Noncommutative Geometry Lecture 3: Cyclic Cohomology

Transcription

1 Noncommutative Geometry Lecture 3: Cyclic Cohomology Raphaël Ponge Seoul National University October 25, / 21

2 Hochschild Cohomology Setup A is a unital algebra over C. Definition (Hochschild Complex) 1 The space of n-cochains, n 0, is C n (A) := { (n + 1)-linear forms ϕ : A n+1 C }, n 0. 2 The coboundary b : C n (A) C n+1 (A) is given by (bϕ)(a 0,, a n+1 ) = ( 1) j ϕ(a 0,, a j a j+1,, a n+1 ) 0 j n + ( 1) n+1 ϕ(a n+1 a 0,, a n ). 2 / 21

3 Hochschild Cohomology Lemma We have b 2 = 0. Definition The cohomology of the complex (C (A), b) is called the Hoschschild cohomology of A and is denoted HH (A). 3 / 21

4 Hochschild Cohomology Example Let C be a k-dimensional current on a compact manifold M. Define a k-cochain on A = C (M) by ϕ C (f 0, f 1,, f k ) = 1 k! C, f 0 df 1 df k. Then bϕ C = 0. In fact, we have Theorem (Hochschild-Kostant-Rosenberg, Connes) There is an isomorphism, HH k (M) D k (M). 4 / 21

5 Cyclic Cohomology (Connes, Tsygan) Definition (Cyclic Cochains) A cochain ϕ C n (A), n 0, is cyclic when ϕ(a 1,, a n, a 0 ) = ( 1) n ϕ(a 0,, a n ) a j A. We denote by Cλ n (A) the space of cyclic n-cochains. Example Let C be a k-dimensional current on a compact manifold M. We saw it defines a Hochschild cocycle, Then C closed (i.e., d t C = 0) = ϕ C cyclic. 5 / 21

6 Cyclic Cohomology Lemma ϕ cyclic = bϕ cyclic. Definition The cohomology of the sub-complex (C λ (A), b) is called the cyclic cohomology of A and is denoted HC (A). 6 / 21

7 Periodic Cyclic Cohomology (Connes, Tsygan) Definition Define B : C n (A) C n 1 (A) by B = AB 0, where B 0 : C n (A) C n 1 (A) and A : C n 1 (A) C n 1 (A) are given by B 0 ϕ(a 0,, a n 1 ) = ϕ(1, a 0,, a n 1 ) ( 1) n ϕ(a 0,, a n 1, 1), Aψ(a 0,, a n 1 ) = ( 1) j(n 1) ψ(a j,, a n 1, a 0,, a j 1 ). Remark 1 If ϕ is a cyclic cochain, then B 0 ϕ = 0, and hence Bϕ = 0. 2 An n-cochain ϕ is cyclic if and only if Aϕ = 1 n+1 ϕ. 7 / 21

8 Periodic Cyclic Cohomology Example Let C be a k-dimensional current on a compact manifold M. It defines a Hochschild cocycle, We then have ϕ C (f 0, f 1,, f k ) = 1 k! C, f 0 df 1 df k. Bϕ C = ϕ d t C. Lemma We have B 2 = 0 and bb + Bb = 0. 8 / 21

9 Periodic Cyclic Cohomology Definition (Even/Odd Cochains) Define C even (A) := k 0 C 2k (A) and { } = ϕ = (ϕ 0, ϕ 2, ); ϕ 2k C 2k (A), ϕ 2k = 0 for large k, C odd (A) := k 0 C 2k+1 (A) { } = ϕ = (ϕ 1, ϕ 3, ); ϕ 2k+1 C 2k+1 (A), ϕ 2k+1 = 0 for large k. 9 / 21

10 Periodic Cyclic Cohomology Proposition We have a 2-periodic complex, C even (A) b+b C odd (A). Definition The cohomology of ( C even/odd (A), b + B ) is called the periodic cyclic cohomology of A and is denoted HC even/odd (A). 10 / 21

11 Periodic Cyclic Cohomology Example Let C = C 0 + C 2 + be an even current on a compact manifold M. Then C defines an even cochain, ϕ C = (ϕ C0, ϕ C2, ), ϕ C2k (f 0, f 1,, f 2k ) = 1 (2k)! C 2k, f 0 df 1 df 2k. Then (b + B)ϕ C = Bϕ C = ϕ d t C. Theorem (Connes) C closed = ϕ C even cyclic cocycle. The map C ϕ C gives rise to isomorphisms, H even/odd (M) HC even/odd (C (M)). 11 / 21

12 Periodic Cyclic Cohomology Remark Assume M is oriented, Riemannian and has even dimension. Then the Â-form Â(R M ) defines an even cyclic cocycle by i.e., ϕâ(r M ) = (ϕ 0, ϕ 2, ), with ϕâ(r M ) = ϕ Â(R M ), ϕ 2k (f 0, f 1,, f 2k ) = 1 f 0 df 1 df 2k (2k)! Â(RM ). M Likewise the Pfaffian Pf(R M ), the L-form L(R M ) and the Todd form Td(R M ) define even cyclic cocycles. 12 / 21

13 Morita Equivalence Definition Let ϕ C n (A). The n-cochain ϕ# Tr on M q (A) = A M q (C) is defined by ϕ# Tr(a 0 µ 0,, a n µ n ) := ϕ(a 0,, a n ) Tr [ µ 0 µ 1 µ n] for all a j A and µ j M q (C). Lemma We have b (ϕ# Tr) = (bϕ)# Tr. Theorem (Connes) The map ϕ ϕ# Tr gives rise to isomorphisms, HC (A) HC (M q (A)). 13 / 21

14 Morita Equivalence Example Let C be a k-dimensional current on a compact manifold M. For the cochain, we have ϕ C (f 0, f 1,, f k ) = 1 k! C, f 0 df 1 df k, ϕ C # Tr(a 0, a 1,, a k ) = 1 k! C, Tr [a 0 da 1 da k] for all a j in M q (C (M)) = C (M, M q (C)). 14 / 21

15 Pairing with K-Theory (A = C (M)) Setup Lemma M is a compact manifold. E = ran e, e = e 2 M q (C (M)). F E is the curvature of the Grassmanian connection E of E. 1 F E = e(de) 2 = e(de) 2 e. 2 Ch(F E ) = ( 1) k k! Tr [ e(de) k]. 15 / 21

16 Pairing with K-Theory (A = C (M)) Proposition Let C = C 0 + C 2 + be a closed even current on M with associated even cocycle ϕ C = (ϕ C0, ϕ C2, ). Then C, E = ( 1) k (2k)! (ϕ C2k # Tr) (e, e, e), k! = ϕ C0 (e) + ( 1) k (2k)! (ϕ C2k # Tr) (e 12 ) k!, e, e. k 1 16 / 21

17 Pairing with K-Theory (General Case) Setup A is a unital algebra over C. Definition A cochain ϕ C n (A), n 1, is normalized when ϕ(a 0, a 1,, a n ) = 0 whenever a j = 1 for some j 1. Lemma Any class in HC even (A) contains a normalized representative. 17 / 21

18 Pairing with K-Theory Example Let C be a k-dimensional current on a compact manifold M with associated cochain, ϕ C (f 0, f 1,, f k ) = 1 k! C, f 0 df 1 df k. Then ϕ C is a normalized cochain. 18 / 21

19 Pairing with K-Theory Definition Let ϕ = (ϕ 0, ϕ 2, ) be an even cyclic cocycle and let E = ea q, e = e 2 M q (A), a finitely generated projective module. The pairing of ϕ and E is ϕ, E := ϕ 0 (e) + ( 1) k (2k)! (ϕ 2k # Tr) (e 12 ) k!, e, e. k 1 Theorem (Connes) The above pairing descends to a bilinear pairing,, : HC even (A) K 0 (A) C. 19 / 21

20 Pairing with K-Theory Example Let C = C 0 + C 2 + be a closed even current on a compact manifold M and let E = ran e, e = e 2 M q (C (M)), so that E = C (M, E) C (M) q. Then ϕ C, E = C, E. 20 / 21

21 The Atiyah-Singer Index Theorem Example Assume M is spin, oriented, Riemannian and has even dimension. 1 For C = Â(R M ) and the Dirac operator, ϕâ(r M ), E = Â(RM ), E and ind /D [E] = ind /D [E]. 2 By the K-theoretic version Atiyah-Singer Index Theorem explained in Lecture 2, ind /D [E] = (2iπ) n 2 Â(R M ), E. 3 The Atiyah-Singer Index Theorem then can be further restated as Theorem ind /D [E] = (2iπ) n 2 ϕâ(r M ), E E K 0(C (M)). 21 / 21