorial geophysics waves of Matsuno [30] and the topological interpretation from [8].
The reader may prefer to read first Section 3 and 4 that present the examples with detailed computations before Section 2 that presents the general but more abstract model.
Appendix A gives a short overview of symbols and pseudo-differential operators. Appendix B gives a short overview of vector bundles over spheres.
This article is made from the lecture notes in French [15].
Acknowledgement 1.1. The author thanks P. Delplace and A. Venaille for interesting discussions about models of geophysical waves.

A general model on R n and index formula
In this Section we propose a general framework that will contains the particular models of molecular physics of Section 3 and of geophysics of Section 4. For this general model we define a spectral index N that corresponds to the number of eigenvalues that move upwards with respect to an external parameter µ and we define a topological index (Chern index) C of a vector bundle that characterizes the (stable) isomorphism class of the model. We establish the index formula N = C.

Admissible family of symbols (H µ ) µ
Let µ ∈]−2, 2[ be a parameter. Let n ∈ N\ {0} and (x, ξ) ∈ T * R n = R n ×R n a point on the cotangent space T * R n called "slow phase space". Let d ≥ 2 be an integer and Herm C d denotes Hermitian operators on C d . We consider a function (µ, x, ξ) → H µ (x, ξ) smooth with respect to µ, x, ξ and valued in Herm C d : called symbol (we suppose that H µ ∈ S m ρ,δ (T * R n ) belongs to the class of Hörmander symbols. This corresponds to suitable hypothesis of regularity at infinity, see Section A).
For fixed values of µ, x, ξ, the eigenvalues of the matrix H µ (x, ξ) are real and are denoted We will assume the following hypothesis 3 for the family of symbols (H µ ) µ . This hypothesis is illustrated on Figure 2.1. 3 Here ∥(µ, x, ξ)∥ := µ 2 + n j=1 x 2 j + ξ 2 j is the Euclidean distance from (µ, x, ξ) to the origin in , for parameters (µ, x, ξ) ∈ R × R n × R n in the green domain, we assume that the spectrum of the hermitian matrix H µ (x, ξ), has r eigenvalues smaller than −C and that the others are greater than C > 0. Equivalently, on figure (b), the spectrum of H µ (x, ξ) for any (x, ξ) is contained in the red domain.

Spectral index N for the family of symbols (H µ ) µ
The reader may read first the appendix A that gives an introduction with examples to pseudo-differential operators (PDO) and pseudo-differential calculus.
Let us introduce a new parameter ϵ > 0 called adiabatic parameter or semi-classical parameter. We define the pseudo-differential operator 4 (PDO) In the red domain, the spectrum is discrete: discrete eigenvalues are shown in blue and depend continuously on µ, ϵ. Consequently we can label the eigenvalues by some increasing number n and label each spectral gap by the index n of the first eigenvalue below it. We define then the spectral index of the family of symbols (H µ ) µ by N = n in − n out . In this example, N = n in − n out = 0 − (−2) = +2 corresponding to the fact that N = +2 eigenvalues are moving upward as µ increases.
As a consequence of Theorem 2.2 we can define the spectral index N as follows, as shown on figure 2.2. Definition 2.3. «Spectral index N of the family of symbols (H µ ) µ ». With assumption 2.1 and from Theorem 2.2, for fixed ϵ, each spectral gap can be labeled as follows. Let (ω n (µ, ϵ)) n∈Z be the eigenvalues of the operatorĤ µ,ϵ that belongs to the interval I α :=] − C + α, +C − α[, labeled by n ∈ Z and sorted by increasing values (this is well defined up to a constant). For a given n, the eigenvalue ω n (µ, ϵ) ∈ R is continuous w.r.t. µ, ϵ. For a point (µ, ω) ∈ (−1 − α, 1 + α) × I α different from an eigenvalue, we associate the index n (µ, ω) ∈ Z of the eigenvalue just below it, i.e. such that ω n (µ, ϵ) < ω < ω n+1 (µ, ϵ). We denote n in := n (−1, 0) the index of the first gap and n out := n (1, 0) the index of the last gap. This defines an integer N := n in − n out ∈ Z (2.4) called the spectral index of the family of symbols (H µ ) µ . This integer N counts the number of eigenvalues that go upwards as µ increases. N is independent on ϵ and more generally invariant under any continuous variation of the symbol (H µ ) µ family satisfying the assumption 2.1. Hence, N is a topological index. Answer: in the next section, with Theorem 2.7, we will see that N is simply related to the degree of a certain map f : S 2n−1 → S 2n−1 that is obtained from the symbol (H µ ) µ .

Chern topological index C and index formula
The reader may read first the appendix B.3 that gives an introduction and general informations about topology of vector bundles over spheres. Let be the unit sphere in the space of parameters. From assumption 2.1, for every parameter (µ, x, ξ) ∈ S 2n , we have a spectral gap between eigenvalues ω r (µ, x, ξ) and ω r+1 (µ, x, ξ).
Then we can define the spectral projector associated to the first r eigenvalues by Cauchy formula where the integration path γ ⊂ C enclosed the segment [ω 1 (µ, x, ξ) , ω r (µ, x, ξ)] and crosses the spectral gaps. The spectral space associated to the first r eigenvalues ω 1 . . . ω r is then the image of this projector The linear space F (µ, x, ξ) ⊂ C d has complex dimension r and defines a smooth complex vector bundle of rank r over the sphere S 2n , that we denote F → S 2n . From remark 2.8 below, we can suppose that r ≥ n. From Bott's theorem B.20, the topology of F → S 2n is characterized by an integer C ∈ Z called Chern index defined in (B.18) from the degree deg (f ) of a map f : S 2n−1 → S 2n−1 in (B.17), by C = deg(f ) (n−1)! , and f is directly obtained from the clutching function of the bundle F → S 2n on the equator S 2n−1 with respect to some local trivialization. In dimension n = 1 this is more simple because C is just the winding number of the clutching function g : S 1 → U (1) ≡ S 1 on the equator S 1 . The physical applications considered later in this paper correspond to dimension n = 1.
• The spectral index ofH µ and H µ are equal, i.e.Ñ = N . This is because is on the constant horizontal line ω = ω 0 < −C, so does not give moving eigenvalues.
• The associated vector bundleF → S 2n isF = F ⊕ T m where T m = S 2n × C m is the trivial bundle and rank F = rank (F ) + m.
This remark shows that the spectral index does not change if one adds a trivial bundle T m to the bundle F . It means that N depends only on the equivalence class of F (or H) in the K-theory groupK (S 2n ), cf [24].
2.4 Special case of matrix symbols that are linear in (µ, x, ξ) In this section, we give a simple but important remark to understand why the model of Matsuno presented in Section 4 does not depend on a small parameter ϵ but nevertheless belongs to the general model presented here. This is the same for the normal form model presented in Section 3.

Suppose thatH
: μ,x,ξ ∈ R 1+2n → H μ,x,ξ ∈ Herm C d is a linear map with respect to μ,x,ξ and consider the quantization rule Op 1 ξ = −i∂x (i.e. with ϵ = 1). For example, see the normal form symbol (3.1) or the Matsuno's symbol (4.4). For any ϵ > 0, we do the change of variables Hence the symbol H (µ, x, ξ) = √ ϵH (µ, x, ξ) satisfies In other words all these models with different ϵ are equivalent up to a scaling of the parameters and the operator (and spectrum). The benefit to consider an additional semiclassical (or adiabatic) parameter ϵ ≪ 1 is that one can perturb the linear symbol to a non linear symbol and still get the index formula N = C from Theorem 2.7.

Proof of the index formula (2.7)
In this section we give a proof of Formula Let us denote F → S 2n the smooth vector bundle of rank r defined from H in (2.5). We will construct a new symbol in the same equivalence class, so having the same indices N H , C H , but that will be easier to handle to show that N H = C H . Let g : S 2n−1 → U (r) be the clutching function on the equator of the bundle F , as defined in (2.6) or appendix B.3.2. We extend g outside of S 2n−1 ⊂ R 2n x,ξ giving a 1-homogeneous functiong : R 2n x,ξ → Mat (C r ) byg Then we define the (new) symbol H µ as follows. For µ ∈ R, (x, ξ) ∈ R 2n , let Lemma 2.9. There are two eigenvalues of H µ (x, ξ) defined in (2.9), given by ω ± (µ, x, ξ) = ± ∥(µ, x, ξ)∥, each with multiplicity r. For (µ, x, ξ) ∈ S 2n , the eigenspace F − (µ, x, ξ) associated to ω − (µ, x, ξ) = −1 defines a vector bundle F − → S 2n of rank r isomorphic to the initial given vector bundle F → S 2n .
Remark 2.10. Eq.(2.9) car be related to a more general construction of a projector from a given vector bundle, see [21, p.14].
From Lemma 2.9, we see that the symbol H µ in (2.9) satisfies the assumption 2.1. As in (2.3) we define the operator and from Theorem 2.2 we can define the spectral index N H in (2.4).
Proof. For simplicity of notation, we denote the operator A := Op ϵ (g).

Some models with topological contact without exchange of states
In Section 2, we have seen a model constructed from a symbol H µ (x, ξ) on a phase space (x, ξ) ∈ R 2n (i.e. n degrees of freedom) and parameter µ ∈ (−2, 2), with a spectral gap for µ < −1 and µ > 1 and with a spectral index N ∈ Z that counts the exchange of discrete energy eigenvalues (or states) between two energy bands, as the parameter µ increases (energy bands are the spectrum below the gap and the spectrum above the gap). We have seen that N is equal to the Chern index C of a vector bundle F → S 2n of rank r that is defined from the symbol.
• If the vector bundle F is trivial, it means that the two bands are not "topologically coupled" and we can perturb continuously the symbol (H µ ) µ so that the gap may exist for every values of µ ∈ (−2, 2), i.e. we can "open the gap".
• If the vector bundle F is non trivial, it means that the two bands are "topologically coupled" with a "topological contact" and we can not "open the gap", or remove the contact between the two bands.
If N = C ̸ = 0 then the bundle F is not trivial and we can not open the gap, since some energy levels pass through it, and this situation cannot be changed by continuous perturbations. From Bott's theorem B.20, if r = rank (F ) ≥ n then C ∈ Z characterizes the topology of F . In other words, if r ≥ n then C = N = 0 ⇔ F is trivial.
However for vector bundles F of smaller ranks, r < n this is not always true (we only have the obvious fact F is trivial ⇒ C = N = 0 but not the converse). There exist some non trivial bundles F → S 2n with Chern index C (F ) = 0. From table 2, the simplest example is for F → S 6 , i.e. n = 3 degrees of freedom, with rank r = 2, because Vect 2 (S 6 ) = Z 2 = {0, 1}. Suppose for example that F → S 6 is non trivial and with topological class [F ] = 1 ∈ Vect 2 (S 6 ) = Z 2 . It means that the two bands have a "topological contact", i.e. that we can not open the gap. Nevertheless N = C = 0, i.e. there is no exchange of states between the two bands at the contact (since the spectrum is discrete, there is some small gap that goes to zero as ϵ → 0). See figure below. ω µ If one adds a second similar contact (at some other value of µ), then since 1 + 1 = 0 in Z 2 , the result is that the two contact annihilate themselves and one can finally "open the gap". See figure below.
These kind of phenomena may occur with vector bundles F → S 2n that are in the "non stable range", where the homotopy groups are very complicated, see the appendix B.3.
For a different example of the role of topology in spectral phenomena, in the paper [20] there is a simple model used molecular physics, for which the energy bands are topological coupled and associated to a rank 2 vector bundle that can not be splitted into two rank 1 vector bundles. This involves Chern numbers C 1 , C 2 and shows the manifestation of algebraic topology in quantum mechanics of molecules or more generally quantum interacting systems.

Spectral flow and index formula for quantum waves in molecules
References for this Section are [17,19,18,14].

Introduction
A small molecule is a set of atoms (electrons and nuclei) and can be considered as an isolated but complex quantum system since many degrees of freedom interact strongly on different time scales: the electrons that are light evolve on very short scales of time τ e ∈ [10 −16 s, 10 −15 s], which are small compared to the time scales of the vibration motion of the atoms τ vib ∈ [10 −15 s, 10 −14 s], themselves small compared to the slower rotation of the molecule τ rot. ∈ [10 −12 s, 10 −10 s]. In quantum mechanics the state of the molecule is described by a multivariate "quantum wave function" and a stationary state of the molecule corresponds to an eigenfunction of the Hamiltonian operator. The corresponding eigenvalue is the energy of this state. If the molecule is sufficiently isolated from its environment, one can experimentally measure its quantum energy levels (discrete spectrum) by spectroscopy. These quantum energy levels correspond to stationary collective states of all the internal interactions between all these different degrees of freedom. It seems to be (and it is) a very complicated problem, but these different time scales allows to approximate the dynamics by some "fiber bundle description". This is called the adiabatic theory. In simple words the fast motion phase space is a fiber bundle over the slow motion phase space. In quantum mechanics (or more generally in wave mechanics, like optics, acoustics ...) one has to as a function of the total angular momentum J ∈ N (rotation energy and which is a preserved quantity). The fine structure of the spectrum corresponds to the slow rotation motion and the broad structure to the faster vibration motion. There are groups of levels and levels that pass between these groups. The index formula gives the exact values of number of levels N j in each group [17,19,18,14].
quantize this fiber bundle description. Although this adiabatic approach does not solve completely the problem it gives a geometric description and some rough (and robust under perturbations) first description of the spectrum can be obtained from topological properties of these fiber bundles. This is the subject of this Section. See figure 3.1.

Simple model (normal form)
References for this section: [17,18]. The following model not only is relevant in molecular physics to illustrate the spectral behavior of rotational / vibrational (slow / fast) energy levels of nuclei, but also plays an important role in the general theory because it is an "elementary topological normal form". Let µ ∈ R be a parameter that is fixed. Let (x, ξ) ∈ T * R ≡ R × R "slow variables" on phase space R 2 . We introduce the "symbol" We will call H = C 2 the fast Hilbert space. The space of "slow Hilbert" is L 2 (R) and corresponds to the quantification of the phase space T * R of "slow variables" x, ξ and replace them by quantum operators. Let ϵ > 0, the "adiabatic parameter" and set where Id : , andx is the multiplication operator x in L 2 (R x ), see Section A for more details.
Remark 3.1. In [17,18] it is shown how this normal form gives a micro-local description of the interaction between the fast vibration motion and the slow rotational motion of the molecule of Figure 3.1. In few words, (x, ξ) are local coordinates on the sphere S 2 of rotation in a vicinity of a point where two spectral bands have a contact, and the C 2 space describes the quantum dynamics of the fast vibrations by restricting to an effective two level problem.

Spectral index N
In the following Theorem, (φ n ) n∈N is the orthonormal basis of Hermite functions of L 2 (R) defined by the Gaussian function and with the operators (so called annihilation and creation operators from quantum optics) eigenvalue transiting upwards, for µ increasing. See figure 3.2.
Remark 3.3. It appears in (3.7) that √ ϵ is a natural parameter of "scaling". See Section 2.4 for a discussion.
For the moment we can not say that (3.8) is a result of topology. For N to be recognized as a "topological index", it would be necessary for this model to belong to a set of models and to show that this number N = +1 is model independent (robust by continuous perturbation within this set). This is done in Section 2.
• Proof. We will see from (3.12) that the operatorĤ µ is elliptic hence 5Ĥ µ has discrete spectrum that we will determine now by different (but similar) methods.
Method 2: This second method explicitly uses a "symmetry" of the problem. We first calculate the spectrum ofĤ 2 µ and then we diagonalizeĤ µ in the eigenspace obtained. Observe that and We havê We deduce that the spectrum ofĤ 2 µ consists of eigenvalues λ n = ω 2 n = (2ϵn + µ 2 ) , n ≥ 0, and the associated eigenspace E n is It remains to diagonalizeĤ µ in each space E n . For n = 0, we observed that and for n ≥ 1, So in the basis of E n ,Ĥ µ is represented by the matrix whose eigenvalues and eigenvectors are ω ± n = ± µ 2 + 2ϵn, and eigenvectors 6

Topological Chern Index C
We can first consult the section B which introduces in simple terms the notion of topology of a complex vector bundle of rank 1 on the sphere S 2 .
There is therefore a degeneracy ω i.e. on the unit sphere in the parameter space, the eigenspace F − (µ, x, ξ) ⊂ C 2 associated with the eigenvalue ω − defines a complex vector bundle of rank 1, denoted F − . Its isomorphism class is characterized by the topological Chern index C (F − ) = +1.
• Proof. We will calculate the index C by two equivalent methods, see section B.
A unit vector v (θ, φ) in the fiber F − over the unit sphere S 2 (except at points where (1 + cos θ) = 0) is given by We will use the curvature integral formula (B.10) that gives with the curvature two form sin θdθ = 1

Conclusion on the model (3.2)
In the model defined by (3.2), we observe from the symbol, a vector bundle F − whose index of Chern is C (F − ) = +1 and we observe that there is N = +1 level transiting (upwards) in the spectrum of the operator. We see in Section 2, Theorem 2.7, that this equality is a special case of a more general result, called the index formula, valid for a continuous family of symbols and for spaces and bundles of larger dimensions. Another equivalent formulation given in [17,19,18] in a more general context: for |µ| ≫ 1, there are two groups of levels j = −, + in the spectrum ofĤ µ . When changing µ = −∞ → +∞ each group has a variation ∆N j ∈ Z of the number of levels. We have the formula where C j is the Chern index of the bundle F j → S 2 .

Spectral flow and index formula for oceanic equatorial waves
In this Section we present the model of Matsuno (1966) [30] for equatorial waves and the topological interpretation given by P. Delplace, J. B. Marston, and A. Venaille in [8].

Matsuno's model
We first present the physical meaning of the Matsuno's model [30]. See also this Document, [38].
The shallow water model: See also Shallow_water_equations on wikipedia.. Let x = (x 1 , x 2 ) ∈ R 2 be local coordinates on the horizontal plane near the equator. x 1 is the longitude and x 2 the latitude. The function (h (x, t) + H) ∈ R with H > 0 represents the depth of water (or of a layer of hot water) at position x and time t ∈ R. The vector u (x, t) = (u 1 (x, t) , u 2 (x, t)) ∈ R 2 represents the (horizontal) velocity of this water. Water is submitted to gravity (g = 9.81 m/s 2 is the g-force) and since the earth is rotating with frequency Ω, there is also an effective Coriolis force. The Navier-Stokes equations with shallow water assumptions give Then (4.1) at first order give the following linear equations With c = √ gH and the change of variables we obtain the dimensionless equations, written without ′ (equivalently we put H = 1, g = 1, β = 1): We will write Then Since the coefficients do not depend on x 1 one can assume the Fourier mode in x 1 : with Fourier variable µ ∈ R and ψ ∈ L 2 R 2 x 2 ,t ⊗ C 3 . In other words, µ is the spatial frequency in x 1 (and λ 1 = 2π µ is the wave length).
For simplicity we replace (x 2 , ξ 2 ) by (x, ξ). This gives the Matsuno model: and its symbol

Spectral index N
The following proposition describes the spectrum of the operatorĤ µ with respect to the µ parameter.
Remarks on the physics of equatorial waves: (from oral explanations by Antoine Venaille).
• The Matsuno model applies either to the ocean or the atmosphere. It can for instance describe the dynamics of the upper oceanic layer called the thermocline (1 km depth), above the abyss (4 km). It can also describe the dynamics of the troposphere (10 km) below the stratosphere (50 km).
• The Matsuno model applies to the ocean (warm water layer of thickness ∼ 1 km) or to the atmosphere (boundary layer between troposphere and stratosphere around 10 km).
• The El Nino phenomenon in the atmosphere-ocean climate system is triggered by a trapped oceanic Kelvin wave propagating across the Pacific ocean. It is symmetric in x 2 , of wavelength λ 1 = 2π/µ and propagates towards Peru. More precisely El Nino is a phenomena that couple ocean and atmosphere. The Kelvin oceanic mode is an essential ingredient for the apparition of high temperature anomalies on the Peru coast and has global consequences.
• From satellites, Yanai modes can sometimes be observed in the form of regular cloud pattern asymmetric with respect to the equator. These clouds reflect the patterns of vertical velocity fields, related to horizontal temperature anomalies. Proof. We will see from (4.7) that the operatorĤ µ is elliptic. So 8Ĥ µ has discrete spectrum. 8 We can do without this argument by noticing at the end of the computation that the found eigenvectors Any vector ψ ∈ L 2 (R) ⊗ C 3 can be written ψ = n≥0   a n φ n b n φ n c n φ n   , a n , b n , c n ∈ C.
We introduce a = 1 We havê We introduce s n := a n + b n , d n := b n − a n ⇔ a n = 1 2 We consider different cases.
form a basis of the Hilbert space.
There is a solution with s 0 = 1, called "Kelvin Wave".

Topological Chern index C
We can first consult the section B which introduces the notion of topology of a complex vector bundle of rank 1 on the sphere S 2 .
Computation of C 1 Let u 0 =   0 0 1   ∈ C 3 a fixed vector and that defines a global section of the bundle F 1 . We consider the sphere S 2 = {(µ, x, ξ) , r = 1}.
Computation of C 3 Let we use 3 j=1 C j = 0 (because the bundle C 3 → S 2 is trivial) giving directly C 3 = −2, we calculate C 3 as we did for C 1 :
A unit vector v (θ, φ) in the fiber F 1 over the unit sphere S 2 is given by (4.13) We use the curvature integral formula (B.10) that gives with the curvature two form We get

Conclusion on the model (3.2)
Formulation given in [17,19,18] in a more general context: for |µ| ≫ 1, there are three groups of levels j = 1, 2, 3 in the spectrum ofĤ µ . When changing µ = −∞ → +∞ each group has a variation ∆N j ∈ Z of the number of levels. We have the formula where C j is the Chern index of the bundle F j → S 2 . Another possible formulation: In the model defined by (3.2), one observes from the symbol, a vector bundle F 1 (or F 1 ⊕ F 2 ) whose index of Chern is C = +2 and we observe that there is N = +2 levels that transits (upwards) in the spectrum of the operator. We see in Section 2, Theorem 2.7, that this equality N = C is a special case of a more general result, called the index formula, valid for a continuous family of symbols and for spaces and bundles of larger dimensions.
A Quantization, pseudo-differential-operators, semi-classical analysis on R 2d
We denote x ∈ R n the "position" and ξ ∈ R n its dual variable, called "momentum". Let ϵ > 0 be a small parameter called semi-classical parameter.
Definition A.1. If a (x, ξ) ∈ S (R n × R n ; C) is a function on phase space T * R n = R 2n called symbol, we associate a pseudo-differential operator (PDO) denoted a = Op ϵ (a) defined on a function ψ ∈ S (R n ) by The operation Op ϵ : a →â = Op ϵ (a) that gives an operatorâ from a symbol a is called Weyl quantization.
Remark A.2. For example, • For a function V (x) (function of x only) we get that Op ϵ (V (x)) = V (x), is the multiplication operator by V . For examplex j = Op ϵ (x j ) is called the position operator.

A.2 Algebra of operators PDO
The following proposition shows that the product of two PDO is a PDO Proposition A.3.

A.3 Classes of symbols
The relations of proposition A.3 are a little bit formal. In order to make them useful, one has to control the remainders in terms of operator norm. For this we need to make some assumption on the symbols that express their "slow variation at the Plank scale dxdξ ∼ ϵ» (i.e. uncertainty principle). We call class of symbol the set of symbols that forms an algebra for the operator of composition ⋆. For example, the following classes of symbols have been introduced by Hörmander [26]. Let M be a smooth compact manifold. For x ∈ R n , we denote ⟨x⟩ := 1 + |x| 2 1/2 ∈ R + , called the Japanese bracket.
For example on a chart, Remark A.6. The geometric meaning of Definition A.5 may be not very clear a priori. Hörmander improved the geometrical meaning in [25,26] by introducing an associated metric on phase space T * M . See also [31], [16].
We will give precise definitions in Section B.3. We begin in Section B.1 and B.2 by a description of vector bundles based on examples and sufficient to understand the case of dimension n = 1 used in this paper.
A complex (or real) vector bundle F → B of rank r is a collection of complex (or real) vector spaces F x of dimension r, called fiber, and continuously parametrized by points x on a manifold B, called "base space". Locally over U ⊂ B, F is isomorphic to a direct product U × C r . The simplest example is the case where the base space is the circle B = S 1 and the rank is r = 1, i.e. each fiber is isomorphic (as a vector space) to the real line R.
One can easily imagine two examples of real fiber space of rank 1 on S 1 : • The trivial bundle S 1 × R that we obtain from the trivial bundle [0, 1] × R on the segment x ∈ [0, 1] (i.e. direct product) and identifying the points (0, t) ∼ (1, t), for all t ∈ R.
• The Moebius bundle, which is obtained from the bundle [0, 1] × R on the segment The Moebius bundle is not isomorphic to the trivial bundle. One way to justify this is that in the case of the trivial bundle, the complement of the null section (s (x) = 0, ∀x) has two connected components, whereas for the bundle of Moebius, the complement has only one component. (Make a paper construction that is cut with scissors according to s (x) = 0 to observe this). Proof. Starting from any bundle F → S 1 of rank 1, we cut the base space S 1 at a point, and we are left with the bundle [0, 1] × R over x ∈ [0, 1]. To reconstruct the initial bundle F , there are two possibilities: for all t ∈ R, identify (0, t) ∼ (1, t), or (0, t) ∼ (1, −t), which gives the trivial or Moebius bundle respectively.
• the Stiefel-Whitney index SW = 0, 1 gives the number of half turns that the fibers make above the base space S 1 . The case SW = 2 (one full turn) is isomorphic to the trivial bundle. We therefore agree that the index SW ∈ Z/ (2Z), i.e. SW is an integer modulo 2.
It is interesting to have the additive structure on the SW indices (1 + 1 = 0 for example).
•Note that in the space R 3 , a ribbon making a turn, i.e. SW = 2, can not be deformed continuously towards the trivial bundle. 10 . This restriction is due to the embedding in the space R 3 (in R 4 , this would be possible), and is not an intrinsic property of the bundle that is nevertheless trivial.  x 1 Base space : Fiber space F 10 Because if we cut this ribbon on the section s = 0, we obtain two ribbons interlaced, whereas the same cut for a trivial ribbon gives two separate ribbons We call zeros of the section s the points x ∈ B such that s (x) = 0. Let us first consider the very simple and instructive case of a real bundle of rank 1 on S 1 . A section is locally like a real value numerical function, so generically, it vanishes transversely at isolated points. Note that "generic" means "except for exceptional case". The following figure shows that we have the following result: Theorem B.4. If F → S 1 is a real bundle of rank 1 on S 1 , and s is a "generic" section, then the topological index SW (F ) is given by where σ s (x) = 1 for a generic zero of section s. The sum is obtained modulo 2, and so SW (F ) ∈ Z 2 = {0, 1}. The result is independent of the chosen section s. We proceed similarly to the previous Section B.1.

B.2.1 Construction of a complex vector bundle of rank 1 on S 2
Let's first see how to build a complex fiber space of rank 1 over S 2 . We cut the sphere S 2 along the equator S 1 , obtaining two hemispheres H 1 and H 2 . We get two trivial bundles F 1 = H 1 × C and F 2 = H 2 × C on each hemisphere. To construct a bundle on S 2 , it is enough to decide how to "connect" or "identify" the fibers of F 1 above the equator with those of F 2 . Note θ ∈ S 1 the angle 11 (longitude) that characterizes a point on the equator. Note φ (θ) ∈ S 1 the angle which means that the fiber F 2 (θ) is identified to the fiber F 1 (θ) after a rotation of angle φ (θ): a v ∈ F 1 (θ) ≡ C is identified with the e iφ(θ) v ∈ F 2 (θ). After gluing that way the two hemispheres and the fibers above the equator, we obtain a complex vector bundle F → S 2 of rank 1. Thus the bundle F that we have just built is defined by its clutching function on the equator It is a continuous and periodic function so: with the integer C ∈ Z that represents the number of revolutions that φ makes when θ goes around. We call C the degree of the application φ : S 1 → S 1 . It is clear that two functions φ, φ ′ are homotopic if and only if they have the same degree C = C ′ , and therefore the bundles F and F ′ are isomorphic if and only if C = C ′ .
' v 2 clutching function Theorem B.5. Any complex fiber bundle F → S 2 of rank 1 is isomorphic to a bundle constructed as above with a clutching function φ on the equator. Its topology is characterized by an integer C ∈ Z called (1st) Chern index given by C = deg (φ).
In other words the equivalence class of rank 1 complex vector bundle on S 2 is We must show that every bundle F is isomorphic to a bundle constructed as above.
Starting from a given bundle F , we cut the base space S 2 along the equator denoted S 1 to obtain two bundles F 1 → H 1 and F 2 → H 2 . Each of these bundles is trivial because[24, corrollaire 1.8 p.21] the base spaces are disks (contractile spaces). The bundle F is thus defined by its clutching function above the equator S 1 , φ : S 1 → S 1 .
Consider the example of the tangent bundle T S 2 of the sphere. T S 2 can be identified with a complex bundle of rank 1 because S 2 is orientable.
Theorem B.6. The tangent bundle T S 2 has Chern index and is therefore non trivial.
Proof. We will calculate the degree C of its recollection function defined by Eq. (B.15). We proceed as in the proof above. We trivialize the bundle above H 1 , and H 2 , and we deduce the degree C of the gluing function. See figure that represents the two hemispheres seen from above and below with a vector field on each. We find C = +2.
Remark B.7. The trivial bundle S 2 × C has the Chern index C = 0.

B.2.2 Topology of the rank 1 vector bundle on S 2 from the zeros of a section
There is a result analogous to Thm. B.4 for a complex bundle F → S 2 of rank 1 on S 2 . Before establishing it, let us notice that a section s of such a bundle is locally like a function with two variables and with values in C, so generically, it vanishes transversely at isolated points. If θ ∈ S 1 parameterizes a small circle of points x θ around a zero x ∈ S 2 of s,then by hypothesis, the value of the section s (x θ ) ∈ F x θ ≡ C is non-zero for all x θ , and we write φ ∈ S 1 his argument. For each zero x of the section s is therefore associated an application φ : θ → φ (θ) whose degree, also called index of the zero (defined by Eq. (B.15)), will be noted σ s (x) ∈ Z. Generically, σ s (x) = ±1. (Note that the sign of σ s (x) depends on the chosen orientation of the base space and the fiber. In the case of the tangent bundle on S 2 , these two orientations are not independent, and the result σ s (x) becomes independent of the choice of orientation).
Theorem B.8. If F → S 2 is a complex bundle of rank 1 on S 2 , and s is a "generic" section, then the topological index of Chern C (F ) is given by where σ s (x) = ±1 characterizes the degree of zero. The result is independent of the chosen section s .
Proof. In the proof of the theorem B.5, we have constructed sections v 1 , v 2 for the respectively bundles F → H 1 , F → H 2 , that never vanish. If we modify these sections v 1 , v 2 to make them coincide on the equator for the purpose of constructing a global section s of the bundle F → S 2 , we can get do this except in points isolated, which will be the zeros of s, and one realizes that the sum of the indices will be equal to the degree of the clutching function φ therefore equal to C (F ).
Example of the bundle T S 2 The following figure shows a vector field on the S 2 sphere. It is a global section of the tangent bundle. This vector field has two zeros with indices +1 each. Thus we find C (T S 2 ) = +2, i.e. Eq. (B.3). x global section Remark B.9. If we want to give an explicit computation we need an explicit global section (or vector field on T S 2 ). We can take the fixed vector in R 3 : V = (0, 0, 1) oriented along the z axis. Then for a given point x ∈ S 2 we choose: where P x : R 3 → T x S 2 is the orthogonal projector given by P x = Id − |x⟩⟨x|.⟩. We get The vector field s (x) vanishes at the north and south pole. At distance ϵ of north pole (0, 0, 1), we use local oriented coordinates (x 1 , x 2 ) ≡ ϵe iθ and get s ( The map e iθ ∈ S 1 → −e iθ ∈ S 1 has degree 1 hence the zero has index σ = +1. At distance ϵ of south pole (0, 0, −1), we use local oriented coordinates (x 2 , x 1 ) ≡ ϵe iθ and get s (x) = (x 1 , x 2 , 0) + O (ϵ 2 ) = e iθ + O (ϵ 2 ). The map e iθ ∈ S 1 → e iθ ∈ S 1 has degree 1 hence the zero has again index σ = +1. Formula (B.4) gives C T S 2 = +1 + 1 = +2.

B.2.3
Topology of the rank 1 vector bundle on S 2 from a curvature integral in differential geometry Let F → S 2 be a complex vector bundle of rank 1 over S 2 . Let us assume 12 that there exists a fixed vector space C d such that for every x ∈ S 2 , the fiber F x ⊂ C d is a linear subspace of C d for some d ≥ 1. For every point x ∈ S 2 , let us denote P x : C d → C d the orthogonal projector onto F x . Then if s ∈ C ∞ (S 2 ; F ) is a smooth section we can consider s ∈ C ∞ S 2 ; C d as a d multi-components function on S 2 . If V ∈ T x S 2 is a tangent vector at point x ∈ S 2 , the derivative V (s) ∈ C d can be projected onto F x . We get called the covariant derivative of s along V at point x. It measures the variations of s within the fibers F . Since V (s) = ds (V ) where ds means the differential 13 , we usually write Ds := P ds for the covariant derivative or Levi-Civita connection (in differential geometry, P ds ∈ C ∞ (S 2 ; Λ 1 ⊗ F ) is a one form valued in F ). Suppose that U ⊂ S 2 and for every point x ∈ U one has v (x) ∈ F x a unitary vector that depends smoothly on x ∈ U . This is called a local unitary trivialization of F → U (as in the proof of Theorem B.6). Since the fiber F can expressed in this basis with one complex component: 13 In local coordinates x = (x 1 , x 2 ) ∈ R 2 on S 2 , if f (x 1 , x 2 ) is a function, then its differential is written and a tangent vector is written V = k V k ∂ ∂x k . Then since df (V ) = V (f ) gives in particular for the function x k that dx k ∂ ∂x l = ∂x k ∂x l = δ k=l , we get that df (V ) = k ∂f ∂x k V k . 14 A is imaginary valued from the fact that ⟨v|v⟩ = 1 hence 0 = d⟨v|v⟩ = ⟨Dv|v⟩ + ⟨v|Dv⟩ = 2Re (⟨v|Av⟩) = 2Re (A) . 15 If s ∈ C S 2 ; F is an arbitrary section, then locally one can write shows that the components of the covariant derivatice Ds with respect to the unitary trivialization v (x) and local coordinates (x k ) k on U are ∂ϕ ∂x k + iA k ϕ k . In quantum physics books it is common to see the expression ∂ϕ ∂x k + iA k ϕ k for a definition of the "covariant derivative" or "minimal coupling", e.g. [27, p.31].
Let Ω := dA (B.8) be the two form 16 called the curvature of the connection.
Lemma B.10. Let F → S 2 be a rank 1 complex vector bundle with F x ⊂ C d . Let v (x) ∈ F x a given local unitary trivialization and Dv = Av with A the connection one form and Ω the curvature two form. Then and Ω does not depend on the trivialization, hence is globally defined on S 2 . Finally the topological Chern index C defined in (B.2) is given by the curvature integral Proof. The orthogonal projector is given by hence the covariant derivative is given by Ds = P ds = P x = |v x ⟩⟨v x |ds⟩ and since by definition Dv = Av we get A = ⟨v|dv⟩ and The second term vanishes since is a symmetric array and (dx l ∧ dx k ) k,l is antisymmetric. If we replace v by another trivialization v ′ (x) = e iα(x) v (x) (this is called a Gauge transformation) then is changed but As in Section B.2.1, let H 1 , H 2 be the north and south hemispheres of S 2 and suppose that for every point x ∈ H 1 , v 1 (x) ∈ F x is a unitary vector that depends smoothly on x, i.e. v 1 is a trivialization of F → H 1 . Suppose that v 2 is a trivialization of F → H 2 (as in the proof of Theorem B.6). Let x ≡ (θ, φ) denotes the spherical coordinates on S 2 . For a given 0 ≤ θ ≤ π 2 on Hemisphere H 1 , let γ θ : φ ∈ [0, 2π] → γ θ (φ) ∈ S 2 be the closed path. Let ψ (1)

under the condition of zero covariant derivative
is called the holonomy of the connection on the closed path γ θ and also called Berry's phase after the paper of M. Berry [3] that shows its natural manifestation in quantum mechanics, see also [15]. We can do the same on the south hemisphere H 2 with v 2 and angles α (2) θ , giving at θ = 0, α with opposite signe because the orientation of γ 0 is reversed. In particular, on the equator θ = 0 that belongs to both Hemisphere, we have for every φ that and by definition of Chern index C, , and since the parallel transport preserves the angles, ψ 1 2π Remark B.11. Formula (B.10) is a special case of a more general Chern-Weil formula formula (B.19) given below for a general vector bundle F → S 2n of rank r.
Example B.12. For the special case of the tangent bundle T S 2 , with fiber T x S 2 ⊂ R 3 , if iΩ is the (2 form) Gauss curvature of the sphere (that is, the curvature of the tangent bundle T S 2 , which is the solid angle), the Gauss-Bonnet formula gives: We say that φ i are trivialization functions, and f ij are transition functions.
Proposition B.14. The transition functions satisfy the cocycle conditions: Conversely functions f ij with cocycle conditions, define a unique vector bundle. 15. Two vector bundles (F, π, B) and (F ′ , π ′ , B) (with same base B) are isomorphic if there exists h : F → F ′ which preserves the fibers and such that h : x is an isomorphism of linear spaces.
We write Vect r C (B) for the isomorphism class of complex vector bundles of rank r over B.
Proposition B.16. Two vector bundles F and F ′ are isomorphic if and only if there exists functions h i : U i → GL (n, C) such that

B.3.2 Complex Vector bundles over spheres S k
Reference: Hatcher [24] p. 22. We treat the case where the base space is a sphere The sphere S k = D k 1 D k 2 can be decomposed in two disks (or hemispheres), the north hemisphere D k 1 where x k+1 ≥ 0 and the south hemisphere D k 2 where x k+1 ≤ 0. The common set is the equator S k−1 = D k 1 ∩D k 2 = x ∈ R k+1 , x k+1 = 0 which is also a sphere S k−1 . So a vector bundle is described by the transition function at the equator: f 21 : S k−1 → GL (r, C), which is called the clutching function. Let us denote [f 21 ] the homotopy class of the map f 21 . The set of homotopy classes is S k−1 , GL (r, C) ≡ S k−1 , U (r) =: π k−1 (U (r)) is called homotopy group of U (r). . In other words the group of equivalence classes of vector bundles coincide with the homotopy groups: Vect r C S k ≡ π k−1 (U (r)) .
π n (S m ) π 1 π 2 π 3 π 4 π 5 π 6 π 7 Table 1: Homotopy groups of the spheres π m (S n ) Homotopy groups of spheres The groups Vect r S k = π k−1 (U (r)) can be obtained from homotopy groups of the spheres π m (S n ) from the fact that U (r) /U (r − 1) ≡ S 2r−1 . (B.14) This is obtained by observing that the unit sphere in C r is S 2r−1 and thus, for f ∈ U (r) and e r = (0, . . . , 0, 1) ∈ C r we have f (e r ) ∈ S 2r−1 ⊂ C r that characterizes f up to U (r − 1), i.e. its action on C r−1 . See table 1. See Hatcher's book.
We have π n (S n ) = Z which is the degree and is computed as follows. which is independent of the choice of the generic point y ∈ S m . In the case f : S 1 → S 1 , the degree deg (f ) is also called «winding number of f ».
For m < n we have π m (S n ) = 0, because the image of f : S m → S n is not onto and therefore gives f : R m → S n which can be retracted to a point because R m is contractible. For m > n, the homotopy groups of the spheres π n (S m ) are quite complicated and are not all known.

Observations on table 2
• Vect 2 (S 5 ) ≡ Z 2 = {0, 1}: means that there is only one class of non trivial bundles of rank 2 over S 5 .
• Vect 1 S k≥2 ≡ 0 means that all vector bundles of rank 1 over S k≥2 are all trivial.
A remarkable observation is the following theorem: (K-theory 17 ) Theorem B.19. «Bott periodicity Theorem 1959». If 2r ≥ k then Vect r S k is independent on r. We denoteK S k := Vect r S k called group of K-theory. Moreover there is the periodicity property: For the proof, see [24].
B.3.3 Topological Chern index C of a complex vector bundle F → S 2n of rank r ≥ n From the table 2, if F → S 2n is a complex vector bundle of rank r, with r ≥ n, then its isomorphism class is characterized by an integer C ∈ Z called topological Chern index.
Here is an explicit expression for C. The equivalence class of the bundle F is characterized by the homotopy class of the clutching function at the equator g = f 21 , g : S 2n−1 → U (r) .
(B. 16) which is the transition function from north hemisphere to south hemisphere.
Clutching function S 2n g : S 2n−1 → U (r) If r > n, we can continuously deform g so that ∀x ∈ S 2n−1 , g x (e r ) = e r , where (e 1 , . . . e r ) is the canonical basis of C r . Cf [5, Section III.1.B, p.271]. Then g restricted to C r−1 ⊂ C r gives a function g : S 2n−1 → U (r − 1). By iteration we get the case r = n with a clutching function g : S 2n−1 → U (n). Then using g we define the function is an integer C ∈ Z (not only a rational number!) and characterizes the topology of F . Namely, if F → S 2n and F ′ → S 2n are fiber bundles of same rank r ≥ n with the same index C then F and F ′ are isomorphic.
Remark B.21. If the vector bundle F → S 2n has a (arbitrary) connection, the Chern-Weil theory permits to express the topological index C from the curvature Ω of the connection, considered as a imaginary valued 2-form on S 2n as follows. We first define Ch (F ) called the Chern Character which is a differential form on S 2n :

B.3.4 A normal form bundle F n → S 2n−1 in each K-isomorphism class
We have seen in Theorem B.19 that for r ≥ n then the isomorphism class of complex vector bundles of rank r over S 2n−1 is Vect r S k ≡ Z. In this section we provide and explicit model for the generator in this class, i.e. giving the topological index C = +1 ∈ Vect r S k ≡ Z.

B.3.6 The index formula on Euclidean space of Fedosov-Hörmander
For the previous canonical vector bundle F n → S 2n with topological index C and clutching function g n we have observed that and that C = +1, meaning that this vector bundle F n is the generator of its equivalence class in K-theory. Since both indices Ind (Op 1 (g n )) and C are additive under direct sum of vector bundles in K-theory, we deduce the next Theorem showing that (B.26) is generally true. We consider F → S 2n , a general complex vector bundle of rank r with topological index C ∈ Z as defined in (B.18) and clutching function g : S 2n−1 → U (r) on the equator S 2n−1 as defined in (B.16). We extend g from S 2n−1 to 1-homogeneous function on R 2n \ {0} by g (z) := |z| g z |z| and consider this extension as a symbol g : R 2n \ {0} → GL (r). Quantization (A.1) gives an operator Op 1 (g).