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Home , Chern class, Line bundle, Tautological bundle

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 |ψi should be described by the density matrix, the projector to the 1D subspace generated by |ψi,

Pψ = |ψihψ|. (I.1)

n−1 All quantum states Pψ comprise the P H of H, and pH ≈ CP . If we exclude zero point from H, there is a natural projection from H − {0} to P H,

π : H − {0} → P H. (I.2)

On the other hand, there is a tautological T (P H) over over P H, where over the

point Pψ the fiber Tψ is the complex line generated by |ψi. Therefore, there is the pullback line bundle π∗L(P H) over H − {0} and the line bundle morphism,

π˜ π∗T (P H) T (P H)

π H − {0} 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 E with zero section 0B excluded. We obtain the PEconsisting of points

(Pψx , x) (I.3) where

|ψxi ∈ Hx, x ∈ 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 T (PE) over PE,

π : T (PE) → PE, (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 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 ∞, (I.7) with f ∗EC ≈ L¯(PE). We recall the basic facts of H∗(CP ∞). For n ≥ 2, we know  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 ring H (CP ) = Z[z]/(z − 1) with z the generator of H (CP ) = ∗ ∞ ∼ 2 ∞ Z. Then, H (CP ) = Z[z] with c ∈ H (CP ). Then, let a be f #z ∈ H2(PE), then ai ∈ H2i(PE). It is easy to see from the univer- ∞ 2i n−1 2 n−1 sality of CP that ai|PEx generates H (PEx ≈ CP ), and therefore 1, a, a , ··· , a , are a basis of H∗(CP n−1) as a free . Furthermore, under the action of the cohomology ring H∗(B) on H∗(PE), induced by

p# : H∗(B) → H∗(PE), (I.9)

the set, {ai, i = 0, 1, ··· , n − 1} 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 {ai, i = 0, 1, ··· , n − 1}. Thus, 3

an has a decomposition in terms of H∗(B) under the basis {ai, i = 0, 1, ··· , n − 1} as

n n X n−i a (E) = − ci(E)a . (I.10) i=1

Here, ci(E) is called the ith 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 B to Heven(B, Z). c(E) 2i has the graded decomposition, c(E) = 1 + c1(E) + ··· + cn(E) with ci(E) ∈ 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 . 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 ∞, Z) → H2(CP 2, Z) is an isomorphism, and the generator z of H2(CP ∞, Z) generates the cohomology ring H∗(CP ∞, Z), the dimension axiom can be

equivalently stated as c1(γ) = z.

0 ∞ 3 . For the universal line bundle γ over CP , c1(γ) = z, with z the generator of H∗(CP ∞, 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 ∞]. Since the tensor product of two line bundles is still a line bundle, L(B) is actually an abelian group. That implies for two classes [f] and [g] in [B,CP ∞], there exists a homotopy class [k] with f ∗γ ⊗ g∗γ ≈ k#γ. So the multiplication in [B,CP ∞] can be according defined as [f][g] = [k], and therefore the gives a group isomorphism between L(B) and [B,CP ∞]. It is easy to see L(−) and [−,CP ∞] are actually isomorphic functors. CP ∞ presents the Eilenberg-MacLane space K(Z, 2), the functor [−,CP ∞] 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) ∈ H (B, Z) with

c1(ξ ⊗ η) = c1(ξ) + c1(η). (I.13)

∗ ∞ 2 ∞ Particularly, H (CP , Z) is generated by c1(γ) ∈ 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 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∗(PE, 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 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. Then, c(E) = c(θn)c(E) = c(E ⊕ θn) = c(E0 ⊕ θm) = c(E0)c(θm) = c(E0).

E. Chern character

The homomorphism, c(E ⊕ E0) = c(E)c(E0), is not satisfactory enough. Moreover, for

two line bundles λ and η, c1(λ ⊗ η) = c1(λ) + c1(η), which is also not satisfactory with ‘×’ 6

and ‘+’ exchanged. If we define ch(λ) = ec1(λ) ∈ Heven(B, R), then ch(λ ⊗ η) = ch(λ)ch(η), P which looks nicer. Furthermore, for a direct sum of line bundles, E = i λi, we define X ch(E) = ch(λi), (I.15) i so that ch(E ⊕ E0) = ch(E) + ch(E0). (I.16)

0 P is satisfied. Moreover, for the tensor product with E = ηj,

0 X c1(λi⊗ηj ) X X X 0 ch(E ⊗ E ) = e = ch(λi)ch(λj) = ch(λi) ch(λj) = ch(E)ch(E ). i,j ij i j (I.17) ch(E) is called the Chern character. From the splitting principle, we know the Chern character is well defined for an arbitrary bundle.

II. THE CHERN-WEIL THEORY

We now consider complex vector bundles over a M. Let ∇ is a connection on

the manifold. Over a locally trivializable chart Ui, it can be presented by the more familiar form, ∇ = d + A (II.1)

P µ with A = Aµdx . Then, the curvature is defined by

F = dA − A ∧ A. (II.2)

On the over lap of two charts Ui ∩ Uj, the connection transforms as

A → UAU −1 + dUU −1 (II.3)

where U is the transition function from Ui to Uj. Then, the curvature transforms as

F → UFU −1. (II.4)

Thus, F is a well-defined 2-form over M, that is F ∈ Ω2(M, End(E)). The curvature 2-form satisfies the Bianchi identity.

dF = A ∧ F − F ∧ A = [A, F ]. (II.5) 7

A. The invariant polynomials of curvature

Let X be an n × n matrices with entries xij. Then, a matrix U ∈ GL(n, C) can act on X by matrix conjugate X → UXU −1. (II.6)

We consider polynomials C[x] with x = {xij} invariant under the conjugate action of GL(n, C). Consider n X q det(I + Xt) = cq(x)t . (II.7) q=0

Then, cq(x) are invariant polynomials. Actually, all invariant polynomials of x are polyno-

mials of cq.

GL(n,C) C[x] = C[c1(x), ··· , cn(x)]. (II.8) Since det(M) = exp Tr log M, we derive

∞ ! X (−t)q det(1 + Xt) = exp − Tr(Xq) . (II.9) q q=1 Then, the following algebras of polynomials are equal

2 n C[c1, ··· , cn] = C[Tr(X), Tr(X ), ··· , Tr(X )]. (II.10)

2 2 As an examples, X = c1(X) and Tr(X ) = c1(x) − 2c2(x).

B. The Chern class

For an invariant polynomial φ(x), we introduce a form φ(F ) ∈ Heven(M, C), which is closed, dφ(F ) = 0. The closeness can be seen from the closeness of Tr(F q). X dTr(F q) = Tr(F idF F j) i+j=q−1 X = Tr(F i[A, F ]F j) i+j=q−1 (II.11) = Tr[A, F q] = 0. Actually, the Chern-Weil theorem states that the cohomology class [φ(F )] is independent of the choice of connection. Therefore φ(F ) maps isomorphism classes of vector bundles over M into Heven(M, C). 8

We then introduce the qth Chern form for a nD vector bundle E,

 F  1 c (E, ∇) = c = c (F ). (II.12) q q 2πi (2πi)q q and  F   F   F  c(E, ∇) = 1 + c + ··· + c = det 1 + . (II.13) 1 2πi n 2πi n 2πi The cohomology class of c(E, ∇) is the image of the total Chern class c(E) under Heven(M, Z) → Heven(M, C). To see this, we just need to check the axioms of Chern classes.

0. c(E) = det(1n + F/(2πi)) = 1 + c1(F/(2πi)) + ··· + cn(F/(2πi))

1. This follows from the Chern-Weil theorem.

0 F F 0 F F 0 0 0 0 2. c(E ⊕ E ) = det(1n + 2πi ) ⊕ (1n + 2πi ) = det(1n + 2πi ) det(1n + 2πi ) = c(E)c(E )

3. For a line bundle λ, c(λ, ∇) = det(1 + F/(2πi)) = 1 + F/(2πi). Particularly, one can 1 1 2 verify that for the γ → CP = S , c1 presents the generator of H2(S2, Z).

The canonical bundle over CP 1 corresponds to the valence band of the Hamiltonian for Weyl fermions, H(k) = −k · σ. Let F be the Berry curvature of the valence band. The monopole Chern is given by 1 Z F = 1. (II.14) 2πi S2

C. The Chern Character

c1(λ) F  For a line bundle λ, the Chern character ch(λ) = e = exp 2πi . Then, for a direct P sum of line bundles j λj,   !   M X Fj M Fj F ch( λ ) = exp = Tr exp = Tr exp (II.15) j 2πi 2πi 2πi j j j L L where Fj is a curvature of λj and F = j Fj is the corresponding curvature of j λj. Thus, for an arbitrary nD bundle E with a connection ∇, the Chern character is presented as

 F  ch(E) = Tr exp . (II.16) 2πi 9

It is easy to verify ch preserves the direct sum and multiplication of vector bundles. For two bundles E and E0 with curvatures F and F 0, respectively, the corresponding curvature of the direct sum E ⊕ E0 is F ⊕ F 0, which leads to

ch(E ⊕ E0) = ch(E) + ch(E0). (II.17)

0 0 The curvature for the direct product E ⊗ E is F ⊗ 1n0 + 1n ⊗ F , so  F   F 0  ch(E ⊗ E0) = Tr exp ⊗ exp 2πi 2πi  F   F 0  = Tr exp Tr exp (II.18) 2πi 2πi = ch(E)ch(E0).