() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 2, 2012 pp. 103-113 Supersymmetry and the Hopf fibration Simon Davis Abstract The Serre spectral sequence of the Hopf fibration S 15 S 7 → S 8 is com- puted. It is used in a study of supersymmetry and actions based on this fibration. 2010 MSC: 14D21, 55F20, 81Q60 Keywords: spectral sequence, Hopf fibration, supersymmetry 1. Introduction There are compactifications of eleven-dimensional supergravity with an SU(3)× SU(2) × U(1) isometry group of the compact space that are known to yield the particle spectrum of the standard model [1], [2]. The fermion multiplets can be included in a spinor space represented by a tensor product of division algebras for each generation. The automorphism group of this product would be G2 × SU(2) × U(1) and it may be demonstrated that there are coset spaces G2×SU(2)×U(1) SU(3)×U(1)′×U(1)′′ yielding particles and antiparticles with the known quantum numbers [3]. The dimensions of the normed real alternative division algebras correspond to the parallelizability of the spheres. The spheres S1, S3 and S7 in the reduc- tion sequence of the unified field theory represent submanifolds of the higher- dimensional coset space. The representation of unit elements in the compo- nents of the spinor space could be related to the fermion bilinears arising in the set of light-like lines in two larger dimensions, yielding S2, S4 and S8. The unit fermions can exist in a fibre of a bundle over the space of light-like lines. Amongst the S7 bundles over S8 is the Hopf fibration S15 S 7 → S8. A classification of physical states described by the Hopf fibrations is given in §2. 104 S. Davis It has been demonstrated previously that an S7 transformation rule cannot be constructed for a pure Yang-Mills theory with the connection taking values only on the four-dimensional base space [4]. Since twistor variables that trans- form under Sp(4; O) can be combined to transform parameterize S7 [5], the problem of constructing a model with this invariance may be considered. This can be done only if the space S8 of lightlike lines of octonionic superparticles is interpreted in terms of fundamental variables in the theory. If the fermion field is allowed to take values in a one-point compactification of the space identified with the division algebra, an equivalence with the bosonic sector given by the light-like lines can provide a basis for a supersymmetry. This approach can be compared to an algebraic description of the supersymmetric Hopf fibration. When the base space super-sphere S2∗, a supersymmetric version of the U(1) theory is found [6]. The spectral sequences of the Hopf fibrations of the superspheres and the homology groups are found to unaltered by the introduction of supersymmetry in §3. The effect of an S7 transformation on fields in the twistor formalism can be elevated to an invariant action directly because there are anomalous terms in the commutators. Although various spinor bilinears and combinations of supertwistors are found to be invariant, there is a associator term with a spinor field, which must be cancelled for invariance under the composition of these transformations. A method for eliminating the additional terms through the commutator of BRST and gauge transformations is suggested in §4. 2. Spectral Sequences and Hopf Fibrations It may be recalled that the homology group of the total space of a fibre bundle may be determined from the Serre spectral sequence. For a filtration X0 ⊂ X1 ⊂ X2 ⊂ ... ⊂ X, let D = ⊕m,nHm(Xn) and E = ⊕m,nHm(Xn,Xn−1) define an exact couple such that im j = ker k where j : D → E and k : E → D. Let D′ = i(D) and d = jk : E → E with d2 = 0. Suppose that E′ = H(E;d), i′ = i|′D and the map j ′ : D′ → E′ is defined by j′(x) is the coset of j(y) in Z(E), where x = i(y) ∈ D′ for y ∈ D. The map k′ completes the exact sequence and (D′,E′, i′,j′,k′) is an exact couple. Further iterations give Dn,En;in,jn,kn) such that dn = jnkn : E n → En, where En+1 = H(En;dn), and D = ⊕p,qDp,q and E = ⊕p,qEp,q. Then i(Dp,q) ⊂ Dp+1,q−1, j(Dp,q) ⊂ Ep,q and k(Ep,q) ⊂ Dp−1,q. If D n p,q = D n ∩ Dp,q and E n p,q = H(E n−1 p,q ;dn), i(Dnp,q) ⊂ D n p+1,q−1, jn(D n p,q) ⊂ E n p−n+1,q+n−1 and kn(E n p,q) ⊂ D n p−1,q, while dn : E n p,q → E n p−n,q+n−1. For a fibre bundle, E F → B, E2p,q = H(E 1 p,q;d1) = Hp(B;Hq(F)) [7]. Consider the Hopf fibration S7 S 3 → S4. E2p,q = Hp(S 4;Hq(S 3)) = { Hp(S 4) q = 0,3 0 otherwise (2.1) = { Z p = 0,4, q = 0,3 0 otherwise Supersymmetry and the Hopf fibration 105 The boundary mapping d4 is injective as there would exist an element y 6= d4x mapped to zero otherwise, implying that E54,0 = H 4(E44,0;d4) 6= 0. This latter statement would imply E54,0 ≃ ... ≃ E ∞ 4,0 6≃ 0 (2.2) contrary to H4(S7) ≃ 0. Also, d4 is surjective, because E 5 0,3 ≃ ... ≃ E ∞ 0,3 ≃ 0 since H3(S 7) ≃ 0. It follows that d4 is an isomorphism and d4 : E 4 4,3 → E 4 0,6 is surjective. Since E54,0 ≃ 0, E 5 3,0 ≃ d4(E 5 4,0) ≃ 0, removing the (0,4) and (3,0) elements in the sequence for E5p,q. The remaining non-zero entries in the E 5 p,q sequence may be deduced from exact sequences derived for filtrations of the total space. Given that Dp,q = 0 for p < 0 and Ep,q = 0 for p < 0 or q < 0, E n p,q = 0 for p < 0, q < 0 and p+q < 0. For large n, there are the following exact sequences: →Enp+n−1,q−n+2 kn → Dnp+n,q−n+2 in → Dnp+n−1,q−n+1 jn → Enp,q kn → Dnp−1,q → ... i ↓ i ↓ (2.3) 0 → Dn+1p+n−2,q−n+2 in+1 → Dn+1p+n−1,q−n+1 jn+1 → En+1p−1,q+1 → ... . There is a related sequence 0 → Dn+1p+n−2,q−n+2 in+1 → Dn+1p+n−1,q−n+1 jn → En+1p,q → ... (2.4) which holds if the domain of jn can be chosen such that jn(D n+1 p+n−1,q−n+1) = En+1p,q . Since D n+1 p,q = i(D n p,q) ⊆ D n p,q, this is feasible, although jn has not been defined to have the range En+1p,q . This could be ensured, however, if En+1p,q = E n p,q. From the sequence (2.4), it follows that the exactness of the sequence Enp+n,q−n+1 dn → Enp,q dn → Enp−n,q+n−1 (2.5) which is equivalent to 0 dn → Enp,q dn → 0 (2.6) for n > p and n > q + 1, implying the constancy of Enp,q for large n. From the complex of sequences 0 → Dn+1p+n−2,q−n+2 in+1 → Dn+1p+n−1,q−n+1 jn → En+1p,q kn → Dn+1p−1,q → ... i ↓ i ↓ 0 → Dn+2p+n−2,q−n+2 in+1 → Dn+2p+n−1,q−n+1 jn → En+2p,q kn+2 → Dn+2p−1,q → ... (2.7) ... 0 → D∞p+n−2,q−n+2 i∞ → D∞p+n−1,q−n+1 j∞ → E∞p,q k∞ → D∞p−1,q → ... 106 S. Davis the last exact sequence does not end. It implies 0 → D∞p−1,q+1 i∞ → D∞p,q j1 → E∞p,q k∞ → D∞p−1,q → ... (2.8) when n = 1 is substituted in the final sequence of Eq.(2.7). For the sequences, Dn+1p+n−1,q−n+1 jn+1 → En+1p−1,q+1 kn+1 → Dn+1p−2,q+1 → ... i ↓ Dn+2p+n−1,q−n+1 jn+2 → En+2p−2,q+2 kn+2 → Dn+2p−3,q+2 → ... i ↓ ... (2.9) D n+p−1 p+n−1,q−n+1 jn+p−1 → E n+p−1 1,p+q−1 kn+p−1 → D n+p−1 0,p+q−1 → ... i ↓ D n+p p+n−1,q−n+1 jn+p → E n+p 0,p+q kn+p → D n+p −1,p+q ≃ 0 and 0 → D n+p p+n−2,q−n+2 in+p → D n+p p+n−1,q−n+1 jn+p → E n+p 0,p+q kn+p → 0 (2.10) yielding eventually the sequence 0 → D∞p+n−2,q−n+2 i∞ → D∞p+n−1,q−n+1 j∞ → E∞0,p+q k∞ → 0. (2.11) With n = 1 in the indices 0 → D∞p−1,q+1 i∞ → D∞p,q j∞ → E∞0,p+q k∞ → 0 (2.12) implying E∞0,p+q ≃ D ∞ p,q/i∞(D ∞ p−1,q+1). Since Hp+q(Xp−1) = D ∞ p−1,q+1 ⊂ Hp+q(Xp) = D ∞ p,q, D ∞ p−1,q+1 ⊂ D ∞ p,q ⊂ D ∞ p+1,q−1 ⊂ ... ⊂ D ∞ p+n,q−n ⊂ .... For n sufficiently large, D∞p+n,q−n = D ∞ ∩ Dp+n,q−n = D ∞ ∩ Hp,q(Xp+n) = D∞ ∩Hp,q(X) is constant, when the exhaustion of X contains a finite sequence of proper subspaces. The sequences Supersymmetry and the Hopf fibration 107 0 → D n+p p+n−2,q−n+2 in+p → D n+p p+n−1,q−n+1 jn+p → E n+p 0,p+q kn+p → ...0 i ↓ i ↓ 0 → D n+p+1 p+n−2,q−n+2 in+p+1 → D n+p p+n−1,q−n+1 jn+p+1 → E n+p+1 0,p+q kn+p+1 → 0 i ↓ i ↓ ... (2.13) 0 → Dn+n ′ p+n−2,q−n+2 in+n′ → Dn+n ′ p+n−1,q−n+1 jn+n′ → En+n ′ 0,p+q kn+n′ → 0 ... 0 → D∞p+n−2,q−n+2 i∞ → D∞p+n−1,q−n+1 j∞ → E∞0,p+q k∞ → 0 yield the isomorphisms E n+p 0,p+q ≃ D n+p p+n−1,q−n+1/in+p(D n+p p++n−2,q−n+2) ... En+n ′ 0,p+q ≃ D n+n′ p+n−1,q−n+1/in+n′(D n+n′ p+n−2,q−n+2) ... E∞0,p+q ≃ D ∞ p+n−1,q−n+1/i∞(D ∞ p+n−2,q−n+2). (2.14) It is apparent that Dn+n ′ p+n−1,q−n+1 ≃ D ′ ∩Dp+n−1,q−n+1 = D ′ ∩Hp+q(Xp+n−1) Dn+n ′ p+n−2,q−n+2 ≃ D ′∩Dp+n−2,q−n+2 = D ′∩Hp+q(Xp+n−2). (2.15) For sufficiently large n, Hp+q(Xp+n−1) ≃ Hp+q(Xp+n−2) ≃ Hp+q(X) and E n+p 0,p+q ≃ [D ′ ∩Hp+q(X)]/in+p(D ′ ∩Hp+q(X)) ... En+n ′ 0,p+q ≃ [D ′∩Hp+q(X)]/in+n′(D ′∩Hp+q(X)) ... E∞0,p+q ≃ [D ′ ∩ Hp+q(X)]/i∞(D ′ ∩ Hp+q(X)) (2.16) are the quotient groups related to Hp+q(X), and D ′ = i(D) = i(⊕p,qDp,q) = i(⊕p,qHp+q(Xp)). 108 S. Davis Fixing p + q, ⊕p+q=constantHp+q(Xp) = Hp+q(X0) ⊕ Hp+q(X1) ⊕ ... ⊕ Hp+q(Xm) ⊕ ... ⊕ Hp+q(X) ≃ Hp+q(X) ⊕p,qHp+q(Xp) ≃ ⊕ ∞ p+q=−∞Hp+q(X) ⊕∞p′+q′=0Hp′+q′(X) ∩ Hp+q(X) ≃ Hp+q(X) (2.17) and the following isomorphisms hold: E n+p 0,p+q ≃ Hp+q(X)/in+p(Hp+q(X)) ... E∞0,p+q ≃ Hp+q(X)/i∞(Hp+q(X)). (2.18) From the sequences (2.8) and 0 → D∞p−1,q+1 ι∞ → D∞p,q j2 → E∞p−1,q+1 k∞ → D∞p−2,q+1 → .. (2.19) isomorphisms of the form E∞p−1,q+1 ≃ E ∞ p,q ≃ ... may be deduced, and E∞p,q ≃ D ∞ p,q/i∞(D ∞ p−1,q+1). By the exact sequence (2.12), i∞(D ∞ p−1,q+1) = ker j∞(D ∞ p,q) consists of the identity element, because j∞ must be an injec- tive homomorphism as j∞(D ∞ p,q) ⊂ E ∞ p,q = E ∞ 0,p+q, and E ∞ p,q = Hp+q(X). It follows that, for the Hopf fibration S7 S 3 → S4, E∞p,q ≃ Hp+q(S 7) and Hr(S 7) ≃ { Z r = 0,7 0 otherwise . (2.20) For the Hopf fibration S3 S 1 → S2, there exist multi-soliton solutions parameter- ized by the homotopy group π3(S 2) [8]. Similarly the homotopy group π7(S 4) could be used to parameterize the Hopf number of soliton solutions to theories based on the next Hopf fibration, as the one-soliton solutions can be combined to give N-soliton solutions. From the Hurewicz theorem [9], the kth homology and homotopy groups of the sphere Sk are isomorphic to Z. By the exact sequence S3 → S7 → S4, it follows that π7(S 4) = π7(S 7) ⊕ π6(S 3) = Z ⊕ Z12 [10] [11], implying that there would be twelve varieties of each N-soliton. For the Hopf fibration S15 S 7 → S8, E2p,q = Hp(S 8;Hq(S 7)) = { Z p = 0,8, q = 0,7, p + q = 15 0 otherwise . (2.21) Since every element in E87,0 is d8x, x ∈ E 8 0,8, And E 9 7,0 ≃ H7(E 8 7,0;d8) ≃ 0. Similarly, d8 : E 7,0 8 → E 0,8 8 . Therefore, E 0,8 9 ≃ H 8(E 0,8 8 ;d8) ≃ 0 (2.22) Supersymmetry and the Hopf fibration 109 Therefore, E90,8 ≃ 0. Again E 9 p,q ≃ ... ≃ E ∞ p,q and E∞p,q ≃ Hp+q(S 15) ≃ { Z p = q = 0, p = 8, q = 7, p + q = 15 0 otherwise , (2.23) which is consistent with Hr(S 15) ≃ Z r = 0, 15 and Hr(S 15) ≃ 0 otherwise. An instanton solution to the Yang-Mills equations related to the last Hopf fibration has been found with the Euler number equal to N ∫ S8 (F ∧ F ∧ F ∧ F ∧ F)dV [12]. However, there is no reference to the gauge instanton in the Euler number, which is a topological invariant that is entirely characteristic of the spheres in the fibration. This result has been explained through the equivalence of this integral with that of the Pfaffian of 1 2π F̂ , where F̂µν is the field of the spinor connection [13]. The expression for the Euler number is derived from the curvature form. However, the formula for the curvature form of the spinor connection is given by Ωµν = eµ ∧eν [13], and it would appear that equivalence with a volume form would follow. The Hopf invariant, given by the integral ∫ S15 α∧dα, where α is a volume form on S7, can be projected to ∫ S8 dαs, where αs is a singular form as a result of the intersections of the seven-spheres, which has integer values. While the Hopf invariant is equal to the number of links of seven-spheres in S15, its integrality is similar, therefore, to that of the Euler class, which is a generator of a homology group isomorphic to Z. Upon deriving an N-soliton configuration from an N-instanton solution, the classification would be given by the homotopy group π15(S 8) = π15(S 15) ⊕ π14(S 7) = Z ⊕ Z120 [11]. 3. Supersymmetric Hopf Fibrations It has been shown that there is a generalization of the Hopf fibration S3 S 1 → S2 to SU(2)∗ U(1)∗ → S2∗, where each of the spaces is a supersphere [14]. The ana- logue of an element of SU(2) represented by s(t) = exp(iT aǫa(t)), T a = σa 2 , is s∗(t,θ) = exp(iT aηa(t,θ)) ηa(t,θ) = ǫa(t) − 2θiξa(t) (3.1) From Eq.(3.1), s∗(t,θ) = 1 + iT a(ǫa(t) − 2θiξa(t)) − 1 2! T a(ǫa(t) − 2θiξa(t))T b(ǫb(t) − 2θiξb(t)) + ... = (1 + θσaξa(t))(1 + iT aǫa(t) − 1 2! T aT bǫa(t)ǫb(t) + ...) (3.2) = (1 + θξ)s(t) ξ = σaξa(t) and s † ∗(t,θ)s∗(t,θ) = s(t) †(1 − θξ)(1 + θξ)s(t) = s(t)†(1 − θξ + θξ + θξθξ)s(t) = s(t)†s(t) = 1 (3.3) 110 S. Davis The action of U(1)∗ on SU(2)∗ is s∗ → s∗e iσ3α and the projection from SU(2) to S2, s(t) → s(t)σ3s(t) † is generalized such that x̂∗ = x̂∗aσ = s∗σ3σ † ∗ (3.4) parameterizes S2∗. This space may be compared with the supersphere S2,2 defined as OSp(1|2)/U(1), which has even coordinates xi and odd coordinates θα satisfying ∑ i xixi + ∑ α,β Cαβθαθβ (3.5) where C = ( 0 1 −1 0 ) [15]. The coordinates of S2 are given by yi = ( 1 + θCθ 2r2 ) (3.6) and ∑ i yiyi = r 2. The action of the U(1)∗ is right multiplication by an unitary group element and therefore identified with the action of U(1). Since the supersymmetry algebra has the form {D∗,D} = 0, where D = ∂ ∂θ − iθ ∂ ∂t (3.7) is similar to the exterior derivative operator, it might be considered useful to determine de Rham cohomology for the supersymmetric Hopf fibration. The graded differential calculus on a supersphere can be constructed such that (ω ∧ ω′) = dω ∧ ω′ + (−1)pω ∧ dω′ ω ∈ Ωp(Sm,n), ω′ ∈ Ωp ′ (Sm,n) (3.8) where Ωp(Sm,n) and Ωp ′ (Sm,n) are exterior form algebra. and d2 = 0. As the dimension of a supermanifold belongs to Z[ǫ]/(ǫ2 − 1) = Z ⊕ Zǫ, there is an isomorphism of the de Rham cohomology of a supermanifold with that of the underlying manifold [16][17]. The de Rham cohomology groups of the spheres have given HkdR(S n) ∼ { R if k = 0,n 0 if k 6= 0,n (3.9) whereas, HkdR(S n; Z) ∼ { Z if k = 0,n 0 if k 6= 0,n (3.10) By de Rham’s theorem, there is an isomorphism between the de Rham coho- mology group HkdR(M) and the cohomology groups H k(M; R) for any smooth manifold. From the commutative diagram of isomorphisms, it follows that the spectral sequences based on the homology groups of spheres could be adapted to the superspheres after specializing to a specific coefficient field. Conse- quently, the results of §2 may be used for the superspheres and each of the Supersymmetry and the Hopf fibration 111 supersymmetric Hopf fibrations, based on the exact sequences. 0 → S3∗ → S 7 ∗ → S 4 ∗ → 0 0 → S7∗ → S 15 ∗ → S 8 ∗ → 0 (3.11) 4. The Action of S7 and its Supersymmetric Generalization Although it has been demonstrated that the principal bundle structure of gauge theories is dependent on a Lie group structure, the action of S7 has been developed for twistor variables. In ten dimensions, the momentum vector of a massless particle can be expressed as p = ψψ†, where ψ is a spinor that traces out S8 . By the action of S7 on S8, consistent with that of the Hopf fibration, there exists a transformation δψα = Tψα = ψαo(α) such that [T,T ′]ψα = o(α)(o′(α)ψα) − o′(α)(oαψα) = ([o(α),o′(α)] − 2[o(α),o′(α),e(α)]ē(α))ψα e(α) = |ψα|−1ψα (4.1) the action is not covariant, and this prevents the construction of an entirely invariant action [18]. For a supersymmetric particle, the variables ξ = ψθ† + θψ† ω = Xψ + iξθ ZA = (ψα,ωα̇) (4.2) may be used to construct invariants under the action of S7, ψ → ψo, ω → ωo, |o| = 1, JMN = [ 1 2 Z†ΓMNZ ] (4.3) where the square brackets refer to the selection of the e0 component. While the components of JMN, ψ1ψ̄2, ω1ω̄2, ψ1ω̄2 and ψ2ω̄1 are separately invariant, the repeated action of the S7 transformations generates additional terms through the nonvanishing associator. . One method for eliminating the extra term may be based on the construction of a charge which could cancel the associator con- taining either e(1) or e(2). If this is included in the S7 transformation, it would cause the transformations to be covariant. The BRST charge, for example, is typically constructed such that an exact term does not affect the invariance of the Lagrangian under local gauge transformations. However, this would de- pend on the associativity of the operations of the gauge transformation and BRST transformation. Through the variables transforming under SL(2; O), an additional term derived from an associator containing the two transformations would be introduced. Since d = jk in the exact couple, a correspondence between spectral se- quences and the theories with an operator satisfying d2 = 0 can be established. The BRST cohomology of quantum field theories has been calculated previ- ously with spectral sequences [19]. Similarly, in supersymmetric models, the anticommutator of the supercharge satisfies {Qα,Q † α̇} = σ µ αα̇Pµ, and the trace will be non-zero except for a vacuum with zero energy. The existence of a 112 S. Davis spectral sequence only for the ground state is indicative of a connection with the index Tr (−1)F . The necessity of the BRST charge in the study of the action of S7 trans- formations on components of the supertwistor and the BRST cohomology of ten-dimensional supergravity and superstring theories implies that the role of the anomaly, which can be determined through spectral sequences, is similar to that of the quantum terms representing lack of closure of the classical S7 algebra on spinor fields. 5. Conclusion The homology groups of the total spaces in the Hopf fibrations are calculated with the spectral sequence. The conditions for the equality of E∞p,q = Hp+q(X) will be satisfied if Enp,q is constant for sufficiently large n. The derived homology groups Enp,q are found to be constant if n ≥ 5 for the fibration S 7 S 3 → S4. and n ≥ 9 for the fibration S15 S 7 → S8. The classification of solitons resulting from these fibrations are given by the homotopy groups π7(S 4) and π15(S 8) respectively. With the introduction of supersymmetry, the spectral sequences for the su- perspheres would have the same form. The homotopy groups determining the classification of the soliton states would follow. The description of the spaces in the third and fourth supersymmetric fibrations may be given together with a derivation of the homology and homotopy groups of the superspheres. 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Balachandran, G. Marmo, B.-S. Skagerstam and A. Stern, Gauge Symme- tries and Fibre Bundles, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983. [15] K. Hasebe and Y. Kimura, Fuzzy Supersphere and Supermonopole, Nucl. Phys. B709 (2005), 94–114. [16] H. Grosse and G. Reiter, The Fuzzy Supersphere, J. Geom. Phys. 28 (1998), 349–383. [17] D. Leites, The Index Theorem for Homogeneous Differential Operators on Supermani- folds, math-ph/0202024. [18] M. Cederwall, Octonionic Particles and the S7 Symmetry, J. Math. Phys. 33 (1992), 388–393. [19] J. A. Dixon, Calculation of BRS Cohomology with Spectral Sequences, Commun. Math. Phys. 139 (1991), 495–526. (Received October 2011 – Accepted September 2012) Simon Davis (sbdavis@resfdnsca.org) Research Foundation of Southern California, 8837 Villa La Jolla Drive #13595, La Jolla, CA 92039, USA. Supersymmetry and the Hopf fibration. By S. Davis