Hopf Cyclic Cohomology and Characteristic Classes

Transcription

1 Hopf Cyclic Cohomology and Characteristic Classes Ohio State University Noncommutative Geometry Festival TAMU, April 30 - May 3, 2014

2 Origins and motivation The version of cyclic cohomology adapted to Hopf algebras emerged from the joint work with of A. Connes on the local index formula for the hypoelliptic signature operator on spaces of leaves of foliations. For foliations of codimension n, we have found a Hopf algebra H n which plays the role of a quantum structure group for their spaces of leaves. While the characteristic classes of foliations are described in terms of Gelfand-Fuks Lie algebra cohomology, the appropriate tool for above problem turned out to be Hopf cyclic cohomology. The two cohomologies were shown to be canonically isomorphic by an explicit, but quite intricate, quasi-isomorphism. The transplantation of the characteristic classes in the Hopf cyclic cohomological framework broadened the scope of their applicability. Thus, the issue of finding more explicit constructions of the Hopf cyclic characteristic classes becomes relevant.

3 Chern-Weil construction of characteristic classes = torsion-free connection on M n, with connection form on ω = (ω i j ) Ω1 (FM) gl n and curvature Ω = dω + ω ω, Ω = (Ω i j ) Ω2 (FM) gl n ; P I (gl n ) = S(gl n) GLn, the form P(Ω) is closed and basic, i.e. P(Ω) Ω (M), hence [P(Ω)] H (M, R). In particular, ( det Id t ) 2πi A = n t k c k (A), k=1 A gl n (C) give the classical Chern forms c k (Ω) Ω 2k (M), and the Pontryagin classes [p k (Ω)] = [c 2k (Ω)] H 4k (M, R).

4 Local Index Formula in Noncommutative Geometry Theorem (A. Connes & HM, 1995) Assume (A, H, D) = spectral triple, such that residue T := Res s=0 Tr(T D 2s ), T Ψ{A, [D, A], D z ; z C}. 1 [(ϕ n ) n=1,3,... ] is a cocycle in the (b, B)-bicomplex of A, ϕ n (a 0,..., a n ) = k c n,k a 0 [D, a 1 ] (k 1)... [D, a n ] (kn) D n 2 k (T ) = [D 2, a], T (k) = k (T ), k = k k n, ( 1) k Γ ( k + n ) 2 c n,k = k 1!... k n!(k 1 + 1)... (k k n + n). 2 [(ϕ n ) n=1,3,... ] = ch (H, F ) HC (A).

5 Relation with Atiyah-Singer Local Index Formula 1 The zeta functions associated to the Dirac spectral triple 2 (C (M m ), L 2 (S), /D) are meromorphic with simple poles. P = 1 (2π) n σ n (P). P ΨDO(M n ); S M (Guillemin-Wodzicki residue) 3 f 0 [/D, f 1 ] (k1)... [/D, f n ] (kn) /D (n+2 k ) = 0, if k > 0 ; 4 f 0 [/D, f 1 ]... [/D, f n ] /D n = ( 2 ) 1 /4πi 2 = c n det f 0 M sinh 2 df 1... df n ; /4πi 5 under the isomorphism (Connes, HKR) HP (C (M m )) = H dr (M, C), ch (H, /D) [(ϕ n ) ] = [Â(R)] Ch (/D).

6 Local index formula for codimension 1 M = S 1, Γ G = Diff(S 1 ) δ, H = L 2 (FS 1 S 1, y 1 dx dy dα) C 2 Q = 2y y α γ ( i y x γ 2 + (y y ) 2 α 2 1 ) γ 3, 4 where γ 1, γ 2, γ 3 M 2 (C) are the Pauli matrices. 1 ( ) 1 ϕ 1 (a 0, a 1 ) = Γ a 0 [Q, a 1 ](Q 2 ) 1/2 2i 2 1 ( ) 3 2 Γ a 0 [Q, a 1 ](Q 2 ) 3/ Γ Γ ( 7 2 ) a 0 3 [Q, a 1 ](Q 2 ) 7/2 0 ( ) 5 a 0 2 [Q, a 1 ](Q 2 ) 5/2 2

7 Local index cocycle 1 ϕ 3 (a 0, a 1, a 2, a 3 ) = 1 ( ) 3 2i 3i Γ a 0 [Q, a 1 ][Q, a 2 ][Q, a 3 ](Q 2 ) 3/2 2 1 ( ) Γ a 0 [Q, a 1 ][Q, a 2 ][Q, a 3 ](Q 2 ) 5/2 2 1 ( ) Γ a 0 [Q, a 1 ] [Q, a 2 ][Q, a 3 ](Q 2 ) 5/2 2 1 ( ) Γ a 0 [Q, a 1 ][Q, a 2 ] [Q, a 3 ](Q 2 ) 5/2 2 = transverse fundamental cocycle + boundary. While the computation is purely symbolical, it requires the symbol σ 4, hence about 10 3 terms!

8 Deciphering the transverse local index formula 1 The Chern character cocycle of the transverse signature operator D is a sum of cochains of the form a 0 [Q, a 1 ] (k 1)... [Q, a q ] (kq) Q q 2k, a i = f i U ϕ i A. 2 By the residue formula, these can be expressed as τ (a 0 h 1 (a 1 ) h q (a q )), where τ (f U ϕ) = f vol P, if ϕ = Id and 0 otherwise, h i are transverse differential operators, e.g. X (f U ϕ) = X (f ) U ϕ. 3 The operators h generate a Hopf algebra H, which acts on A and gives rise to a characteristic map χ q Γ : C q (H) H... H C q (A). 4 The Chern character of D is in the range of the induced map in cyclic cohomology χ Γ : HC (H) HC (A Γ ) (1).

9 Hopf algebra H 1 Action on A = C (F + S 1 ) G, ϕ(x, y) = (ϕ(x), ϕ (x) y), Y (fu ϕ) = y f y U ϕ, X (fu ϕ) = y f x U ϕ Y (f ϕ) = Y (f ) ϕ = Y (ab) = Y (a) b + a Y (b). X (f ϕ) = (X (f ) ϕ) + y ϕ (x) (Y (f ) ϕ) (x, y) ϕ (x) = X (ab) = X (a) b + a X (b) + δ 1 (a) Y (b). ( ) δ n := [X, δ n 1 ] = δ n (fuϕ) = y n dn dx log dϕ n dx fuϕ. As algebra= generated by {X, Y, δ 1, δ 2,...} subject to relations: [Y, X ] = X, [Y, δ k ] = kδ k, [X, δ k ] = δ k+1, [δ k, δ l ] = 0.

10 As coalgebra: (Y ) = Y Y, (X ) = X X + δ 1 Y, (δ 1 ) = δ δ 1, (δ 2 ) = δ δ 1 δ δ 2, (δ 3 ) = δ δ 2 δ 1 + 3δ 1 δ 2 + δ 2 1 δ δ 3, etc. Counit: ɛ(x ) = ɛ(y ) = ɛ(δ k ) = 0, ɛ(1) = 1. Antipode: S(1) = 1, S(X ) = X + δ 1 Y, S(Y ) = Y, S(δ 1 ) = δ 1, S(δ 2 ) = δ 2 1 δ 1,... Character δ H 1 : δ(y ) = 1, δ(x ) = 0, δ(δ n) = 0. Twisted antipode S δ (h) = δ(h (1) ) S(h (2) ) is involutive: S 2 δ = Id.

11 Hopf cyclic cohomology Cylic structure: H (δ,1) = {C n (H; C δ ) = H n } n 0 0 (h 1... h n 1 ) = 1 h 1... h n 1, j (h 1... h n 1 ) = h 1... h j... h n 1 n (h 1... h n 1 ) = h 1... h n 1 1 σ i (h 1... h n+1 ) = h 1... ε(h i+1 )... h n+1 τ n (h 1... h n ) = S δ (h 1 ) h 2... h n 1 ( n+1 n ) Bicomplex : b = ( 1) i i, B = ( 1) ni τn i σ n 1 τ n. i=0 i=0 Example (HP (H 1 ; C δ )) TF = X Y Y X δ 1 Y Y GV = δ 1 (fundamental class) (Godbillon-Vey class)

12 Relative Hopf cyclic cohomology K = Hopf subalgebra of H, C := H K C, with K acting on H by right multiplication and on C by the counit. As left H-module C H/HK +, where K + = Ker ε K, via h + HK + ḣ = h K 1 H K C. Cylic structure: {C n (H, K; C σ δ ) = C n } n 0 0 (c 1... c n 1 ) = 1 c c n 1, i (c 1... c n 1 ) = c 1... c(1) i ci (2)... cn 1, n (c 1... c n 1 ) = c 1... c n 1 1 ; σ i (c 1... c n+1 ) = c 1... ε(c i+1 )... c n+1, τ n (ḣ 1 c 2... c n ) = S δ (h 1 ) (c 2... c n 1).

13 Transverse Index Theorem Theorem (A. Connes & HM, 1998) There are canonical constructions for the following entities: 1 a Hopf algebra H n associated to Diff(R n ), with modular character δ, and modular pair (δ, 1); 2 an isomorphism κ between the Gelfand-Fuks cohomology H GF (a n, O n ) and HP (H n, O n ; C δ ); 3 an action of H n on A G (F R n ), and a characteristic map χ(h 1... h n )(a 0,..., a n ) = τ(a 0 h 1 (a 1 )... h n (a n )) χ : HP (H n, O n ; C δ ) HP (1) (A G(PR n )), PR n := F R n /O n 4 ch (D) (1) = χ (L), L HP (H n, O n ; C δ ) = H GF (a n, O n ).

14 Diff-invariant de Rham cohomology de Rham complex of invariant forms on -jet bundle {Ω (P M) G, d} for M n = smooth manifold, G = Diff(M) δ F k M = k jets at 0 of local diffeos ρ : R n M M F 1 M F 2 M F M := lim F k M P k M = F k M/O n M P 1 M P 2 M P M := lim P k M G action : φ G, ρ F k M = φ j 0 (ρ) := j 0 (φ ρ) Gelfand-Fuks Lie algebra cohomology complex of formal vector fields {C (a n ), d} a n = {v = j0 ( d dt ) t=0 ρ t, t ρt : R n R n } ( ) d ṽ j 0 (φ)= j0 dt t=0 (φ ρ t ) ; ω(ṽ 1,..., ṽ m ) = ω(v 1,..., v m ) DGA-isomorphism ω C (a n, O n ) ω Ω (P M) G

15 Diff-equivariant de Rham cohomology Simplicial manifold G M = { G M[p] = G p+1 M} p 0, i (ρ 0,..., ρ p, x) = (ρ 0,..., ˇρ i,..., ρ p ), 0 i p, σ i (ρ 0,..., ρ p, x) = (ρ 0,..., ρ i, ρ i,..., ρ p, x), 0 i p. Geometric realization G M = p G M[p]/ Dupont complex of (covariant) compatible forms {Ω G M, d} ω = {ω p } p 0, ω p Ω ( p G M[p]) (µ Id) ω q = (Id µ ) ω p Ω ( p G M[q]) ω(ρ 0 ρ,..., ρ p ρ) = ρ ω(ρ 0,..., ρ p ), ρ, ρ i G Bott complex { C (G, Ω (M)), δ, d} c(ρ 0,..., ρ p ) Ω q (M) c(ρ 0 ρ,..., ρ p ρ) = ρ c(ρ 0,..., ρ p ), ρ, ρ i G p δ c(ρ 0,..., ρ p ) = ( 1) i c(ρ 0,..., ˇρ i,..., ρ p ). i=0

16 Differentiable cohomology (à la Haefliger) Differentiable cochain ω C p d (G, Ωq (M)) if locally, ω(ρ 0,..., ρ p, x) = ) f I (x, jx k (ρ 0 ),..., jx k (ρ p ) dx I. Differentiable compatible form ω = {ω p } p 0 Ω d ( G M ) if ω p (t; ρ 0,..., ρ p, x) = ) f I,J (t; x, jx k (ρ 0 ),..., jx k (ρ p ) dt I dx J. Theorem (Differentiable analogue of Dupont s Theorem) The chain map : Ω Φ d ( G M ) C d (G, Ω (M)) induces an isomorphism H d ( GM, R) = H d,g (M, R).

17 Explicit van Est-Haefliger quasi-isomorphism = torsion free connection = cross-section to π 1 : F M FM σ (u) = j 0 (exp x u); σ φ = φ 1 σ φ, φ G, σ p (t; ρ 0,..., ρ p, u) = σ p 0 t i ρ i (u); ˆσ = {σ p } p 0 : G FM F M. Theorem The chain map C (ω) = ˆσ ( ω) Ω d ( G FM ) induces quasi-iso of DG-algebras C : C (a n, On) Ω d ( G (PM, On) ). Corollary The composition D = is quasi-isomorphism. Φ C : C (a n, On) tot C d (G, Ω (PM))

18 Hopf cyclic analogue of van Est isomorphism This involves Connes map Φ : C (G, Ω p (F R n )) CC (Cc (F R n ) G). If λ Im(D ) where now M = R n, and = flat connection, then Φ(λ) is of the form Φ(λ)(a 0,..., a l ) = α τ(a 0 h 1 α(a 1 )... h q α(a q )), h i α H n, with the tensor α h1 α... h q α H q n uniquely determined by λ. One obtains a chain map Υ : Im(D ) CC tot (H n, C δ ), Υ(λ) = α h 1 α... h q α H q n. Theorem (AC & HM 1998, 2001) The composition Υ D : C (a n, On) CC tot (H n, O n ; C δ ) is quasi-isomorphism.

19 Characteristic cocycles by simplicial Chern-Weil The universal connection ϑ = (ϑ i j ), where ϑi j ( n k=1 ξk k ) = j ξ i, and curvature forms R = (Rj i), where Ri j = dϑ i j + ϑi k ϑk j in C (a n ) generate a DG-subalgebra CW (a n ). By Gelfand-Fuks Thm. CW (a n ) C (a n ) is quasi-isomorphism. CW (a n ) = Ŵ (gl n) = W (gl n )/I 2n, where W (gl n ) = gl n S(gl n) and I 2n = ideal generated by elements of S(gl n) of deg > 2n. Lemma For any torsion-free connection, σ ( ϑ i j ) = ωi j and σ ( R i j ) = Ωi j. Simplicial connection and curvature: ˆω p (t; ρ 0,..., ρ p ) := p i=0 t iρ i (ω) Ω1 d ( G FM ) ˆΩ := d ˆω + ˆω ˆω Ω 2 d ( G FM ), ˆΩ p (t; ρ 0,..., ρ p ) = p i=0 dt i ρ i (ω)+ p i=0 t ( i ρ i (Ω) ρ i (ω) ρ i (ω)) + p i,j=0 t it j ρ i (ω) ρ j (ω).

20 Vey basis in differentiable Dupont algebra The forms ˆω i j and ˆΩ i j generate a DG-subalgebra CW d ( G FM ), and C gives isomorphism between CW (a n ) Ŵ (gl n ) and CW d ( G FM ). c k (ˆΩ) = d(tc k (ˆω)), with Tc k (ˆω) = k 1 0 c k ˆΩ t = t ˆΩ + (t 2 t)ˆω ˆω. (ˆω, ˆΩ t,..., ˆΩ t ) dt, By restriction to O n -basic elements, C induces an isomorphism of CW (a n, On)) Ŵ (gl n, On) onto CW d ( G PM ). c 2k (ˆΩ) remain but c 2k 1 (ˆΩ) = d(tc 2k 1 (ˆω)), with Tc 2k 1 (ˆω) = (2k 1) ) 1 0 c 2k 1 (s(ˆω), ˆΩ t,..., ˆΩ t dt, ˆΩ t = ts(ˆω) + o(ˆω) + (t 2 1)s(ˆω) s(ˆω). {Tc I (ˆω) c J (ˆΩ)} (I,J) Vn, resp. {Tc I (ˆω) c J (ˆΩ)} (I,J) VOn, represents a basis in cohomology.

21 Vey basis in differentiable Bott complex Corollary The cocycles obtained by their integration along fibers, C I,J ( ) := Tc I (ˆω) c J (ˆΩ), (I, J) VO n, Φ form a complete set of representatives for a basis of H d,g (PM, R). For example, C,{2k} ( ) = C 2k (ˆΩ) = c 2k (ˆΩ) is the cocycle Φ C 2k (ˆΩ) = {C (p) 2k (ˆΩ)} p 0 with components ) C (p) 2k (ˆΩ)(φ 0,..., φ p ) = ( 1) p c 2k (ˆΩ(t; φ 0,..., φ p ) = Φ p = ( 1) p ( 1) µ ˆΩi 1µ(i1 1 i 1 <...<i 2k n µ S p ) ˆΩ iq µ(i 2k ) (t; φ 0,..., φ p ). 2k

22 Generators of the Hopf algebra H n H n acts on A = C c (F R n ) G where G = Diff(R n ) δ by X k = y µ k x µ, Y j i (f U φ ) = Y j i (f ) U φ, X k (f U φ ) = X k(f ) U φ, δ i jk (f U φ ) = γi jk (φ) f U φ, Y j i = y µ i y µ ; j δ i jk l 1...l r := [X lr,... [X l1, δ i jk ]...] ; U φ Y j i U φ = Y j i ; U φ X k U φ = X k γ i jk (φ) Y j i, δ i jk l 1...l r (f U φ ) := γi jk l 1...l r (φ) f U φ, where γ i jk l 1...l r (φ) := X lr X l1 ( γ i jk (φ) ) Basic algebra generators : {X k, Y j i, δi jk }

23 Transfer to Hopf cyclic cohomology Let M = R n, G = Diff(R n ) and = trivial connection. Identify F R n = R n GL n (R). One has ω i j := (y 1 ) i µ dy µ j = ( y 1 dy ) i j, φ (ω i j ) = ω i j + γ i jk (φ) θk, φ G γ i j k (φ)(x, y) = ( y 1 φ (x) 1 µ φ (x) y ) i ˆω(t; φ 0,..., φ p ) i j = ˆΩ(t; φ 0,..., φ p ) = + p t r φ r (ωj i ) = r=0 i, j = 1,..., n j y µ k p t r γjk i (φ r ) θ k r=0 p dt r φ r (ω) r=0 p t r t s φ r (ω) φ s (ω). r,s=0 p t r φ r (ω) φ r (ω) i=0

24 Vey basis in Hopf cyclic complex Theorem (1) The cocycles Υ(C I,J ( )), with (I, J) V n, form a complete set of representatives for the periodic Hopf cyclic cohomology HP (H n ; C δ ). (2) The cocycles Υ(C I,J ( )), with (I, J) VO n, form a complete set of representatives for HP (H n, On; C δ ). (3) Every cohomology class in HP (H n ; C δ ) and HP (H n, On; C δ ), can be represented by cocycles manufactured out of the algebra generators {X k, Y i j, δi jk } of H n. Proof. In addition to the δjk i from the previous slide, the map Φ, when applied to monomials a = f Uφ A, brings in the operators X k and Yj i, via df = n k=1 X k(f )θ k + n i,j=1 Y j i(f ) ωj i.

25 Example: Chern cocycle The top component of {C q (p) (ˆΩ)} p 0, q = 2k, in the simplified cyclic model κ q (q) (ˆΩ) = 0 θ j q η i 1 η iq µ(i q),j q. 1 i s,j t n µ S q ( 1) µ θ j1 µ(i 1 ),j 1 0 The lower components κ (p) q ( ˆΩ) are given by similar expressions, with coefficients of the form t k 1 1 tkp p dt 1 dt p = p 1 (k 1 + 1) (k k p + p) ; note the resemblance with the coefficients appearing in the local index formula.