LECTURE 26: THE CHERNWEIL THEORY


 Reginald Dickerson
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1 LECTURE 26: THE CHERNWEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a ktensor T k V is called symmetric if T v σ1,, v σk ) = T v 1,, v k ), σ S k. We will denote the space of all symmetric ktensors on V by S k V. Let T S k V be any symmetric ktensor. Then T induces a degree k homogeneous polynomial on V, P T v) := T v,, v). Conversely, it is not hard to see that T is completely determined by P T, since we have the following polarization formula k T v 1,, v k ) := 1 P T t 1 v t k v k ). k! t 1 t k Like wedge product, we can define a symmetric product : S k V S l V S k+l V via 1 T 1 T 2 v 1,, v k+l ) = T 1 v 1,, v k )T 2 v k+1,, v k+l ). k + l)! σ S k+l As usual we put S 0 V = R. Of course P T defined as above is not really a polynomial. At least one should need a conception of products of vectors on V before we can talk about polynomials on V.) However, we can associate to each T S k V a true polynomial as follows: We fix a basis {e 1,, e n } of V. Let R[x 1,, x n ] be the ring of polynomials in the variable x 1,, x n, and let R[x 1,, x n ] k be the subset of homogeneous polynomials of degree k. We define a map P : S k V R[x 1,, x n ] k by T p T x 1,, x n ) := P T x i e i ) = T x i e i,, x i e i ) Obviously P is a linear map and satisfies PT 1 T 2 ) = PT 1 )PT 2 ). Lemma 1.1. The map P is a linear isomorphism. In particular, dim S k V = n+k 1 n 1 Proof. By the polarization formula above we see P is injective. To show that P is surjective, we just notice that for k = 1, P maps T i S 1 V = V to the polynomial p Ti x 1,, x n ) = x i, where T i is the map T i x i e i ) := x i. For general k, by repeatedly using the rule PT 1 T 2 ) = PT 1 )PT 2 ) one immediately see that P is surjective. Remark. The fact PT 1 T 2 ) = PT 1 )PT 2 ) implies that P : S V R[x 1,, x n ] is in fact a ring isomorphism. ).) 1
2 2 LECTURE 26: THE CHERNWEIL THEORY Now let V = g be the Lie algebra of a Lie group G. Then the adjoint action of G on g induces a Gaction on S k g ) by g T )X 1,, X k ) := T Ad g 1X 1,, Ad g 1X k ), X i g, g G. Note that for the case that G is a linear Lie group, which we will always assume below, one has g T )X 1,, X k ) = T gx 1 g 1,, gx k g 1 ), where g is an invertible r r matrix while X i s are arbitrary r r matrices. Definition 1.2. T S k g ) is called invariant if g T = T, g G. The set of all Ginvariant elements in S k g ) is denoted by I k G). So I G) = I k G) is a subring of S g ). Note that by definition and the polarization formula, T is invariant if and only if P T is invariant. Example. Consider G = GLr, R). For any positive integer k, we let p k/2 denote the degree k homogeneous polynomial which is the coefficient of λ r k for the following polynomial in λ det λi 1 ) r 2π A = p k/2 A,, A)λ r k, A gln, R). k=0 Obviously p k/2 is invariant. So p k/2 I k GLr, R)). They are called the k/2th Pontrjagin polynomials. Note that for G = Or) GLr, R), the Lie algebra or) consists of skewsymmetric matrices, and hence det λi 1 ) 2π A = det λi + 1 ) 2π A, A or). This implies that p k/2 = 0 for k odd. So for On) one only need to study p k I 2k On)). Proposition 1.3. For any T I k G) and any X, X 1,, X k g, we have T [X, X 1 ], X 2,, X k ) + + T X 1,, X k 1, [X, X k ]) = 0. Proof. This follows from d dt T e tx X 1 e tx,, e tx X k e tx ) = 0. t=0 For T I k G) one can also define T F 1,, F k ) for F i Ω U) g, by extending linearly the relation T ω 1 X 1,, ω k X k ) := ω 1 ω k )T X 1,, X k ). Note that if ω is a 2forms, or more generally is any evenform, then for any η one has ω η = η ω. As a consequence we see
3 LECTURE 26: THE CHERNWEIL THEORY 3 Corollary 1.4. If F 1,, F k Ω even U) g, then for any T I k G) and A Ω U) g one has T [A, F 1 ], F 2,, F k ) + + T F 1,, F k 1, [A, F k ]) = ChernWeil Theory Let G be a Lie group and for simplicity, assume G is a linear Lie group), with Lie algebra g. Let P be a principal Gbundle over M, which is defined by an open cover {U α ) and corresponding transition maps g αβ : U α U β G. Let A be a collection A α Ω 1 U α ) g which is a connection in P. Recall that the curvature F A of the connection A is given locally by the collection For any T I k G) we can define F α = da α [A α, A α ] Ω 2 U α ) g. p T F α ) := T F α,, F α ) Ω 2k U α ). Since T is Adinvariant and since F β = g βα F α g 1 βα, we get p T F α ) = p T F β ) on U α U β. In other words, there is a globallydefined 2kform p T F A ) Ω 2k M) so that p T F A ) = p T F α ) on U α. Note F A is not an element in Ω 2 M): it sits in Ω 2 M; AdP ).) It turns out that these p T F A ) s play a crucial role in modern math and physics. Theorem 2.1 ChernWeil). Let P be a principal Gbundle over M. Then 1) For any T I k G) and any connection A in P, the form p T F A ) is closed. 2) The de Rham class [p T F A )] HdR 2k M) is independent of the choices of A. 3) The ChernWeil map CW : I G), ) HdR M), ) that maps T to [p T F A )] is a ring homomorphism. Proof. 1) According to the Bianchi identity df α = [A α, F α ]. So by corollary 1.4, dp T F A ) = dt F α,, F α ) = T df α,, F α ) + + T F α,, df α ) = 0. This proves p T F A ) is closed. 2) To prove that the de Rham class [p T F A )] is independent of the choices of connection A, we let A 0 and A 1 be two connections defined by the collections A 0 α and A 1 α. By definition it is easy to check that the collection Ã α = 1 s)a 0 α + sa 1 α Ω 1 U α R) g defines a connection on a new principal bundle P R over M R This new principal bundle is defined over open sets {U α R} by using the same set of transition functions g αβ ). If we let ι 1 and ι 0 be the same as in the following lemma, then ι 0Ãα = A 0 α and ι 1Ãα = A 1 α. As a result, we see ι 0FÃ = F A 0 and ι 1FÃ = F A 1.
4 4 LECTURE 26: THE CHERNWEIL THEORY So according to the next lemma, p T F A 0) p T F A 1) = ι 0p T FÃ) ι 1p T FÃ) = dqp T FÃ)) + Qdp T FÃ)). But we just proved that p T FÃ) is closed. So [p T F A 0)] = [p T F A 1)]. Lemma 2.2. Let ι 0, ι 1 : M M R be the inclusions ι 0 x) = x, 0), ι 1 x) = x, 1). Then there exists a collection of linear operators Q : Ω k M R) Ω k 1 M) so that ι 0ω ι 1ω = dqω) Qdω), ω Ω k M R). Proof. One just repeat the first four lines of the proof of theorem 2.6 in lecture 19, to get some linear map Q : Ω k M R) Ω k 1 M R) so that ω φ 1ω = d Qω + Qdω. It follows that ι 0ω ι 1ω = ι 0ω ι 0φ 1ω = ι 0d Qω + ι 0 Qdω = dι 0 Q)ω + ι 0 Q)dω. So the conclusion holds for Q = ι Q 0 : Ω k M R) Ω k 1 M). 3) It is straightforward to show that for T S k G) and S S l G), p T S F A ) = p T F A ) p S F A ). Details left as an exercise. Example. Consider the trivial principal Gbundle P = M G over M. In this case one can take U α = M, i.e. the open cover contains only one element. Then there is only one g αα, which equals e G at each point x M. For such a covering data one can take A α to be identically zero. It follows that F A = 0 and thus for any T I G),the class [p T F A )] = 0. Definition 2.3. For any T I G), the cohomology class [p T F A )]] is called the characteristic class for P corresponding to T. Remark. Obviously the above construction works if we replace the principal bundle P by a vector bundle E. So one can also talk about the charactersitic classes of vector bundle E over M associated with T I G), where G is the structural group of E. Example. Let E be any vector bundle over M. By choosing a Riemannian metric one can always reduce the structural group of E from GLr, R) to Or). Let p k be the Pontrjagin polynomial that we alluded above. Then p k E) := [p k F A )] H 4k drm) is called the kth Pontrjagin characteristic class of E. The Pontrjagin class p k M) of M is defined to be the Pontrjagin class of T M. They are important topological invariants to study manifolds. Example. Suppose r = 2p and consider G = SOr). For any A = a i j) sor) we let P fa) = 1 4π) r r/2)! σ S r 1) σ a σ1) σ2) aσ3) σ4) aσr 1) σr). One can check that P f is an Adinvariant homogeneous polynomial of degree r/2. It is called the Pfaff polynomial.
5 LECTURE 26: THE CHERNWEIL THEORY 5 Now suppose E is an oriented vector bundle over M of rank r. Then the structural group of E can be reduced to SOr). Thus we get a characteristic class ee) := [P ff A )] H r drm) which is called the Euler characterstic class of E. Again the Euler class em) of a smooth manifold M is defined to be the Euler class of T M. It generalizes the classical notion of Euler characteristic. Example. All discusstions above are over R, but they can be generalized to objects over C. For example, one can talk about complex vector bundle E over smooth manifold M: They are vector bundles over M whose fibers are C r and whose structural group is GLr, C). For any A glr, C) = the set of all r r complex matrices), one can define c k to be such that det λi r 1 ) 2πi A = c k A)λ n k. Again c k is a homogeneous polynomial of degree k and is Adinvariant. As a result, for any complex vector bundle E of complex rank r, one gets a complexvalued de Rham cohomology class c k E) := c k F A ) HdRM; 2k C) which is called the kth Chern class of E. If one fix an Hermitian metric on E i.e. fix a Hermitian inner product on each fiber of E which vary smoothly with respect to base points this is always possible by using partition of unity), then one can reduce the structural group to Un). But for A un) = the set of all r r complex matrices with A + A T = 0), one has det λi r 1 ) 2πi A = det λi r 1 ) 2πi A. It follows that each c k is a realvalued polynomial and thus each c k E) H 2k dr M), i.e. c ke) is a realvalued) de Rham cohomology class. It is of fundamental importance in algebraic topology, differential geometry, algebraic geometry and mathematical physics. The End
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