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The Chern Characteristic Classes

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The Chern Characteristic Classes The Chern characteristic classes Y. X. Zhao I. FUNDAMENTAL BUNDLE OF QUANTUM MECHANICS Consider a nD Hilbert space H. Qauntum states are normalized vectors in H up to phase factors. Therefore, more exactly a quantum state j i should be described by the density matrix, the projector to the 1D subspace generated by j i, P = j ih j: (I.1) n−1 All quantum states P comprise the projective space P H of H, and pH ≈ CP . If we exclude zero point from H, there is a natural projection from H − f0g to P H, π : H − f0g ! P H: (I.2) On the other hand, there is a tautological line bundle T (P H) over over P H, where over the point P the fiber T is the complex line generated by j i. Therefore, there is the pullback line bundle π∗L(P H) over H − f0g and the line bundle morphism, π~ π∗T (P H) T (P H) π H − f0g P H . Then, a bundle of Hilbert spaces E over a base space B, which may be a parameter space, such as momentum space, of a quantum system, or a phase space. We can repeat the above construction for a single Hilbert space to the vector bundle E with zero section 0B excluded. We obtain the projective bundle PEconsisting of points (P x ; x) (I.3) where j xi 2 Hx; x 2 B: (I.4) 2 Obviously, PE is a bundle over B, p : PE ! B; (I.5) n−1 where each fiber (PE)x ≈ CP . Then, there is the tautological bundle T (PE) over PE, π : T (PE) ! P E; (I.6) ∗ and the pullback π T (PE) of T (PE) over E − 0B, corresponding to the bundle morphism π~ π∗T (PE) T (PE) π E − 0B PE . The line bundle T (PE) has an associated (complex) conjugate line bundle T¯(PE), and their tensor product T¯(PE) ⊗ T (PE) is isomorphic to the trivial line bundle PE × C over PE. As a line bundle, T¯(PE) corresponds to a classifying function, f : PE ! CP 1; (I.7) with f ∗EC ≈ L¯(PE). We recall the basic facts of H∗(CP 1). For n ≥ 2, we know 8 <>Z p even and 0 ≤ p ≤ 2(n − 1) Hp(CP n−1) = (I.8) :>0 otherwise ∗ n−1 ∼ n 2 n−1 ∼ and the cohomology ring H (CP ) = Z[z]=(z − 1) with z the generator of H (CP ) = ∗ 1 ∼ 2 1 Z. Then, H (CP ) = Z[z] with c 2 H (CP ). Then, let a be f #z 2 H2(PE), then ai 2 H2i(PE). It is easy to see from the univer- 1 2i n−1 2 n−1 sality of CP that aijPEx generates H (PEx ≈ CP ), and therefore 1; a; a ; ··· ; a , are a basis of H∗(CP n−1) as a free abelian group. Furthermore, under the action of the cohomology ring H∗(B) on H∗(PE), induced by p# : H∗(B) ! H∗(PE); (I.9) the set, fai; i = 0; 1; ··· ; n − 1g generate H∗(PE), and the action of H∗(B) is free. In other words, H∗(PE) is a Z-graded free H∗(B)-module with basis fai; i = 0; 1; ··· ; n − 1g. Thus, 3 an has a decomposition in terms of H∗(B) under the basis fai; i = 0; 1; ··· ; n − 1g as n n X n−i a (E) = − ci(E)a : (I.10) i=1 Here, ci(E) is called the ith Chern class of the vector bundle E, and the total Chern class c is defined as c = 1 + c1 + c2 + ··· + cn: (I.11) Put this more clearly. The ring H∗(PE) can be presented as ∗ ∗ H (PE) = H (B)[a]=Ihc;ai; (I.12) Pn n−i Pn n−i where Ihc;ai = ( i=0 cia ) is the ideal generated by i=0 cia , consisting of all polyno- Pn n−i mials divisible by i=0 cia . The total chern class satisfies the following axioms 0. c is a map from vector bundles E over a paracompact space B to Heven(B; Z). c(E) 2i has the graded decomposition, c(E) = 1 + c1(E) + ··· + cn(E) with ci(E) 2 H (B; Z), and ci(E) = 0 for i > dimCE. 1. Naturality. c is a natural transformation from E(−) to H∗(−; Z), where E(X) is the set of isomorphism classes of vector bundles over B. That is, if E and E0 are isomorphic over B, c(E) = c(E0), and if f : B0 ! B, f #(c(E)) = c(f #(E)). 2. Semigroup homomorphism. c(E ⊕ E0) = c(E)c(E0). Note that Heven(B; Z) is an abelian semigroup under the cup multiplication. 1 2 3. Dimension axiom. For the canonical line bundle γ over CP = S , c1(λ) is the 2 2 ∼ generator of H (S ; Z) = Z. Since the restriction i# : H2(CP 1; Z) ! H2(CP 2; Z) is an isomorphism, and the generator z of H2(CP 1; Z) generates the cohomology ring H∗(CP 1; Z), the dimension axiom can be equivalently stated as c1(γ) = z. 0 1 3 . For the universal line bundle γ over CP , c1(γ) = z, with z the generator of H∗(CP 1; Z). Before providing elaborations on the axioms, we first look into the classification of line bundles in detail. 4 A. The classification of line bundles Let L(B) be all isomorphism classes of line bundles over B. L(B) has a one-to-one correspondence with [B; CP 1]. Since the tensor product of two line bundles is still a line bundle, L(B) is actually an abelian group. That implies for two homotopy classes [f] and [g] in [B; CP 1], there exists a homotopy class [k] with f ∗γ ⊗ g∗γ ≈ k#γ. So the multiplication in [B; CP 1] can be according defined as [f][g] = [k], and therefore the universal bundle gives a group isomorphism between L(B) and [B; CP 1]. It is easy to see L(−) and [−;CP 1] are actually isomorphic functors. CP 1 presents the Eilenberg-MacLane space K(Z; 2), the functor [−;CP 1] is isomorphic to H2(−; Z). Hence, we see L(−) and H2(−;G) are isomorphic contravariant functors from the homotopy category of paracompact spaces to the category of abelian groups. Actually, the natural isomorphism is given by the first 2 Chern class c1, which implies for any bundle ξ; η over B, c1(ξ); c1(η) 2 H (B; Z) with c1(ξ ⊗ η) = c1(ξ) + c1(η): (I.13) ∗ 1 2 1 Particularly, H (CP ; Z) is generated by c1(γ) 2 H (CP ; Z). And the first Chern class of any line bundle is the pullback of c1(γ) by the classifying map. More discussion should be given for the operation of complex conjugate on line bundles, which is just the inversion of group elements. B. The dimension axiom The dimension axiom trivially holds for the definition of Chern classes above. Since for a line bundle λ, the P λ is just the base space, and T (P λ) is just λ itself. Then, letting f # # the classifying map of λ, a = f z, and hence c1(λ) = f z. For the universal bundle, the classifying map is just the identity map, and therefore c1(γ) = z. The naturality follows from the definition. The Semigroup homomorphism axiom can be proved based on the splitting principle. C. The Splitting principle For a vector bundle E over B, f : B0 ! B is called a splitting map if f ∗E is a sum of line bundles and f # : H∗(B; Z) ! H∗(B0; Z) is a monomorphism. Actually for any bundle 5 over a paracompact base space B, there exists a splitting map. This can be see from the exact sequence of vector bundles over PE, 0 ! L(PE) ! p∗E ! σ ! 0; (I.14) which means p∗E ≈ L(PE) ⊕ σ. As we see above, p# makes H∗(P E; Z) a free H∗(B; Z) module, and therefore p# is a monomorphism. We can inductively repeat this for σ until σ becomes one-dimensional, which proves the existence of a splitting map. Then, by a splitting map and the naturality of the Chern class c, we see for the Chern classes are uniquely characterized by the axioms. Moreover, the above arguments can easily be generalized to see that, for any two bundles E and E0 over B, there is a splitting map f : B0 ! B simultaneously splits E and E0 into direct sums of line bundles. A key technical step is to prove that the semigroup homomorphism axiom holds for a direct sum of line bundles. Given the homomorphism axion for direct sums of line bundles, it is easy to verify our definition of Chern classes satisfies the homomorphism axiom for a direct sum of any two vector bundles. In conclusion, the defined total Chern class is the unique one satisfying all the axioms for the total Chern class. D. Stability A first reflection on the axioms of Chern classes is that all Chern classes vanish on trivial bundles. The reason is that a trivial bundle is the pullback of a vector space over a point, but for a point Hp(pt; Z) = 0 for p > 0. Then, it follows immediately that two stably isomorphic bundles have the same Chern classes. Recall that E and E0 are stably isomorphic if E⊕θn ≈ E0⊕θm for some trivial bundles θn and θm.
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