Scalar, Vector and Tensor Harmonics on the Flat Compact Orientable Three-Manifolds
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Scalar, Vector and Tensor Harmonicson the Flat Compact OrientableThree-Manifolds
Zhi-Peng Peng, a Lee Lindblom, b , c and Fan Zhang, a , d , a Gravitational Wave and Cosmology Laboratory, Department of Astronomy,Beijing Normal University, Beijing 100875, China b Center for Astrophysics and Space Sciences, University of California at San Diego,La Jolla, CA 92093, USA c Center for Computational Mathematics, University of California at San Diego,La Jolla, CA 92093, USA d Department of Physics and Astronomy, West Virginia University,Morgantown, WV 26506, USAE-mail: [email protected], [email protected], [email protected]
Abstract.
Observations suggest that our universe is spatially flat on the largest observable scales.Exactly six di ff erent compact orientable three-dimensional manifolds admit flat metrics. These sixmanifolds are therefore the most natural choices for building cosmological models based on thepresent observations. This paper briefly describes these six manifolds and the harmonic basis func-tions previously developed for representing arbitrary scalar fields on them. The principal focus of thispaper is the development of new harmonics for representing arbitrary vector and second-rank tensorfields on these manifolds. These new harmonics are designed to be useful tools for analyzing thedynamics of electromagnetic and gravitational fields on these spaces. Corresponding author. a r X i v : . [ g r- q c ] N ov ontents Current observations of the cosmic microwave radiation, together with observations of lower-redshiftdistance indicators, show that the large-scale average density of the universe di ff ers by no more than0.4% (the present observational error estimate) from the critical value that implies the geometry ofour universe is spatially flat on these largest observable scales [1]. There are eighteen di ff erent three-dimensional manifolds that admit flat metrics [2–4], so these are the most natural manifolds on whichto construct realistic cosmological models. Ten of these flat manifolds are orientable, while eightare non-orientable. Spacetimes having non-orientable spatial slices are not parallelizable (i.e. theydo not admit collections of smooth non-vanishing linearly independent vector fields), and (conse-quently) such manifolds do not admit spinor structures [5, 6]. Representations of fermions dependon the existence of these spinor structures, so unless (or perhaps until) new theoretical approachesfor defining spinors are developed, the non-orientable three-manifolds appear to be less physicallyrelevant than the orientable ones. A purely emperical approach to the analysis of cosmological ob-servations could justify the inclusion of the non-orientable cases anyway. But given the theoreticalspinor structure argument for excluding them, we have chosen to limit our analysis here to the ori-entable cases. Of the ten orientable flat three-manifolds, six are compact while the remaining fourare non-compact. Current observations do not allow us to see the entire universe, so we have no wayof knowing whether or not our universe is spatially compact. For computational convenience, wechoose to limit consideration here to models having compact spatial slices, i.e. models with finitespatial volumes.Each of the six compact orientable three-dimensional manifolds that admits a flat metric canbe obtained as a quotient E / Γ of three-dimensional Euclidean space E by an isometry group Γ ofsymmetries of E . The classification of these spaces has long been known [2–4]. We use the notation E , E , ..., E to refer to these spaces. The group action that defines each of these spaces can bethought of as a particular representation of E as a periodic lattice of polytopes. The individual flatcompact manifolds can be thought of as one of these polytopes with identifications between its faces.Figure 1 of Ref. [7] illustrates these polytope-with-identifications representations of these manifolds.The spaces E , E and E are based on rectangular lattice representations of E . E is the simplethree-torus, obtained by identifying opposite faces of a rectangular solid. E and E are obtainedby identifying the opposite x and y faces of a rectangular sold, but twisting by π before identifyingopposite z faces for the half-turn space E , or by π/ E . E and E are– 1 –btained from representations of E as a lattice of hexagonal prisms. These hexagonal prisms havesix rectangular faces, and two hexagonal faces. The opposing rectangular faces of these prisms areidentified in E and E , while the hexagonal faces are twisted by 2 π/ E , orby π/ E . The Hantzsche-Wendt space, E , is based on a representation of E by a lattice of rhombic dodecahedrons. The space E is formed by a particular identification of therhombic faces of one of these dodecahedrons. Explicit descriptions of the symmetries used to createeach of the spaces, E , E , ..., E , are given in Sec. 2 as part of our discussion of the scalar harmonicson these spaces.We use the term harmonics in this paper to refer to the eigenfunctions of the covariant Laplaceoperator. These eigenfunctions form a complete set of smooth functions on these compact orientablemanifolds, and can therefore be used as a basis for representing arbitrary square integrable functionson them. These harmonics can be thought of as generalizations of the Fourier basis functions used toconstruct representations of functions: f ( x ) = (cid:80) k f k e i k · x . Scalar harmonics have been developed foreach of the six flat compact orientable three-dimensional manifolds, and these harmonics have beenused to model the temperature variations in the cosmic microwave background radiation that would beobserved on these spaces [7]. Scalar harmonics, however, are not adequate to model the full dynamicsof the gravitational or the electromagnetic fields. For example, an analysis of the polarization prop-erties of the cosmic microwave background radiation would require representations of the full vectorstructure of the electromagnetic field. A number of studies have been carried out on the vector andtensor harmonics on the (less physically relevant) manifolds with the topology of the three-sphere, S [8–14]. But little attention has been paid to such matters on the less-familiar three-manifolds thatadmit flat metrics. There has been some work on constructing second-rank tensor basis functions for E in the context of studying the e ff ects of inhomogeneous initial conditions on inflation [15], or thepossibility of gravitational wave turbulence in the early universe [16]. Here we significantly gener-alize these studies by constructing complete vector harmonic and second-rank tensor harmonic basisfunctions on all six flat compact orientable three-manifolds. To facilitate the analysis of the dynam-ics of the gravitational and electromagnetic fields on these manifolds, we have organized these newharmonics into subsets that maximize the number of classes having vanishing divergence and trace.These new harmonics were also constructed to satisfy nice orthonormality conditions that make iteasy to represent arbitrary scalar and second-rank tensor fields on these manifolds.The remainder of this paper is organized as follows. Explicit expressions for the symmetriesused to construct each of the flat compact orientable three-manifolds, E , E , ..., E , are given inSec. 2, along with explicit expressions for the (suitably re-normalized) scalar harmonics developedin Ref. [7]. The analogous vector and anti-symmetric second-rank tensor harmonics are developedin Sec. 3. Two classes of these new vector harmonics are divergence free, so they provide a naturalway to represent the transverse parts of dynamical electromagnetic fields. Symmetric second-ranktensor harmonics are developed in Sec. 4. These new tensor harmonics include five classes of trace-free harmonics, two of which are also divergence-free. These new tensor harmonics are well suitedtherefore for representing the transverse-traceless parts (i.e. the dynamical gravitational wave parts)of the gravitational fields on these flat compact orientable three-manifolds. Section 5 contains abrief summary and discussion of the new results. Several useful technical lemmas needed in theconstruction of the new vector and tensor harmonics are given in an Appendix. The scalar harmonics are defined here to be eigenfunctions of the covariant Laplace operator: ∇ a ∇ a Y = − κ Y . (2.1)– 2 –e use the notation Y [ E j ] k to denote the harmonics on the manifold E j (for j = , , ..., k = k a = ( k , k , k ) are parameters that identify a particular harmonic. On the three-torus, E , theseharmonics are (up to normalizations) just the Fourier basis functions: Y [ E ] k = e i k · x √ L L L . (2.2)where L , L , and L , are the lengths of the three principal axes of E . The parameters k are chosento ensure that the harmonics have the appropriate periodicities on E to make them smooth functionson E : k = k a = π (cid:32) n L , n L , n L (cid:33) . (2.3)This requires the n a (for a = , ,
3) to be integers, n a ∈ Z . The eigenvalues for these solutions toEq. (2.1) are given by κ = k a k a = k + k + k . (2.4)The normalization has been chosen in Eq. (2.2) to ensure that these harmonics satisfy the orthonor-mality conditions, (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / Y [ E ] k Y [ E ] ∗ k (cid:48) dx dy dz = δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (2.5)The harmonics for the remaining flat manifolds, E , ..., E , were derived in Ref. [7] by usingthe fact that each scalar harmonic Y k of E / Γ lifts to a Γ − periodic harmonic Y k of E , i.e. to ascalar harmonic of E that is invariant under the action of the isometry group Γ . Finding the scalarharmonics of the flat space E / Γ is equivalent, therefore, to finding the Γ -periodic scalar harmonics of E . Each element of an isometry group Γ of Euclidean space E , can be written as a rotation / reflection M followed by a translation T : i.e. these isometries map points x ∈ E to the points x (cid:48) = M · x + T ,or equivalently in component notation x (cid:48) a = M ab x b + T a . These transformations are isometries sothey preserve the forms of the metric g = g ab = diag(1 , ,
1) and the inverse metric g − = g ab = diag(1 , , g ab = M ca M db g cd and g ab = M ac M bd g cd , and consequently that κ = k a k a = g − ( k , k ) and κ = k a M ac k b M bd g cd = g − ( kM , kM ) for these isometries.The isometry group Γ of the half-turn space, E , consists of pure translations by L , L or L inthe principal directions, plus a compound rotation-translation defined by M [ E ] = − − , T [ E ] = L . (2.6)The scalar harmonics invariant under these transformations are given by [7], Y [ E ] k = √ (cid:104) Y [ E ] k + ( − n Y [ E ] kM [ E ] (cid:105) for (cid:0) n ∈ Z + , n , n ∈ Z (cid:1) or (cid:0) n = , n ∈ Z + , n ∈ Z (cid:1) , (2.7) Y [ E ] (0 , , k ) = Y [ E ] (0 , , k ) for ( n = n = , n ∈ Z ) , (2.8)where the Y [ E ] k are the basic E harmonics given in Eq. (2.2), and k = ( k , k , k ) = π (cid:16) n L , n L , n L (cid:17) . These harmonics have the same eigenvalues as the E harmonics, Eq. (2.4), and satisfy orthonormalityconditions analogous to Eq. (2.5). – 3 –he isometry group of the quarter-turn space, E , consists of pure translations by L , L or L in the principal directions, plus a compound rotation-translation defined by M [ E ] = − , T [ E ] = L . (2.9)These symmetries imply that L = L in the E case. The scalar harmonics invariant under thesetransformations are given by [7], Y [ E ] k = (cid:104) Y [ E ] k + i n Y [ E ] k M [ E ] + i n Y [ E ] k M [ E ] + i n Y [ E ] k M [ E ] (cid:105) for (cid:0) n ∈ Z + , n ∈ Z + ∪ { } , n ∈ Z (cid:1) , (2.10) Y [ E ] (0 , , k ) = Y [ E ] (0 , , k ) for ( n = n = , n ∈ Z ) , (2.11)where the Y [ E ] k are the basic E harmonics given in Eq. (2.2), and k = ( k , k , k ) = π (cid:16) n L , n L , n L (cid:17) . These harmonics have the same eigenvalues as the E harmonics, Eq. (2.4), and satisfy orthonormalityconditions analogous to Eq. (2.5).The spaces E and E are constructed from a hexagonal prism lattice representation of E thatis generated by the four pure-translation symmetries T = L , T = − L √ L , T = − L − √ L , T = L . (2.12)The isometry group of the third-turn space, E , consists of the pure translations given in Eq. (2.12)plus a compound rotation-translation defined by M [ E ] = − − √ √ −
00 0 1 , T [ E ] = L . (2.13)These symmetries imply that L = L in the E case. The scalar harmonics invariant under thesetransformations are given by [7], Y [ E ] k = √ (cid:104) Y [ E ] k + ω n Y [ E ] kM [ E ] + ω n Y [ E ] kM [ E ] (cid:105) for (cid:0) n ∈ Z + , n ∈ Z + ∪ { } , n ∈ Z (cid:1) , (2.14) Y [ E ] (0 , , k ) = Y [ E ] (0 , , k ) for ( n = n = , n ∈ Z ) , (2.15)where the Y [ E ] k are the basic E harmonics given in Eq. (2.2), and ω = e i π/ . To preserve thehexagonal translation symmetry in this case we must also take k = ( k , k , k ) = π (cid:18) − n L , n − n √ L , n L (cid:19) .These harmonics have the same eigenvalues as the E harmonics, Eq. (2.4), and satisfy orthonormalityconditions analogous to Eq. (2.5).The isometry group of the sixth-turn space, E , consists of the pure hexagonal lattice transla-tions given in Eq. (2.12), plus a compound rotation-translation defined by M [ E ] = − √ √
32 12
00 0 1 , T [ E ] = L . (2.16)– 4 –hese symmetries imply that L = L in the E case. The scalar harmonics invariant under thesetransformations are given by [7], Y [ E ] k = √ (cid:104) Y [ E ] k + ω n Y [ E ] kM [ E ] + ω n Y [ E ] k M [ E ] + ω n Y [ E ] k M [ E ] + ω n Y [ E ] k M [ E ] + ω n Y [ E ] kM [ E ] (cid:105) for (cid:0) n ∈ Z + , n ∈ Z + ∪ { } , n < n , n ∈ Z (cid:1) , (2.17) Y [ E ] (0 , , k ) = Y [ E ] (0 , , k ) for ( n = n = , n ∈ Z ) , (2.18)where the Y [ E ] k are the basic E harmonics given in Eq. (2.2), and ω = e i π/ . To preserve thehexagonal translation symmetry in this case we must also take k = ( k , k , k ) = π (cid:18) − n L , n − n √ L , n L (cid:19) .These harmonics have the same eigenvalues as the E harmonics, Eq. (2.4), and satisfy orthonormalityconditions analogous to Eq. (2.5).The isometry group of the Hantzsche-Wendt space, E , consists of pure translations by L , L or L in the principal directions, plus compound rotation-translations defined by M [ E ] = − − , T [ E ] = L L , M [ E ] = − − , T [ E ] = L L , M [ E ] = − − , T [ E ] = L L . (2.19)The scalar harmonics invariant under these transformations are given by [7], Y [ E ] k = (cid:104) Y [ E ] k + ( − n − n Y [ E ] kM [ E ] + ( − n − n Y [ E ] kM [ E ] + ( − n − n Y [ E ] kM [ E ] (cid:105) for (cid:0) n , n ∈ Z + , n ∈ Z (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) , (2.20) Y [ E ] ( k , , = √ (cid:104) Y [ E ] ( k , , + Y [ E ] ( − k , , (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) , (2.21) Y [ E ] (0 , k , = √ (cid:104) Y [ E ] (0 , k , + Y [ E ] (0 , − k , (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) , (2.22) Y [ E ] (0 , , k ) = √ (cid:104) Y [ E ] (0 , , k ) + Y [ E ] (0 , , − k ) (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) , (2.23)where the Y [ E ] k are the basic E harmonics given in Eq. (2.2), and k = ( k , k , k ) = π (cid:16) n L , n L , n L (cid:17) . These harmonics have the same eigenvalues as the E harmonics, Eq. (2.4), and satisfy orthonormalityconditions analogous to Eq. (2.5). This section constructs vector harmonics on the six flat compact orientable three-manifolds. Whilethese harmonics are not unique and can be chosen in a variety of di ff erent way, our goal here is to– 5 –hoose harmonics having three useful properties: First, the vector harmonics constructed here will beeigenfunctions of the covariant Laplace operator: ∇ b ∇ b Y a = − κ Y a . (3.1)This ensures that these harmonics will form a complete basis for the (square integrable) vector fieldson these manifolds. We use the notation Y [ E j ] a ( A ) k (for A = , ,
2) to denote the three linearly inde-pendent classes of vector harmonics that satisfy Eq. (3.1) on the space E j (for j = , , ..., Y [ E j ] a ( A ) k and Y [ E j ] a ( A (cid:48) ) k (cid:48) will be orthogonal under the standard L inner product unless k = k (cid:48) and A = A (cid:48) . Andthird, the vector harmonics constructed here in classes A = A = A = Y [ E j ] a (0) k = κ − ∇ a Y [ E j ] k , (3.2)where ∇ a = g ab ∇ b and κ is the corresponding eigenvalue of the covariant scalar Laplace operator,Eq. (2.4). Since the scalar harmonics Y [ E j ] k are smooth eigenfunctions of the Laplace operator,their gradients are automatically smooth eigenfunctions having the same eigenvalues on these flatmanifolds. The normalization factor in Eq. (3.2) is chosen to ensure that these harmonics satisfy thenice orthonormality conditions (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ab Y [ E j ] a (0) k Y [ E j ] b ∗ (0) k (cid:48) dx dy dz = δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (3.3)The divergences of these class A = ∇ a Y [ E j ] a (0) k = − κ Y [ E j ] k . (3.4)The construction of the class A = A = ff erent approaches that produce vector harmonics satisfying the three usefulproperties listed above. The first approach produces the simplest expressions for the vector harmon-ics, but this approach only works in the space E . A somewhat more general approach can be usedto derive fairly simple expressions for these vector harmonics in the spaces E , ..., E , but not in E .We have also developed an even more general approach capable of constructing vector harmonics inall the E j spaces, but the resulting expressions produced in this way are quite complicated. Here wereport the results of this general approach only for the otherwise intractable E case.The first approach to constructing the needed vector harmonics is based on the fact that anycovariently constant vector field c a is invariant under the pure translation symmetry group of thethree-torus, E . Any vector field of the form c a Y [ E ] k is an eigenfunction of the covariant Laplaceoperator, and therefore a candidate vector harmonic. The choices for three linearly independentvectors c a will define the three classes of vector harmonics for this case. The class A = E have this form, Y [ E ] a (0) k = i κ − g ab k b Y [ E ] k with c a = i κ − g ab k b . All that is needed to complete this simple approach is to choose two additionalconstant vectors to define the class A = A = k a will do. One choice is (cid:96) a = ( − k − k , k − k , k + k ) and m a = g ab (cid:15) bcd k c (cid:96) d , where (cid:15) abc is thecovariantly constant, ∇ a (cid:15) bcd =
0, totally antisymmetric tensor volume element. For convenience,these vectors can be normalized by setting ˆ k a = κ − k a , ˆ (cid:96) a = λ − (cid:96) a , and ˆ m a = κ − λ − m a , where– 6 – = (cid:96) a (cid:96) a = ( k + k ) + ( k − k ) + ( k + k ) . Vector harmonics defined in terms of the orthonormalconstant vectors ˆ k a = g ab ˆ k b , ˆ (cid:96) a = g ab ˆ (cid:96) b , and ˆ m a = g ab ˆ m b are given by Y [ E ] a (0) k = i ˆ k a Y [ E ] k , (3.5) Y [ E ] a (1) k = i ˆ (cid:96) a Y [ E ] k , (3.6) Y [ E ] a (2) k = i ˆ m a Y [ E ] k , (3.7)Special forms for ˆ k a , ˆ (cid:96) a , and ˆ m a are needed when κ = λ =
0. In the κ = k a = (1 , , (cid:96) a = (0 , ,
0) and ˆ m a = (0 , , λ = κ (cid:44) k a = √ (1 , − ,
1) sowe can simply define ˆ (cid:96) a = √ (1 , ,
0) and ˆ m a = √ ( − , , − κ , and satisfy the following orthonormality conditions, (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ab Y [ E ] a ( A ) k Y [ E ] b ∗ ( B ) k (cid:48) dx dy dz = δ AB δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (3.8)The divergences of these vector harmonics on the space E satisfy, ∇ a Y [ E ] a (0) k = − κ Y [ E ] k , (3.9) ∇ a Y [ E ] a (1) k = ∇ a Y [ E ] a (2) k = . (3.10)The divergences of these class A = A = (cid:96) a and ˆ m a arechosen to be orthogonal to ˆ k a .The second approach to constructing vector harmonics uses the fact that the unit vector alongthe z -axis, ˆ z a , is the only unit vector field invariant under all the symmetry groups of the manifolds E , ..., E . Therefore ˆ z a Y [ E j ] k (for j = , ...,
5) is an eigenfunction of the covariant Laplace operator,and can be used in the construction of the vector harmonics on these spaces in much the same waythe constant vectors c a were used in the first approach. We note that ˆ z a ∇ a Y [ E j ] k = i k Y [ E j ] k in allof these cases. The vector harmonics for these manifolds can therefore be taken to be Y [ E j ] a (0) k = κ ∇ a Y [ E j ] k , (3.11) Y [ E j ] a (1) k = κ k √ κ − k (cid:16) κ ˆ z a ˆ z b − k g ab (cid:17) ∇ b Y [ E j ] k , (3.12) Y [ E j ] a (2) k = √ κ − k (cid:15) abc ˆ z b ∇ c Y [ E j ] k , (3.13)so long as κ (cid:44) k (cid:44)
0, and κ (cid:44) k . When κ =
0, the vector harmonics must be spatiallyconstant vector fields. Three linearly independent spatially constant vector fields exist in the space E , so any orthonormal set can be used as κ = E , ..., E the only spatially constant unit vector field is ˆ z a , so it becomes the only κ = k = κ (cid:44) z , i.e. they depend only on x and y . In this case the class A = A = A = Y [ E j ] a (1) k = ˆ z a Y [ E j ] k . Finally if κ = k and κ (cid:44) x and y , i.e. the harmonics depend only on z . Inthis case there is only the single class of vector harmonics given by Eq. (3.11). It is straightforwardto show that the vector harmonics defined in Eqs. (3.11)–(3.13) are eigenfunctions of the covariantLaplace operator with eigenvalue − κ , satisfy orthonormality conditions that are the analogs of thosegiven in Eq. (3.8), and satisfy divergence identities that are the analogs of those given in Eqs. (3.9)and (3.10). – 7 –he third (most general) approach to constructing vector harmonics on these compact orientableflat spaces, E j , is based on the fact that these spaces are quotients, E / Γ , of Euclidean space E and an isometry group Γ . Therefore the problem of finding vector harmonics on the E j spaces isequivalent to finding the Γ -invariant vector harmonics on E . This can be done using the methoddeveloped in Ref. [7] to derive expressions for the scalar harmonics. The Vector Action Lemma andthe
Vector Invariance Lemma described in the Appendix to this paper provide the tools needed toconstruct linear combinations of the E vector harmonics that are invariant under all the elements ofthe symmetry groups Γ . The vector harmonics obtained using this method on the spaces E , ..., E are more complicated than those derived using the first two approaches. So here we report only thevector harmonics Y [ E ] a ( A ) k obtained in this way for the otherwise intractable E case: Y [ E ] a ( A ) k = (cid:104) Y [ E ] a ( A ) k + ( − n − n M [ E ] ab Y [ E ] b ( A ) kM [ E ] + ( − n − n M [ E ] ab Y [ E ] b ( A ) kM [ E ] + ( − n − n M [ E ] ab Y [ E ] b ( A ) kM [ E ] (cid:105) , for (cid:0) n , n ∈ Z + , n ∈ Z (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) , (3.14) Y [ E ] a ( A ) ( k , , = √ (cid:104) Y [ E ] a ( A ) ( k , , + Y [ E ] a ( A ) ( − k , , (cid:105) , for (cid:0) n ∈ Z + , n = n = (cid:1) , (3.15) Y [ E ] a ( A ) (0 , k , = √ (cid:104) Y [ E ] a ( A ) (0 , k , + Y [ E ] a ( A ) (0 , − k , (cid:105) , for (cid:0) n ∈ Z + , n = n = (cid:1) , (3.16) Y [ E ] a ( A ) (0 , , k ) = √ (cid:104) Y [ E ] a ( A ) (0 , , k ) + Y [ E ] a ( A ) (0 , , − k ) (cid:105) , for (cid:0) n ∈ Z + , n = n = (cid:1) , (3.17)where Y [ E ] a ( A ) k are the vector harmonics on E described above, and A = , ,
2. We note that theexpressions in Eqs. (3.14)–(3.17) for the A = − κ , satisfy orthonormality conditions that are the analogs of thosegiven in Eq. (3.8), and satisfy divergence identities that are the analogs of those given in Eqs. (3.9)and (3.10). The proofs of these properties use the fact that the matrices M that define the symmetriesof these spaces preserve the inner product of vectors, e.g. for arbitrary vectors u and v , u · v = g ( u , v ) = g ( M · u , M · v ) = ( M · u ) · ( M · v ).Anti-symmetric tensor fields, w ab = − w ba , on orientable three-manifolds are dual to the vectorfields v a . Thus, for every w ab there exists a vector field v a so that w ab = (cid:15) abc v c . Therefore, anyanti-symmetric tensor field can be represented as a sum of vector harmonics. This section constructs symmetric second-rank tensor harmonics on the six flat compact orientablethree-manifolds. While these harmonics are not unique and can be chosen in a variety of ways,our goal here is to choose harmonics having three useful properties: First, the tensor harmonicsconstructed here will be eigenfunctions of the covariant Laplace operator: ∇ c ∇ c Y ab = − κ Y ab . (4.1)This ensures that these harmonics will form a complete basis for the (square integrable) symmetricsecond-rank tensor fields on these manifolds. We use the notation Y [ E j ] ab ( A ) k (for A = , ...,
5) todenote the six linearly independent classes of tensor harmonics that satisfy Eq. (4.1) on the space E j (for j = , ..., Y [ E j ] ab ( A ) k and Y [ E j ] ab ( A (cid:48) ) k (cid:48) will be orthogonal under the standard L inner product– 8 –nless k = k (cid:48) and A = A (cid:48) . And third, the tensor harmonics constructed here in classes A = , ..., A = A = A = , ..., Y [ E j ] ab (0) k = √ g ab Y [ E j ] k , (4.2) Y [ E j ] ab (1) k = √ (cid:16) κ − ∇ a ∇ b Y [ E j ] k + g ab Y [ E j ] k (cid:17) , (4.3) Y [ E j ] ab (2) k = κ √ (cid:16) ∇ a Y [ E j ] b (1) k + ∇ b Y [ E j ] a (1) k (cid:17) , (4.4) Y [ E j ] ab (3) k = κ √ (cid:16) ∇ a Y [ E j ] b (2) k + ∇ b Y [ E j ] a (2) k (cid:17) , (4.5)where κ is the corresponding eigenvalue of the covariant scalar Laplace operator, Eq. (2.4). Sincethe scalar harmonics Y [ E j ] k are smooth eigenfunctions of the Laplace operator, their gradients areautomatically smooth eigenfunctions having the same eigenvalues on these flat manifolds. The nor-malization factors in Eqs. (4.2)–(4.5) are chosen to ensure that these harmonics satisfy the nice or-thonormality conditions (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ac g bd Y [ E j ] ab ( A ) k Y [ E j ] cd ∗ ( B ) k (cid:48) dx dy dz = δ AB δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) , (4.6)for A = , ..., B = , ...,
3. The traces of these tensor harmonics are given by, g ab Y [ E j ] ab (0) k = √ Y [ E j ] k , (4.7) g ab Y [ E j ] ab ( A ) k = , for A = , ..., , (4.8)while the divergences are given by ∇ a Y [ E j ] ab (0) k = κ √ Y [ E j ] b (0) k , (4.9) ∇ a Y [ E j ] ab (1) k = − κ √ Y [ E j ] b (0) k , (4.10) ∇ a Y [ E j ] ab (2) k = − κ √ Y [ E j ] b (1) k , (4.11) ∇ a Y [ E j ] ab (3) k = − κ √ Y [ E j ] b (2) k . (4.12)The construction of the class A = A = A = A = ff erentapproaches that produce tensor harmonics satisfying the three useful properties listed above. The firstapproach produces the simplest expressions for the tensor harmonics, but this approach only workson the space E . A somewhat more general approach can be used to derive fairly simple expressionsfor these tensor harmonics on the spaces E , ..., E , but not on E . We have also developed an evenmore general approach capable of constructing tensor harmonics in all the E j spaces, but the resultingexpressions produced in this way are quite complicated. Here we report the results of this generalapproach only for the otherwise intractable E case.The first approach to constructing the needed tensor harmonics is based on the fact that anycovariantly constant tensor field c ab is invariant under the pure translation symmetry group of thethree-torus, E . Therefore any tensor field of the form c ab Y [ E ] k is an eigenfunction of the covariant– 9 –aplace operator, and therefore a candidate tensor harmonic. The choices for six linearly independentsymmetric tensors c ab will define the six classes of tensor harmonics for this case. The class A = A = E have this form: Y [ E ] ab (0)( k = √ g ab Y [ E ] k and Y [ E ] ab (0)( k = √ (cid:16) g ab − k a ˆ k b (cid:17) Y [ E ] k . All that is needed to complete this simpleapproach are choices for four additional constant tensors to define the class A = , ..., k a , ˆ (cid:96) a , and ˆ m a constructed inSec. 3: Y [ E ] ab (0) k = √ (cid:16) ˆ k a ˆ k b + ˆ (cid:96) a ˆ (cid:96) b + ˆ m a ˆ m b (cid:17) Y [ E ] k , (4.13) Y [ E ] ab (1) k = √ (cid:16) ˆ (cid:96) a ˆ (cid:96) b + ˆ m a ˆ m b − k a ˆ k b (cid:17) Y [ E ] k , (4.14) Y [ E ] ab (2) k = − √ (cid:16) ˆ k a ˆ (cid:96) b + ˆ k b ˆ (cid:96) a (cid:17) Y [ E ] k , (4.15) Y [ E ] ab (3) k = − √ (cid:16) ˆ k a ˆ m b + ˆ k b ˆ m a (cid:17) Y [ E ] k , (4.16) Y [ E ] ab (4) k = √ (cid:16) ˆ (cid:96) a ˆ (cid:96) b − ˆ m a ˆ m b (cid:17) Y [ E ] k , (4.17) Y [ E ] ab (5) k = √ (cid:16) ˆ (cid:96) a ˆ m b + ˆ (cid:96) b ˆ m a (cid:17) Y [ E ] k , (4.18)The class A = , ..., − κ , and satisfy the following orthonormality conditions, (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ac g bd Y [ E ] ab ( A ) k Y [ E ] cd ∗ ( B ) k (cid:48) dx dy dz = δ AB δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (4.19)The traces of these tensor harmonics on E are given by, g ab Y [ E ] ab (0) k = √ Y [ E ] k , (4.20) g ab Y [ E ] ab ( A ) k = , for A = , ..., , (4.21)and their divergences are given by, ∇ a Y [ E ] ab (0) k = κ √ Y [ E ] b (0) k , (4.22) ∇ a Y [ E ] ab (1) k = − κ √ Y [ E ] b (0) k , (4.23) ∇ a Y [ E ] ab (2) k = − κ √ Y [ E ] b (1) k , (4.24) ∇ a Y [ E ] ab (3) k = − κ √ Y [ E ] b (2) k , (4.25) ∇ a Y [ E ] ab (4) k = ∇ a Y [ E ] ab (5) k = . (4.26)The second approach to constructing tensor harmonics uses the fact that the vector ˆ z a , the metric g ab and the tensor ˆ z a ˆ z b (where ˆ z a is the unit vector along the z -axis) are the only covariantly constantvector and tensor fields invariant under all the symmetry groups of the manifolds E , ..., E . Thereforetensors like g ab Y [ E j ] k , ˆ z a ˆ z b Y [ E j ] k , and ˆ z a Y [ E j ] b k + ˆ z b Y [ E j ] a k (for j = , ...,
5) are eigenfunctions ofthe covariant Laplace operator that can be used in the construction of the tensor harmonics on thesespaces. We note that ˆ z a ∇ a Y [ E j ] k = i k Y [ E j ] k and ˆ z a ∇ a Y [ E j ] b ( A ) k = i k Y [ E j ] b ( A ) k in all of these cases.– 10 –he tensor harmonics for these manifolds can therefore be taken to be Y [ E j ] ab (0) k = √ g ab Y [ E j ] k , (4.27) Y [ E j ] ab (1) k = κ √ (cid:16) ∇ a ∇ b Y [ E j ] k + g ab κ Y [ E j ] k (cid:17) , (4.28) Y [ E j ] ab (2) k = κ √ (cid:16) ∇ a Y [ E j ] b (1) k + ∇ b Y [ E j ] a (1) k (cid:17) , (4.29) Y [ E j ] ab (3) k = κ √ (cid:16) ∇ a Y [ E j ] b (2) k + ∇ b Y [ E j ] a (2) k (cid:17) , (4.30) Y [ E j ] ab (4) k = κ k √ κ − k ) (cid:20) k (cid:16) κ + k (cid:17) (cid:18) ∇ a Y [ E j ] b (1) k + ∇ b Y [ E j ] a (1) k (cid:19) + i κ (cid:16) κ − k (cid:17) (cid:16) ˆ z a Y [ E j ] b (1) k + ˆ z b Y [ E j ] a (1) k (cid:17) + κ (cid:113) κ − k (cid:16) κ ˆ z a ˆ z b − k g ab (cid:17) Y [ E j ] k (cid:21) , (4.31) Y [ E j ] ab (5) k = κ √ ( κ − k ) (cid:104) k (cid:16) ∇ a Y [ E j ] b (2) k + ∇ b Y [ E j ] a (2) k (cid:17) − i κ (cid:16) ˆ z a Y [ E j ] b (2) k + ˆ z b Y [ E j ] a (2) k (cid:17)(cid:105) , (4.32)so long as the vector harmonics Y [ E j ] a (1) k and Y [ E j ] a (2) k are well defined, and so long as κ (cid:44) k (cid:44)
0, and κ (cid:44) k . When κ =
0, the tensor harmonics must be spatially constant tensor fields. Sixlinearly independent spatially constant tensor fields exist in the space E , so any orthonormal set canbe used as κ = E , ..., E the only spatially constant tensorfields are g ab and g ab − z a ˆ z b , so (suitably normalized) they are the only κ = k = κ (cid:44) z , i.e. they dependonly on x and y . In this case the expressions for the vector harmonics are given in Sec. 3 and theexpressions for the A = , ..., A = κ = k and κ (cid:44) x and y , i.e. the harmonics depend only on z . Inthis case the class A = A = A = A = − κ , satisfy orthonormality conditions that are the analogs of those given in Eq. (4.19),satisfy trace identities that are the analogs of those given in Eqs. (4.20) and (4.21), and divergenceidentities that are the analogs of those given in Eqs. (4.22)–(4.26).The third (most general) approach to constructing tensor harmonics on these compact orientableflat spaces, E j , is based on the fact that these spaces are quotients, E / Γ , of Euclidean space E and an isometry group Γ . Therefore the problem of finding tensor harmonics on the E j spaces isequivalent to finding the Γ -invariant tensor harmonics on E . This can be done using the methoddeveloped in Ref. [7] to derive expressions for the scalar harmonics. The Tensor Action Lemma andthe
Tensor Invariance Lemma described in the Appendix to this paper provide the tools needed toconstruct linear combinations of the E tensor harmonics that are invariant under all the elements ofthe symmetry groups Γ . The tensor harmonics obtained using this method on the spaces E , ..., E are more complicated than those derived using the first two approaches. So here we report only thetensor harmonics Y [ E ] ab ( A ) k with A = , ..., E – 11 –ase: Y [ E ] ab ( A ) k = (cid:20) Y [ E ] ab ( A ) k + ( − n − n M [ E ] ac M [ E ] bd Y [ E ] cd ( A ) k M [ E ] + ( − n − n M [ E ] ac M [ E ] bd Y [ E ] cd ( A ) k M [ E ] + ( − n − n M [ E ] ac M [ E ] bd Y [ E ] cd ( A ) k M [ E ] (cid:21) for (cid:0) n , n ∈ Z + , n ∈ Z (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) or (cid:0) n = , n , n ∈ Z + (cid:1) , (4.33) Y [ E ] ab ( A ) ( k , , = √ (cid:104) Y [ E ] ab ( A ) ( k , , + Y [ E ] ab ( A ) ( − k , , (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) , (4.34) Y [ E ] ab ( A ) (0 , k , = √ (cid:104) Y [ E ] ab ( A ) (0 , k , + Y [ E ] ab ( A ) (0 , − k , (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) , (4.35) Y [ E ] ab ( A ) (0 , , k ) = √ (cid:104) Y [ E ] ab ( A ) (0 , , k ) + Y [ E ] ab ( A ) (0 , , − k ) (cid:105) for (cid:0) n ∈ Z + , n = n = (cid:1) . (4.36)where Y [ E ] ab ( A ) k are the tensor harmonics on E described above. We note that the expressions inEqs. (4.33)–(4.36) for the A = , ..., − κ , satisfy orthonormality conditions that are the analogs of thosegiven in Eq. (4.19), satisfy the trace conditions given in Eqs. (4.20)–(4.21), and satisfy divergenceidentities that are the analogs of those given in Eqs. (4.22)–(4.26). The proofs of these properties usethe fact that the matrices M ab that define the symmetries of these spaces preserve the structure of themetric: g ab = M ca M db g cd and g ab = M ac M bd g cd . This paper introduces a uniform notation for the scalar, vector and tensor harmonics on the flat com-pact orientable three-manifolds E , E , ..., E . The Y [ E j ] k represent the scalar harmonics on the space E j (for j = , ...,
6) with parameters k = k a = ( k , k , k ). To enforce the appropriate periodici-ties, these parameters must be given by k a = π (cid:16) n L , n L , n L (cid:17) in the spaces E , E , E and E , and k a = π (cid:18) − n L , n − n √ L , n L (cid:19) in the spaces E and E , where the n , n and n are integers and L , L and L are the periodicity lengths in each dimension. (The lengths L and L must also be equal in thespaces E , E and E to preserve the symmetries.) Explicit expressions for these scalar harmonics,constructed originally in Ref. [7], are summarized in Sec. 2. The Y [ E j ] a ( A ) k represent the three classesof vector harmonics, with A = , ,
2, on the space E j . Explicit expressions for these harmonicsare constructed in Sec. 3. Finally the Y [ E j ] ab ( A ) k represent the six classes of symmetric second-ranktensor harmonics, with A = , , ...,
5, on the space E j . Explicit expressions for these harmonics areconstructed in Sec. 4. All these harmonics satisfy a number of useful properties:The scalar harmonics on the six compact orientable flat three-manifolds E j (for j = , , ..., ∇ b ∇ b Y [ E j ] k = − κ Y [ E j ] , (5.1)and satisfy the orthonormality conditions (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / Y [ E j ] k Y [ E j ] ∗ k (cid:48) dx dy dz = δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (5.2)– 12 –herefore it is easy to represent any (square integrable) scalar field on these manifolds in terms ofthese basis functions: f ( x ) = (cid:88) n (cid:88) n (cid:88) n f k Y [ E j ] , (5.3)where the coe ffi cients f k are given by f k = (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / f ( x ) Y [ E j ] ∗ k dx dy dz . (5.4)The vector harmonics on the six compact orientable flat three-manifolds E j (for j = , , ..., ∇ b ∇ b Y [ E j ] a ( A ) k = − κ Y [ E j ] a ( A ) k , (5.5)and satisfy the orthonormality conditions (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ab Y [ E j ] a ( A ) k Y [ E j ] b ∗ ( B ) k (cid:48) dx dy dz = δ AB δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) . (5.6)Therefore it is easy to represent any (square integrable) vector field on these manifolds in terms ofthese basis functions: v a ( x ) = (cid:88) n (cid:88) n (cid:88) n (cid:88) A v ( A ) k Y [ E j ] a ( A ) k , (5.7)where the coe ffi cients v ( A ) k are given by v ( A ) k = (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ab v a ( x ) Y [ E j ] b ∗ ( A ) k dx dy dz . (5.8)These vector harmonics also satisfy the divergence identities, ∇ a Y [ E j ] a (0) k = − κ Y [ E j ] k , (5.9) ∇ a Y [ E j ] a (1) k = ∇ a Y [ E j ] a (2) k = , (5.10)on each of the six flat compact orientable three-manifolds. The vanishing divergences of the class A = A = E j (for j = , , ..., ∇ c ∇ c Y [ E j ] ab ( A ) k = − κ Y [ E j ] ab ( A ) k , (5.11)and satisfy the orthonormality conditions (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ac g bd Y [ E j ] ab ( A ) k Y [ E j ] cd ∗ ( B ) k (cid:48) dx dy dz = δ AB δ n n (cid:48) δ n n (cid:48) δ n n (cid:48) , (5.12)for A = , ..., B = , ...,
5. Therefore it is easy to represent any (square integrable) symmetricsecond-rank tensor field on these manifolds in terms of these basis functions: t ab ( x ) = (cid:88) n (cid:88) n (cid:88) n (cid:88) A t ( A ) k Y [ E j ] ab ( A ) k , (5.13)– 13 –here the coe ffi cients t ( A ) k are given by t ( A ) k = (cid:90) L / − L / (cid:90) L / − L / (cid:90) L / − L / g ac g bd t ab ( x ) Y [ E j ] cd ∗ ( A ) k dx dy dz . (5.14)The traces of these tensor harmonics are given by, g ab Y [ E j ] ab (0) k = √ Y [ E j ] k , (5.15) g ab Y [ E j ] ab ( A ) k = , for A = , ..., , (5.16)while the divergences are given by ∇ a Y [ E j ] ab (0) k = κ √ Y [ E j ] b (0) k , (5.17) ∇ a Y [ E j ] ab (1) k = − κ √ Y [ E j ] b (0) k , (5.18) ∇ a Y [ E j ] ab (2) k = − κ √ Y [ E j ] b (1) k , (5.19) ∇ a Y [ E j ] ab (3) k = − κ √ Y [ E j ] b (2) k . (5.20) ∇ a Y [ E j ] ab (4) k = ∇ a Y [ E j ] ab (5) k = . (5.21)The vanishing traces and divergences of the class A = A = A Vector and Tensor Action and Invariance Lemmas
All multi-connected three-dimensional flat spaces E j are quotients, E / Γ , of Euclidean space E byan isometry group Γ . The problem of finding the scalar, vector and tensor harmonics on E / Γ isequivalent to the problem of finding the Γ -invariant harmonics of E . Two technical lemmas weredeveloped in Ref. [7] to facilitate the construction of the scalar harmonics on these spaces. The firstof these, the Action Lemma , determines how an element of the isometry group Γ transforms onescalar harmonic on E into another. The second, the Invariance Lemma , constructs a harmonic that isinvariant under the repeated action of any element of the isometry group. Together these lemmas wereused in Ref. [7] to derive the explicit expressions for the scalar harmonics summarized in Sec. 2 ofthis paper. Those lemmas for scalar harmonics are generalized here to action and invariance lemmasfor the vector and tensor harmonics on these flat spaces.Every isometry γ ∈ Γ on these manifolds consists of a reflection / rotation followed by a transla-tion, i.e. they are transformations of the form x (cid:48) = M · x + T where M is a unitary matrix and T is avector, or in component notation x (cid:48) a = M ab x b + T a .– 14 – emma 1 (Vector Action Lemma) The natural action of an isometry γ ∈ Γ of Euclidean space E takes a vector harmonic u Y k ( x ) = u e i k · x to another vector harmonic ( M · u ) e i k · T Y kM ( x ) , where u isany constant vector. Proof
The action of the isometry γ on u Y k is given by: u Y k (cid:55)→ γ (cid:16) u e i k · x (cid:17) = γ ( u ) γ (cid:16) e i k · x (cid:17) = ( M · u ) e i k · ( M · x + T ) = ( M · u ) e i k · T e i kM · x = ( M · u ) e i k · T Y kM ( x ) , where we have used γ ( u ) = u (cid:48) = M · u . (cid:3) Lemma 2 (Vector Invariance Lemma) If γ is an isometry of E with matrix part M and translationalpart T , if u Y k is a vector harmonic on E , and if n is the smallest positive integer such that k = kM n (typically n is simply the order of the matrix M ), then the action of γ
1. preserves the n-dimensional space of harmonics spanned by (cid:110) u Y k , ( M · u ) Y kM , · · · , (cid:16) M n − · u (cid:17) Y kM n − (cid:111) as a set, and2. leaves invariant the harmonic, a u Y k + a ( M · u ) Y kM + · · · + a n − (cid:16) M n − · u (cid:17) Y kM n − , if and onlyif a j + = e i kM j · T a j for each j (mod n). Proof
Both parts are immediate corollaries of Lemma 1. Specifically, the action of γ takes the linearcombination a u Y k + a ( M · u ) Y kM + · · · + a n − (cid:16) M n − · u (cid:17) Y kM n − + a n − (cid:16) M n − · u (cid:17) Y kM n − , (A.1)into a ( M · u ) e i k · T Y kM + a (cid:16) M · u (cid:17) e i k · MT Y kM + · · · + a n − (cid:16) M n − · u (cid:17) e i kM n − T Y kM n − + a n − u e i kM n − · T Y k . (A.2)Therefore the n -dimensional subspace spanned by (cid:110) u Y k , ( M · u ) Y kM , · · · , (cid:16) M n − · u (cid:17) Y kM n − (cid:111) is pre-served as a set. Comparing the coe ffi cients of the expressions in Eqs. (A.1) and (A.2), it follows thatthey are identical if only and if a j + = e i kM j · T a j for each j (mod n ). (cid:3) Lemma 3 (Tensor Action Lemma) The natural action of an isometry γ of Euclidean space E takesa tensor harmonic u ⊗ v Y k = u ⊗ v e i k · x to another tensor harmonic ( M · u ) ⊗ ( M · v ) e i k · T Y kM ( x ) ,where u and v are arbitrary constant vectors. Proof
The action of the isometry γ on u ⊗ v Y k is given by: u ⊗ v Y k (cid:55)→ γ ( u ⊗ v Y k ) = γ ( u ) ⊗ γ ( v ) γ ( Y k ) = ( M · u ) ⊗ ( M · v ) e i k · ( M · x + T ) = ( M · u ) ⊗ ( M · v ) e i k · T Y kM ( x ) , (A.3)where we have used γ ( u ⊗ v ) = u (cid:48) ⊗ v (cid:48) = ( M · u ) ⊗ ( M · v ). (cid:3) Lemma 4 (Tensor Invariance Lemma) If γ is an isometry of E with matrix part M and translationalpart T , if u ⊗ v Y k is a tensor harmonic on E , and if n is the smallest positive integer such that k = kM n (typically n is simply the order of the matrix M ), then the action of γ
1. preserves the n-dimensional space of harmonics spanned by (cid:110) u ⊗ v Y k , ( M · u ) ⊗ ( M · v ) Y kM , · · · , (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) Y kM n − (cid:111) as a set, and – 15 – . leaves invariant the harmonic, a u ⊗ v Y k + a ( M · u ) ⊗ ( M · v ) Y kM + · · · + a n − (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) Y kM n − , if and only if a j + = e i kM j · T a j for each j (mod n). Proof
Both parts are immediate corollaries of Lemma 3. Specifically, the action of γ takes the linearcombination a u ⊗ v Y k + a ( M · u ) ⊗ ( M · v ) Y kM + · · · + a n − (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) Y kM n − + a n − (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) Y kM n − , (A.4)into a ( M · u ) ⊗ ( M · v ) e i k · T Y kM + a (cid:16) M · u (cid:17) ⊗ (cid:16) M · v (cid:17) e i k · MT Y kM + · · · + a n − (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) e i kM n − T Y kM n − + a n − u ⊗ v e i kM n − · T Y k . (A.5)Therefore the n -dimensional subspace spanned by (cid:110) u ⊗ v Y k , ( M · u ) ⊗ ( M · v ) Y kM , · · · , (cid:16) M n − · u (cid:17) ⊗ (cid:16) M n − · v (cid:17) Y kM n − (cid:111) is preserved as a set. Compar-ing the coe ffi cients of the expressions in Eqs. (A.4) and (A.5), it follows that they are identical if onlyand if a j + = e i kM j · T a j for each j (mod n ). (cid:3) Acknowledgments
We thank James Nester for helpful conversations concerning this work. L.L. thanks the Morning-side Center for Mathematics, Academy of Mathematics and Systems Science, Chinese Academyof Sciences, Beijing 100190, China for their hospitality during a visit in which a portion of thisresearch was performed. This research was supported in part by the National Natural Science Foun-dation of China grants 11503003 and 11633001, the Interdisciplinary Research Funds of Beijing Nor-mal University, the Strategic Priority Research Program of the Chinese Academy of Sciences grantXDB23000000, and by the National Science Foundation, USA grants PHY-1604244, DMS-1620366,and PHY-1912419.
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