488 Characteristic Classes

Gilmore R and Letellier C (2006) The Symmetry of Chaos Alice Ott E (1993) Chaos in Dynamical Systems. Cambridge: Cambridge in the Land of Mirrors. Oxford: Oxford University Press. University Press. Gilmore R and Pei X (2001) The topology and organization of Solari HG, Natiello MA, and Mindlin GB (1996) Nonlinear unstable periodic orbits in Hodgkin–Huxley models of receptors Physics and Its Mathematical Tools. Bristol: IoP Publishing. with subthreshold oscillations. In: Moss F and Gielen S (eds.) Tufillaro NB, Abbott T, and Reilly J (1992) An Experimental Handbook of Biological Physics, Neuro-informatics, Neural Approach to Nonlinear Dynamics and Chaos. Reading, MA: Modeling, vol. 4, pp. 155–203. Amsterdam: North-Holland. Addison-Wesley.

Characteristic Classes P B Gilkey, University of Oregon, Eugene, OR, USA Frames R Ivanova, University of Hawaii Hilo, Hilo, HI, USA A frame s := (s , . . . , s ) for V Vect (M, F) over an S Nikcˇ evic´ , SANU, Belgrade, Serbia and Montenegro 1 k k open set M is a collection2of k smooth sections ª 2006 Elsevier Ltd. All rights reserved. O  to V so that {s1(P), . . . , sk(P)} is a basis for the jO fiber VP of V over any point P . Given such a frame s, we can construct a local trivializat2 O ion which Vector Bundles identifies Fk with V by the mapping O Â jO Let Vectk(M, F) be the set of isomorphism classes of P; 1; . . . ;  1s1 P  s P real (F = R) or complex (F = C) vector bundles of ð kÞ ! ð Þ þ Á Á Á þ k kð Þ rank k over a smooth connected m-dimensional Conversely, given a local trivialization of V, we can manifold M. Let take the coordinate frame

Vect M; F Vect M; F si P P 0; . . . ; 0; 1; 0; . . . ; 0 ð Þ ¼ kð Þ ð Þ ¼ Â ð Þ k [ Thus, frames and local trivializations of V are equivalent notions. Principal Bundles – Examples Let H be a Lie group. A fiber bundle Simple Covers  : M An open cover { } of M, where ranges over some P ! indexing set A, Ois said to be a simple cover if any with fiber H is said to be a principal bundle if there finite intersection 1 k is either empty or is a right action of H on which acts transitively on contractible. O \ Á Á Á \ O P the fibers, that is, if =H = M. If H is a closed Simple covers always exist. Put a Riemannian P subgroup of a Lie group G, then the natural metric on M. If M is compact, then there exists a projection G G=H is a principal H bundle over uniform 0 so that any geodesic ball of radius is !  >  the homogeneous space G=H. Let O(k) and U(k) geodesically convex. The intersection of geodesically denote the orthogonal and unitary groups, respec- convex sets is either geodesically convex (and hence k k 1 tively. Let S denote the unit sphere in R þ . Then contractible) or empty. Thus, covering M by a finite we have natural principal bundles: number of balls of radius  yields a simple cover. The argument is similar even if M is not compact O k O k 1 Sk ð Þ  ð þ Þ ! where an infinite number of geodesic balls is used 2k 1 U k U k 1 S þ and the radii are allowed to shrink near . ð Þ  ð þ Þ ! 1 Let RPk and CPk denote the real and complex k 1 Transition Cocycles projective spaces of lines through the origin in R þ k 1 and C þ , respectively. Let Let Hom(F, k) be the set of linear transformations of Fk and let GL(F, k) Hom(F, k) be the group of all Z2 Id O k  ¼ fÆ g  ð Þ invertible linear transformations. 1 S  Id :  1 U k Let {s } be frames for a vector bundle V over some ¼ f Á j j ¼ g  ð Þ open cover { } of M. On the intersection , 1 One has Z2 and S principal bundles: O O \ O one may express s = s , that is k 1 k 1 Z2 S À RP À j ! ! s ;i P ; P s ;j P 1 2k 1 k 1 ð Þ ¼ i ð Þ ð Þ S S À CP À 1 j k ! ! X  Characteristic Classes 489

The maps : GL(F, k) satisfy Spin Structures O \ O ! For k 3, the fundamental group of SO(k) is Z2. Id on  ¼ O 1 Let Spin(k) be the universal cover of SO(k) and let on ½ Š ¼ O \ O \ O  : Spin k SO k ð Þ ! ð Þ Let G be a Lie group. Maps belonging to a be the associated double cover; set Spin(2) = S1 and collection { } of smooth maps from to G let () = 2. An oriented bundle V is said to be spin which satisfy eqn [1] are said to Obe \transO ition if the transition functions can be lifted from SO(k) cocycles with values in G; if G GL(F, k), they to Spin(k); this is possible if and only if the second can be used to define a vector bundl e by making Stiefel–Whitney class of V, which is defined later, appropriate identifications. vanishes. There can be inequivalent spin structures, which are parametrized by the cohomology group 1 Reducing the Structure Group H (M; Z2). If G is a subgroup of GL(F, k), then V is said to have a G-structure if we can choose frames so the The Tangent Bundle of Projective Space transition cocycles belong to G; that is, we can The tangent bundle TRPm of real projective space is reduce the structure group to G. orientable if and only if m is odd; TRPm is spin if Denote the subgroup of orientation-preserving and only if m 3 mod 4. If m 3 mod 4, there are linear maps by two inequivalen t spin structures on this bundle as 1 m H (RP ; Z2) = Z2. GLþ R; k : GL R; k : det > 0 ð Þ ¼ f 2 ð Þ ð Þ g The tangent bundle TCPm of complex projective space is always orientable; TCPm is spin if and only If V Vectk(M, R), then V is said to be orientable if we can2 choose the frames so that if m is odd.

GLþ R; k Principal and Associated Bundles 2 ð Þ Let H be a Lie group and let Not every real vector bundle is orientable; the first Stiefel–Whitney class sw (V) H1(M; Z ), which is 1 2  : H defined later, vanishes if and 2only if V is orientable. O \ O ! In particular, the Mo¨ bius line bundle over the circle be a collection of smooth functions satisfying the is not orientable. compatibility conditions given in eqn [1]. We define Similarly, a real (resp. complex) bundle V is a principal bundle by gluing H to H P O Â O Â said to be Riemannian (resp. Hermitian) if we can using : reduce the structure group to the orthogonal group P; h P;  P h for P O(k) GL(R, k) (resp. to the unitary group ð Þ  ð ð Þ Þ 2 O \ O U(k)  GL(C, k)).  Because right multiplication and left multiplication We can use a partition of unity to put a positive- commute, right multiplication gives a natural action definite symmetric (resp. Hermitian symmetric) fiber of H on : metric on V. Applying the Gram–Schmidt process P then constructs orthonormal frames and shows that P; h h~ : P; h h~ the structure group can always be reduced to O(k) ð Þ Á ¼ ð Á Þ (resp. to U(k)); if V is a real vector bundle, then the The natural projection =H = M is an H fiber structure group can be reduced to the special bundle. P ! P orthogonal group SO(k) if and only if V is Let  be a representation of H to GL(F, k). For orientable.  ,  Fk, and h H, define a gluing 2 P 2 2 1 ;   hÀ ;  h  Lifting the Structure Group ð Þ  ð Á ð Þ Þ The associated vector bundle is then given by Let  be a representation of a Lie group H to GL(F, k). One says that the structure group of V can k k  F : F = be lifted to H if there exist frames {s } for V and P Â ¼ P Â  smooth maps  : H, so  = Clearly, { } are the transition cocycles of the where eqn [1] holds forO \.O ! vector bundle Fk. P Â 490 Characteristic Classes

Frame Bundles Homotopy

If V is a vector bundle, the associated principal Two smooth maps f0 and f1 from N to M are GL(F, k) bundle is the bundle of all frames; if V is said to be homotopic if there exists a smooth map given an inner product on each fiber, then the F : N I M so that f0(P) = F(P, 0) and so that  ! associated principal O(k) or U(k) bundle is the bundle f1(P) = F(P, 1). If f0 and f1 are homotopic maps from of orthonormal frames. If V is an oriented Riemannian N to M, then f1ÃV is isomorphic to f2ÃV. vector bundle, the associated principal SO(k) bundle is Let [N, M] be the set of all homotopy classes the bundle of oriented orthonormal frames. of smooth maps from N to M. The association V f V induces a natural map ! à Direct Sum and Tensor Product N; M Vect M; F Vect N; F ½ Š  kð Þ ! kð Þ Fiber-wise direct sum (resp. tensor product) defines the If M is contractible, then the identity map is direct sum (resp. tensor product) of vector bundles: homotopic to the constant map c. Consequently, k V = IdÃV is isomorphic to cÃV = M F . Thus, any : Vectk M; F Vectn M; F Â È ð Þ Â ð Þ vector bundle over a contractible manifold is trivial. Vectk n M; F In particular, if { } is a simple cover of M and if ! þ ð Þ : Vect M; F Vect M; F V Vect(M, F), thenO V is trivial for each . This k n ð Þ Â ð Þ shows2 that a simple coverjO is a trivializing cover for Vectkn M; F ! ð Þ every V Vect(M, F). The transition cocycles of the direct sum (resp. 2 tensor product) of two vector bundles are the direct Stabilization sum (resp. tensor product) of the transition cocycles of the respective bundles. Let l Vect1(M, F) denote the isomorphism class of the trivial2 line bundle M F over an m-dimensional The set of line bundles Vect1(M, F) is a group manifold M. The map V V l induces a stabili- under . The unit in the group is the trivial line ! È bundle l := M F; the inverse of a line bundle L is zation map the dual line bundl e L := Hom(L, F) since à s : Vectk M; F Vectk 1 M; F ð Þ ! þ ð Þ L Là l ¼ which induces an isomorphism

Vectk M; R Vectk 1 M; R for k > m ð Þ ¼ þ ð Þ 2 Pullback Bundle Vectk M; C Vectk 1 M; C for 2k > m ½ Š ð Þ ¼ þ ð Þ Let  : V M be the projection associated with These values of k comprise the stable range. V Vect !(M, F). If f is a smooth map from N to M, 2 k then the pullback bundle f ÃV is the vector bundle The K-Theory over N which is defined by setting The direct sum and tensor product make È f ÃV : P; v N V : f P  v Vect(M, F) into a semiring; we denote the associated ¼ fð Þ 2  ð Þ ¼ ð Þg ring defined by the Grothendieck construction by The fiber of f V over P is the fiber of V over f (P). à KF(M). If V Vect(M, F), let [V] KF(M) be the Let {s } be local frames for V over an open cover 2 2 1 corresponding element of K-theory; KF(M) is gener- { } of M. For P f À ( ), define O 2 O ated by formal differences [V1] [V2]; such formal À f Ãs P : P; s f P differences are called virtual bundles. f gð Þ ¼ ð ð ð ÞÞÞ The Grothendieck construction (see K-theory) This gives a collection of frames for f ÃV over the m 1 introduces nontrivial relations. Let S denote the open cover {f À ( )} of N. Let m 1 O standard sphere in R þ . Since f à : f ¼  T Sm l m 1 l be the pullback of the transition functions. Then ð Þ È ¼ ð þ Þ we can easily see that [TSm] = m[ l ] in KR(Sm), m f Ãs P P; f P s f P despite the fact that T(S ) is not isomorphic to ml f gð Þ ¼ ð ð ð ÞÞ ð ð ÞÞÞ for m 1, 3, 7. f à f Ãs P ¼ fð Þð Þgð Þ Let 6¼L denote the nontrivial real line bundle over This shows that the pullback of the transition RPk. Then TRPk l = (k 1)L, so functions for V are the transition functions of the È þ TRPk k 1 L l pullback f Ã(V). ½ Š ¼ ð þ Þ½ Š À ½ Š Characteristic Classes 491

The map V Rank(V) extends to a surjective respectively. The topology on these spaces is the map from KF(M!) to Z. We denote the associated weak or inductive topology. The Grassmannians are ideal of virtual bundles of virtual rank 0 by called classifying spaces. The isomorphisms of eqn [4] are compatible with the inclusions of eqn [5] KF M : ker Rank ð Þ ¼ ð Þ and we have In the stable range, V [V] k[ l ] identifies M; Gr F; Vect M; F 6 f ! À ½ kð 1ފ ¼ kð Þ ½ Š Vectk M; R KR M if k > m ð Þ ¼ ð Þ 3 Vect M; C KC M if 2k > m ½ Š kð Þ ¼ gð Þ Spaces with Finite Covering Dimension These groups contain nontrivial torsion. Let L be the g k A metric space X is said to have a covering nontrivial real line bundle over RP . Then dimension at most m if, given any open cover { } U k  k of X, there exists a refinement { } of the cover so KR RP Z L l =2 ð ÞZ L l O ð Þ ¼ Á f½ Š À ½ Šg f½ Š À ½ Šg that any intersection of more than m 1 of the { } where (k) is the Adams number. is empty. For example, any manifoldþof dimensiOon g m has covering dimension at most m. More Classifying Spaces generally, any m-dimensional cell complex has covering dimension at most m. Let Grk(F, n) be the Grassmannian of k-dimensional n n The isomorphisms of [2]–[4], and [6] continue to subspaces of F . By mapping a k-plane  in F to the hold under the weaker assumption that M is a metric corresponding orthogonal projection on , we can space with covering dimension at most m. identify Grk(F, n) with the set of orthogonal projec- tions of rank k:

n 2  Hom F :  ; à ; tr  k Characteristic Classes of Vector f 2 ð Þ ¼ ¼ ð Þ ¼ g There is a natural associated tautological k-plane Bundles bundle The Cohomology of Gr (F, ) k 1 V F; n Vect Gr F; n ; F The cohomology algebras of the Grassmannians are kð Þ 2 kð kð Þ Þ polynomial algebras on suitably chosen generators: whose fiber over a k-plane  is the k-plane itself: V F; n : ; x Hom Fn Fn : x x Hà Grk R; ; Z2 Z2 sw1; . . . ; swk kð Þ ¼ fð Þ 2 ð Þ Â ¼ g ð ð 1Þ Þ ¼ ½ Š 7 Hà Grk C; ; Z Z c1; . . . ; ck ½ Š Let [M, Grk(F, n)] denote the set of homotopy ð ð 1Þ Þ ¼ ½ Š equivalence classes of smooth maps f from M to Grk(F, n). Since [f1] = [f2] implies that f1ÃV is The Stiefel–Whitney Classes isomorphic to f2ÃV, the association Let V Vectk(M, R). We use eqn [6] to find f f ÃVk F; n Vectk M; F É : M 2Gr (R, ) which classifies V; the map É ! ð Þ 2 ð Þ ! k 1 induces a map is uniquely determined up to homotopy and, using eqn [7], one sets M; Grk F; n Vectk M; F ½ ð ފ ! ð Þ i sw V : ÉÃsw H M; Z2 This map defines a natural equivalence of functors ið Þ ¼ i 2 ð Þ in the stable range: The total Stiefel–Whitney class is then defined by

M; Gr R;  k Vect M; R for  > m sw V 1 sw1 V swk V ½ kð þ ފ ¼ kð Þ 4 ð Þ ¼ þ ð Þ þ Á Á Á þ ð Þ M; Grk C;  k Vectk M; C for 2 > m ½ Š The Stiefel–Whitney class has the properties: ½ ð þ ފ ¼ ð Þ n n 1 The natural inclusion of F in F þ induces natural 1. If f : X1 X2, then f Ã(sw(V)) = sw(f ÃV). inclusions 2. sw(V W!) = sw(V)sw(W). È 1 3. If L is the Mo¨ bius bundle over S , then sw1(L) Grk F; n Grk F; n 1 1 1 ð Þ  ð þ Þ 5 generates H (S ; Z2) = Z2. Vk F; n Vk F; n 1 ½ Š ð Þ  ð þ Þ The cohomology algebra of real projective space Let Grk(F, ) and Vk(F, ) be the direct limit is a truncated polynomial algebra: spaces under1these inclusions;1 these are the infinite- k k 1 dimensional Grassmannians and classifying bundles, Hà RP ; Z2 Z2 x =x þ 0 ð Þ ¼ ½ Š ¼ 492 Characteristic Classes

k 4i Since TRP l = (k 1)L, one has classes pi(V) H (X; Z) are characterized by the È þ properties: 2 k k 1 sw TRP 1 x þ ð Þ ¼ ð þ Þ k 1 k 1. p(V) = 1 p1(V) p (V). 1 kx ð þ Þ x2 8 þ þ Á Á Á þ k ¼ þ þ 2 þ Á Á Á ½ Š 2. If f : X1 X2, then f Ã(p(V)) = p(f ÃV). 3. p(V W!) = p(V)p(W) mod elements of order 2. È 2 4. 2 p1(TCP ) = 3. Orientability and Spin Structures CP R The Stiefel–Whitney classes have real geometric We can complexify a real vector bundle V to meaning. For example, sw1(V) = 0 if and only if V construct an associated complex vector bundle VC. is orientable; if sw1(V) = 0, then sw2(V) = 0 if and We have only if V admits a spin structure. With reference to i p V : 1 c2 V the discussion on the tangent bundle or projective ið Þ ¼ ðÀ Þ ið CÞ space, eqn [8] yields Conversely, if V is a complex vector bundle, we can construct an underlying real vector bundle by k 0 if k 0 mod 2 VR sw1 TRP ð Þ ¼ x if k  1 mod 2 forgetting the underlying complex structure. Mod- &  ulo elements of order 2, we have Thus, RPk is orientable if and only if k is odd. Furthermore, p VR c V c Và ð Þ ¼ ð Þ ð Þ k k 0 if k 3 mod 4 Let TCP be the real tangent bundle of complex sw2 TRP ð Þ ¼ x if k  1 mod 4 projective space. Then &  k k 2 k 1 Thus, TRP is spin if and only if k 3 mod 4. p TCP 1 x þ  ð Þ ¼ ð À Þ Chern Classes Let V Vect (M, C). We use eqn [6] to find k Line Bundles É : M 2Gr (C, ) which classifies V; the map É ! k 1 is uniquely determined up to homotopy and, using Tensor product makes Vect1(M, F) into an abelian eqn [7], one sets group. One has natural equivalences of functors which are group homomorphisms: 2i ci V : ÉÃci H M; Z ð Þ ¼ 2 ð Þ 1 sw1 : Vect1 M; R H M; Z2 The total Chern class is then defined by ð Þ ! ð Þ 2 c1 : Vect1 M; C H M; Z c V : 1 c1 V c V ð Þ ! ð Þ ð Þ ¼ þ ð Þ þ Á Á Á þ kð Þ The Chern class has the properties: A real line bundle L is trivial if and only if it is orientable or, equivalently, if sw1(L) vanishes. A 1. If f : X1 X2, then f Ã(c(V)) = c(f ÃV). complex line bundle L is trivial if and only if 2. c(V W!) = c(V)c(W). È c1(L) = 0. There are nontrivial vector bundles with 3. Let L be the classifying line bundle over vanishing Stiefel–Whitney classes of rank k > 1. For 2 1 k S = CP . Then S2 c1(L) = 1. example, swi(TS ) = 0 for i > 0 despite the fact that À k The cohomologyR algebra of complex projective TS is trivial if and only if k = 1, 3, 7. space also is a truncated polynomial algebra

k k 1 Hà CP ; Z Z x =x þ ð Þ ¼ ½ Š Curvature and Characteristic Classes where x = c1(L) and L is the complex classifying line k k de Rham Cohomology bundle over CP = Gr1(C, k 1). If T CP is the þ c complex tangent bundle, then We can replace the coefficient group Z by C at the cost

k k 1 of losing information concerning torsion. Thus, we c TcCP 1 x þ 4i ð Þ ¼ ð þ Þ may regard pi(V) H (M; C) if V is real or ci(V) H2i(M; C) if V is2 complex. Let M be a smoot2h manifold. Let C ÃpM be the space of smooth The Pontrjagin Classes 1 p-forms and let Let V be a real vector bundle over a topological p p 1 space X of rank r = 2k or r = 2k 1. The Pontrjagin d : C1Ã M C1Ã þ M þ ! Characteristic Classes 493 be the exterior derivative. The de Rham cohomology The curvature operator can also be computed groups are then defined by locally. Let (si) be a local frame. Expand

p p 1 j p ker d : C1Ã M C1Ã þ M si !i sj HdeR M : ð 1 ! Þ ð Þ ¼ im d : C Ãp M C ÃpM r ¼ j ð 1 À ! 1 Þ X The de Rham theorem identifies the topological to define the connection 1-form !. One then has cohomology groups Hp(M; C) with the de Rham p cohomology groups H (M) which are given 2 j k j deR si d! ! ! sk differential geometrically. r ¼ i À i ^ k   Given a connection on V, the Chern–Weyl theory and so enables us to compute Pontrjagin and Chern classes in de Rham cohomology in terms of curvature. j d! j !k ! j i ¼ i À i ^ k If s˜ = jsj is another local frame, we compute Connections i 1 1 1 Let V be a vector bundle over M. A connection !~ dggÀ g!gÀ and ~ g gÀ ¼ þ ¼ : C1 V C1 TÃM V Although the connection 1-form ! is not tensorial, the r ð Þ ! ð Þ curvature is an invariantly defined 2-form-valued on V is a first-order partial differential operator endomorphism of V. which satisfies the Leibnitz rule, that is, if s is a smooth section to V and if f is a smooth function on M, Unitary Connections fs df s f s Let ( , ) be a nondegenerate Hermitian inner product rð Þ ¼ þ r on VÁ. WÁ e say that is a unitary connection if If X is a tangent vector field, we define r s X; s s1; s2 s1; s2 d s1; s2 rX ¼ h r i ðr Þ þ ð r Þ ¼ ð Þ where , denotes the natural pairing between the Such connections always exist and, relative to a tangenthÁandÁi cotangent spaces. This generalizes to the local orthonormal frame, the curvature is skew- bundle setting the notion of a directional derivative symmetric, that is, and has the properties: Ã 0 1. s = f s. þ ¼ rfX rX 2. X(fs) = X(f )s f Xs. Thus, can be regarded as a 2-form-valued element r þ r of the Lie algebra of the structure group, O(V) in the 3. X1 X2 s = X1 s X2 s. r þ r þ r 4. (s1 s2) = s1 s2. real setting or U(V) in the complex setting. rX þ rX þ rX

The Curvature 2-Form Projections  Let !p be a smooth p-form. Then We can always embed V in a trivial bundle 1 of dimension ; let V be the orthogonal projection on p p 1 : C1 à M V C1 à þ M V V. We project the flat connection to V to define a r ð Þ ! ð Þ can be extended by defining natural connection on V. For example, if M is embedded isometrically in the Euclidean space R, ! s d! s 1 p! s this construction gives the Levi-Civita connection on rð p Þ ¼ p þ ðÀ Þ p ^ r the tangent bundle TM. The curvature of this In contrast to ordinary exterior differentiation, 2 connection is then given by need not vanish. We set r s : 2s V dV dV ð Þ ¼ r ¼ This is not a second-order partial differential Let VP be the fiber of V over a point P M. The n 2 operator; it is a zeroth-order operator, that is, inclusion i : V R defines the classifying map  f : P Grk(R, n) where we set fs ddf s df s df s f 2s ! ð Þ ¼ À ^ r þ ^ r þ r f s f P i VP ¼ ð Þ ð Þ ¼ ð Þ 494 Characteristic Classes

Chern–Weyl Theory Other Characteristic Classes Let be a Riemannian connection on a real vector The Chern character is defined by the exponential bundler V of rank k. We set function. There are other characteristic classes which appear in the index theorem that are defined 1 p : det I using other generating functions that appear in ð Þ ¼ þ 2 index theory. Let x : (x , ) be a collection of   = 1 . . . indeterminates. Let ( ) be the th elementary T s x  Let denote the transpose matrix of differential symmetric function; form. Since T = 0, the polynomials of odd þ degree in vanish and we may expand 1 x 1 s1 x s2 x  ð þ Þ ¼ þ ð Þ þ ð Þ þ Á Á Á p 1 p1 p Y ð Þ ¼ þ ð Þ þ Á Á Á þ rð Þ For a diagonal matrix A := diag(1, . . . ), denote the where k = 2r or k = 2r 1 and the differential forms normalized eigenvalues by xj := p 1j=2. Then 4i þ À pi( ) C1Ã (M) are forms of degree 4i. Changi2 ng the gauge (i.e., the local frame) replaces p 1 ffiffiffiffiffiffiffi 1 c A det 1 À A 1 s1 x by g gÀ and hence p( ) is independent of the 2 ð Þ ¼ þ ffiffiffiffiffiffiffi ! ¼ þ ð Þ þ Á Á Á local frame chosen. One can show that dpi( ) = 0; let [pi( )] denote the corresponding element of de Thus, the Chern class corresponds in a certain sense Rham cohomology. This is independent of the to the elementary symmetric functions. particular connection chosen and [pi( )] represents Let f (x) be a symmetric polynomial or more 4i pi(V) in H (M; C). generally a formal power series which is symmetric. Similarly, let V be a complex vector bundle of We can express f (x) = F(s1(x), . . . ) in terms of the rank k with a Hermitian connection . Set elementary symmetric functions and define r f ( ) = F(c1( ), . . . ) by substitution. For example, p 1 the Chern character is defined by the generating c : det I À 2 function ð Þ ¼ þ ffiffiffiffiffiffi ffi ! k 1 c1 ck ¼ þ ð Þ þ Á Á Á þ ð Þ f x : ex ð Þ ¼ Again c ( ) is independent of the local gauge and  1 i X¼ dci( ) = 0. The de Rham cohomology class [ci( )] The Todd class is defined using a different 2i represents ci(V) in H (M; C). generating function:

x 1 td x : x 1 eÀ À The Chern Character ð Þ ¼  ð À Þ Y The total Chern character is defined by the formal 1 td1 x ¼ þ ð Þ þ Á Á Á sum If V is a real vector bundle, we can define p 1 =2 some additional characteristic classes similarly. Let ch : tr e À ð Þ ¼ ð Þ { p 11, . . . } be the nonzero eigenvalues of a pffiffiffiffiffi 1  Æ À  skew-symmetric matrix A. We set xj = j=2 ð À Þ tr ffiffiffiffiffiffiffi ¼ 2 ! ð Þ and define the Hirzebruch polynomial L andÀ the Aˆ  ð ffiffiffiffiffiffiÞ ffi X genus by ch0 ch1 ¼ ð Þ þ ð Þ þ Á Á Á x Let ch(V) = [ch( )] denote the associated de Rham L x : ð Þ ¼  tanh x cohomology class; it is independent of the particular Y ð Þ connection chosen. We then have the relations 1 L1 x L2 x ¼ þ ð Þxþ ð Þ þ Á Á Á A^ x :  ch V W ch V ch W ð Þ ¼ 2 sinh 1=2 x ð È Þ ¼ ð Þ þ ð Þ  ðð Þ Þ ch V W ch V ch W Y 1 A^ 1 x A^ 2 x ð Þ ¼ ð Þ ð Þ ¼ þ ð Þ þ ð Þ þ Á Á Á The Chern character extends to a ring isomorph- The generating functions ism from KU(M) Q to He(M; Q), which is a x x natural equivalence of functors; modulo torsion, and tanh x 2 sinh 1=2 x K theory and cohomology are the same functors. ð Þ ðð Þ Þ Characteristic Classes 495

are even functions of x, so the ambiguity in the If M is an even-dimensional manifold, let em(M) := choice of sign in the eigenvalues plays no role. This em(TM). If we reverse the local orientation of M, defines characteristic classes then em(M) changes sign. Consequently, em(M) is a

4i 4i measure rather than an m-form; we can use the Li V H M; C and A^ i V H M; C ð Þ 2 ð Þ ð Þ 2 ð Þ Riemannian measure on M to regard em(M) as a scalar. Let Rijkl be the components of the curvature of Summary of Formulas the Levi-Civita connection with respect to some local orthonormal frame field; we adopt the convention We summarize below some of the formulas in terms 3 that R = 1 on the standard sphere S2 in R . If of characteristic classes: 1221 "I,J := (eI, eJ) is the totally antisymmetric tensor, then p 1tr( ) 1. c1( ) = À , I;J " Ri1i2j2j1 Rim 1imjmjm 1 2 e : Á Á Á À À ffiffiffiffiffiffiffi 2n 8 n 1 2 2 ¼ I; J  n! 2. c2( ) = {tr( ) tr( ) }, ð Þ 82 À X 1 2 Let := Rijji and ij := Rikkj be the scalar curvature 3. p1( ) = tr( ), R À 82 and the Ricci tensor, respectively. Then 2 c1 2c2 4. ch(V) = k c1 À (V), 1 þ þ 2 þ Á Á Á e2 & 2 ' ¼ 4 R c1 (c1 c2) c1c2 5. td(V)= 1 þ (V), 1 2 2 2 þ 2 þ 12 þ 24 þ ÁÁÁ e4 4  R & ' ¼ 322 ðR À j j þ j j Þ 2 p1 7p 4p2 6. Aˆ (V) = 1 1 À (V), À 24 þ 5760 þ Á Á Á & ' 2 Characteristic Classes of Principal p1 7p2 p 7. L(V) = 1 À 1 (V), þ 3 þ 45 þ Á Á Á Bundles & ' 8. td(V W) = td(V)td(W), Let g be the Lie algebra of a compact Lie group G. È 9. Aˆ (V W) = Aˆ (V)Aˆ (W), Let  : M be a principal G bundle over M. For È  , Plet! 10. L(V W) = L(V)L(W). 2 P È  : ker  : T TM and  : ? V ¼ Ã P ! H ¼ V The Euler Form be the vertical and horizontal distributions of the So far, this article has dealt with the structure groups projection , respectively. We assume that the metric O(k) in the real setting and U(k) in the complex on is chosen to be G-invariant and such that P setting. There is one final characteristic class which  :  TM is an isometry; thus,  is a Rieman- Ã H ! arises from the structure group SO(k). Suppose k = 2n nian submersion. If F is a tangent vector field on M, is even. While a real antisymmetric matrix A of shape let F be the corresponding vertical lift. Let  be H V 2n 2n cannot be diagonalized, it can be put in block orthogonal projection on the distribution . The V offÂ2-diagonal form with blocks, curvature is defined by

0  F1; F2  F1 ; F2  ð Þ ¼ V ½Hð Þ Hð ފ  0 the horizontal distribution is integrable if and only if  À  H The top Pontrjagin class p (A) = x2 x2 is a perfect the curvature vanishes. Since the metric is G-invariant, n 1 Á Á Á n square. The Euler class (F1, F2) is invariant under the group action. We may use a local section s to P over a contractible coordinate e2 A : x1 x 1 nð Þ ¼ Á Á Á n chart to split À = G. This permits us to identifyO with TG Oand OtoÂregard as a g-valued is the square root of pn. If V is an oriented vector V bundle of dimension 2n, then 2-form. If we replace the section s by a section s˜, then 1 ˜ = g gÀ changes by the adjoint action of G on g. 2n e2n V H M; C If V is a real or complex vector bundle over M, ð Þ 2 ð Þ is a well-defined characteristic class satisfying we can put a fiber metric on V to reduce the 2 structure group to the orthogonal group O(r) in the e2n(V) = pn(V). If V is the underlying real oriented vector bundle real setting or the unitary group U(r) in the complex of a complex vector bundle W, setting. Let V be the associated frame bundle. A Riemannian Pconnection on V induces an invariant r e2n V cn W splitting of T = and defines a natural ð Þ ¼ ð Þ PV V È H 496 Chern–Simons Models: Rigorous Results

metric on V ; the curvature of the connection Bott R and Tu LW (1982) Differential forms in algebraic defined hereP agrees with the definition previously.r topology. Graduate Texts in Mathematics, p. 82. New York– Let (G) be the algebra of all polynomials on Berlin: Springer-Verlag. Q Chern S (1944) A simple intrinsic proof of the Gauss–Bonnet g which are invariant under the adjoint action. If formula for closed Riemannian manifolds. Annals of Mathe- Q (G), then Q( ) is well defined. One has matics 45: 747–752. dQ2( Q) = 0. Furthermore, the de Rham cohomology Chern S (1945) On the curvatura integra in a Riemannian class Q(P) := [Q( )] is independent of the particular manifold. Annals of Mathematics 46: 674–684. connection chosen. We have Conner PE and Floyd EE (1964) Differentiable periodic maps. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 33. New York: Academic Press; Berlin–Go¨ ttingen– U k C c1; . . . ; c Qð ð ÞÞ ¼ ½ kŠ Heidelberg: Springer-Verlag. SU k C c2; . . . ; ck de Rham G (1950) Complexes a` automorphismes et home´omorphie Qð ð ÞÞ ¼ ½ Š diffe´rentiable (French). Ann. Inst. Fourier Grenoble 2: 51–67. O 2k C p1; . . . ; p Qð ð ÞÞ ¼ ½ kŠ Eguchi T, Gilkey PB, and Hanson AJ (1980) Gravitation, gauge O 2k 1 C p1; . . . ; p theories and differential geometry. Physics Reports 66: 213–393. Qð ð þ ÞÞ ¼ ½ kŠ 2 Eilenberg S and Steenrod N (1952) Foundations of Algebraic SO 2k C p1; . . . ; p ; e =e p Qð ð ÞÞ ¼ ½ k kŠ k ¼ k Topology. Princeton, NJ: Princeton University Press. SO 2k 1 C p1; . . . ; pk Greub W, Halperin S, and Vanstone R (1972) Connections, Qð ð þ ÞÞ ¼ ½ Š Curvature, and Cohomology. Vol. I: De Rham Cohomology Thus, for this category of groups, no new character- of Manifolds and Vector Bundles. Pure and Applied Mathe- istic classes ensue. Since the invariants are Lie- matics, vol. 47. New York–London: Academic Press. algebra theoretic in nature, Hirzebruch F (1956) Neue topologische Methoden in der algebraischen Geometrie (German). Ergebnisse der Mathema- Spin k SO k tik und ihrer Grenzgebiete (N.F.), Heft 9. Berlin–Go¨ ttingen– Qð ð ÞÞ ¼ Qð ð ÞÞ Heidelberg: Springer-Verlag. Other groups, of course, give rise to different Husemoller D (1966) Fibre Bundles. New York–London–Sydney: characteristic rings of invariants. McGraw-Hill. Karoubi M (1978) K-theory. An introduction. Grundlehren der Mathematischen Wissenschaften, Band 226. Berlin–New York: Acknowledgments Springer-Verlag. Kobayashi S (1987) Differential Geometry of Complex Vector Research of P Gilkey was partially supported by Bundles. Publications of the Mathematical Society of Japan, 15. the MPI (Leipzig, Germany), that of R Ivanova by Kanoˆ Memorial Lectures, 5. Princeton, NJ: Princeton University the UHH Seed Money Grant, and of S Nikcˇevic´ by Press; Tokyo: Iwanami Shoten. Milnor JW and Stasheff JD (1974) Characteristic Classes. Annals MM 1646 (Serbia), DAAD (Germany), and Dierks of Mathematics Studies, No. 76. Princeton, NJ: Princeton von Zweck Stiftung (Esen, Germany). University Press; Tokyo: University of Tokyo Press. Steenrod NE (1962) Cohomology Operations. Lectures by NE See also: Cohomology Theories; Gerbes in Quantum Steenrod written and revised by DBA Epstein. Annals of Mathe- Field theory; Instantons: Topological Aspects; K-Theory; matics Studies, No. 50. Princeton, NJ: Princeton University Press. Mathai-Quillen Formalism; Riemann Surfaces. Steenrod NE (1951) The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. Princeton, NJ: Princeton University Press. Further Reading Stong RE (1968) Notes on Cobordism Theory. Mathematical Notes. Princeton, NJ: Princeton University Press; Tokyo: Besse AL (1987) Einstein manifolds. Ergebnisse der Mathematik University of Tokyo Press. und ihrer Grenzgebiete (3) [Results in Mathematics and Weyl H (1939) The Classical Groups. Their Invariants and Related Areas (3)], p. 10. Berlin: Springer-Verlag. Representations. Princeton, NJ: Princeton University Press.

Chern–Simons Models: Rigorous Results A N Sengupta, Louisiana State University, challenges for mathematicians. Most of the tremen- Baton Rouge, LA, USA dous amount of mathematical activity generated by ª 2006 Elsevier Ltd. All rights reserved. Witten’s discovery has been concerned primarily with issues that arise after one has accepted the functional integral as a formal object. This has left, as an important challenge, the task of giving rigorous Introduction meaning to the functional integrals themselves and to The relationship between topological invariants and rigorously derive their relation to topological invar- functional integrals from quantum Chern–Simons iants. The present article will discuss efforts to put the theory discovered by Witten (1989) raised several functional integral itself on a rigorous basis.