Secondary invariants for two-cocycle twists

Transcription

1 Secondary invariants for two-cocycle twists Sara Azzali (joint work with Charlotte Wahl) Institut für Mathematik Universität Potsdam Villa Mondragone, June 17, 2014 Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

2 Overview Outline and keywords context: index theory of elliptic operators primary: index,indexclass secondary: eta,rho,analytictorsion... Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

3 Overview Outline and keywords context: index theory of elliptic operators primary: index,indexclass secondary: eta,rho,analytictorsion... on the universal covering M of a closed manifold projectively invariant operators 2-cocycle twists Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

4 Overview Outline and keywords context: index theory of elliptic operators primary: index,indexclass secondary: eta,rho,analytictorsion... on the universal covering M of a closed manifold projectively invariant operators 2-cocycle twists in physics: magnetic fields, quantum Hall effect in geometry: main ideas (Gromov, Mathai) c H 2 (BΓ, R) natural C -bundles of small curvature pairing without the extension properties Joint with Charlotte Wahl: define η and ρ for Dirac operators twisted by a 2-cocycle use ρ to distinguish geometric structures (positive scalar curvature..) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

5 Introduction Spectral invariants of elliptic operators Primary: the index D elliptic, Dirac type on M, closed manifold in particular: 1 d + d on a Riemannian manifold M 2 D/ the Dirac on a spin manifold M. spec D = {λ j } j N primary ind D := dim Ker D dim Coker D Z Theorem (Atiyah Singer 1963) ind D + = Â(M) ch E/S M Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

6 Primary: the index Introduction Spectral invariants of elliptic operators D elliptic, Dirac type on M, closed manifold in particular: 1 d + d on a Riemannian manifold M 2 D/ the Dirac on a spin manifold M. spec D = {λ j } j N primary ind D := dim Ker D dim Coker D Z Theorem (Atiyah Singer 1963) ind D + = Â(M) ch E/S 1 d + d on Λ +/ ind(d + )=sign(m) = 2 D/ on spinors ind D/ + = Â(M) M M M L(M) Hirzebruch s theorem Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

7 Introduction Secondary: the eta invariant Spectral asymmetry Atiyah Patodi Singer, 1974: for D = D (dim M =odd) η(d, s) = 0 λ j spec D η(d) :=η(d, s) s=0 = sign(λ j ) λ j s 0 λ j spec D sign(λ j ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

8 Introduction Secondary: the eta invariant Spectral asymmetry Atiyah Patodi Singer, 1974: for D = D (dim M =odd) M = W η(d, s) = 0 λ j spec D η(d) :=η(d, s) s=0 = sign(λ j ) λ j s 0 λ j spec D sign(λ j ) Atiyah Patodi Singer theorem (1974) ind D W = Â(W ) ch E/S 1 2 (η(d W )+dim Ker D W ) W η(d) spectral contribution, non-local! M = S 1, η( i d dx + a) = { 0, a Z 2a 1, a (0, 1) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

9 Secondary: rho invariants Introduction rho invariants and classification of positive scalar curvature metrics Atiyah Patodi Singer: α: Γ U(k) flatbundle ρ α (D) :=η(d k ) η(d α ) Cheeger Gromov: ρ Γ (D) :=η Γ ( D) η(d) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

10 Introduction rho invariants and classification of positive scalar curvature metrics Secondary: rho invariants Atiyah Patodi Singer: α: Γ U(k) flatbundle ρ α (D) :=η(d k ) η(d α ) Cheeger Gromov: ρ Γ (D) :=η Γ ( D) η(d) Property: ρ can distinguish geometric structures M closed spin D/ 2 g = scal g then scal g > 0 D/ g invertible Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

11 Secondary: rho invariants Introduction rho invariants and classification of positive scalar curvature metrics Atiyah Patodi Singer: α: Γ U(k) flatbundle ρ α (D) :=η(d k ) η(d α ) Cheeger Gromov: ρ Γ (D) :=η Γ ( D) η(d) Property: ρ can distinguish geometric structures M closed spin D/ 2 g = scal g then scal g > 0 D/ g invertible (g t ) t [0,1] R + (M) :={g metric on TM scal g > 0} 0= W Â(W )+ 1 2 η(d/ g 0 ) 1 2 η(d/ g 1 ) 0= W Â(W ) ch α η(d/α g 0 ) 1 2 η(d/α g 1 ) it gives a map ρ(d/): π 0 (R + (M)) R Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

12 Introduction Role of primary vs. secondary invariants Role of index and rho: positive scalar curvature M closed spin manifold D/ 2 g = scal g. R + (M) ={g metric on TM scal g > 0} ind D/ is an obstruction (ind D/ 0 R + (M) = ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

13 Introduction Role of primary vs. secondary invariants Role of index and rho: positive scalar curvature M closed spin manifold D/ 2 g = scal g. R + (M) ={g metric on TM scal g > 0} ind D/ is an obstruction (ind D/ 0 R + (M) = ) ρ(d/) can distinguish non-cobordant metrics, assuming R + (M), Theorem: (Piazza Schick, Botvinnik Gilkey) R + (M) dim M =4k +3, k > 0 π1(m) hastorsion Theorem: (Piazza Schick) infinitely many non-cobordant {g j } j N R + (M) ρ(d/ gi ) ρ(d/ gj ) i j π1(m) torsionfree,andsatisfiesbaum Connes ρ(d/ g )=0forg R + (M) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

14 Projectively invariant operators Projective actions π : M M universal covering, Γ = π1 (M) Exemple: R n R n /Z n = T n Γ-invariant: Dγ = γ D γ Γ Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

15 Projectively invariant operators Projective actions π : M M universal covering, Γ = π1 (M) Exemple: R n R n /Z n = T n Γ-invariant: Dγ = γ D γ Γ projectively Γ-invariant : BT γ = T γ B γ Γ, where T γ T γ = σ(γ,γ )T γγ, T e = I then σ :Γ Γ U(1) is a multiplier, i.e.[σ] H 2 (Γ, U(1)) σ(γ 1,γ 2 )σ(γ 1 γ 2,γ 3 )=σ(γ 1,γ 2 γ 3 )σ(γ 2,γ 3 ) σ(e,γ)=σ(γ,e) =1 Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

16 Typical construction Projectively invariant operators Example: the magnetic Laplacian π : M M universal covering, Γ = π1 (M). On the trivial line L = M C M consider = d + ia, whereda Ω 2 ( M, R) isγ-invariant:da = π ω, ω Ω 2 (M, R) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

17 Projectively invariant operators Example: the magnetic Laplacian Typical construction π : M M universal covering, Γ = π1 (M). On the trivial line L = M C M consider Then: = d + ia, whereda Ω 2 ( M, R) isγ-invariant:da = π ω, ω Ω 2 (M, R) γ A A = dψ γ σ(γ,γ )=exp(iψ γ (γ x 0 )) is a multiplier H A =(d + ia) (d + ia), called the magnetic Laplacian, is projectively invariant with respect to T γ u =(γ 1 ) e iψγ u, u L 2 ( M) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

18 Projectively invariant operators Example: the magnetic Laplacian Typical construction π : M M universal covering, Γ = π1 (M). On the trivial line L = M C M consider Then: = d + ia, whereda Ω 2 ( M, R) isγ-invariant:da = π ω, ω Ω 2 (M, R) γ A A = dψ γ σ(γ,γ )=exp(iψ γ (γ x 0 )) is a multiplier H A =(d + ia) (d + ia), called the magnetic Laplacian, is projectively invariant with respect to T γ u =(γ 1 ) e iψγ u, u L 2 ( M) Remark: the construction can be done starting from c H 2 (BΓ, R) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

19 Projectively invariant operators Two-cocycle twists Properties and the L 2 -index Properties of B = D : (Gromov, Mathai) BT γ = T γ B B is affiliated to B(L 2 ( M)) T N(Γ,σ) L 2 (F) Q B(L 2 ( M)) T, tr Γ,σ (Q) = F k Q(x, x) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

20 Projectively invariant operators Two-cocycle twists Properties and the L 2 -index Properties of B = D : (Gromov, Mathai) BT γ = T γ B B is affiliated to B(L 2 ( M)) T N(Γ,σ) L 2 (F) Q B(L 2 ( M)) T, tr Γ,σ (Q) = F k Q(x, x) at the level of Schwartz kernels, if QT γ = T γ Q,then e iψγ(x) k Q (γx,γy)e iψγ(y) = k Q (x, y) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

21 Projectively invariant operators Two-cocycle twists Properties and the L 2 -index Properties of B = D : (Gromov, Mathai) BT γ = T γ B B is affiliated to B(L 2 ( M)) T N(Γ,σ) L 2 (F) Q B(L 2 ( M)) T, tr Γ,σ (Q) = F k Q(x, x) at the level of Schwartz kernels, if QT γ = T γ Q,then e iψγ(x) k Q (γx,γy)e iψγ(y) = k Q (x, y) Atiyah s L 2 -theorem (Gromov) ind Γ,σ B = Â(M) ch(e/s) e ω where ω = f c, f : M BΓ classifyingmap M Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

22 Projectively invariant operators Bundles of small curvature Making the curvature small Mathai s constructions: given c H 2 (BΓ, R) Mischchenko-type C (Γ,σ)-bundle V C (Γ,σ) M whose curvature is ω I idea: pass to σ s = e isψ corresponds to curvature = sω I Theorem: (Mathai) For c H 2 (BΓ, R), sign(m, c) := Proof: tr Γ,σ ind(d sign VC (Γ,σ) )= j M L(M) f c are homotopy invariants. s j L(M) ω j j! M Hilsum Skandalis s theorem: ind(d sign VC (Γ,σ) ) is a homotopy invariant, for s small enough Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

23 Projectively invariant operators What are eta and rho? Eta and rho for 2-cocycle twists Questions: 1 define η c (D)? (easy) 2 prove an index theorem for manifolds with boundary 3 find interesting ρ c (D) 4 relate them with higher rho-invariants (defined by Lott, Leichtnam Piazza) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

24 Definitions Projectively invariant operators What are eta and rho? Let τ s be a family of positive (finite) traces on C (Γ,σ s )s.t. τ s (δ γ )=τ s (χ(γ)δ γ ), for any homomorphism χ: Γ U(1) ητ c s (D) := 1 π 0 Tr τ (D VC (Γ,σ) e td V C (Γ,σ) ) dt t ρ c τ s (D) = same expression, with τ s delocalized (τ s (δ e ) = 0) assume here D VC (Γ,σ s ) is invertible Examples of such traces on C (Γ,σ) 1 tr Γ,σ ( a γ γ)=a e 2 on Γ = Γ 1 Γ 2,withΓ 1 perfect and σ = π 2 σ,forσ H 2 (Γ 2, U(1)), take any τ 1 tr Γ,σ Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

25 Projectively invariant operators What are eta and rho? Positive scalar curvature Lemma (Mathai) M spin, g R + (M). For s small enough, D VC (Γ,σ s ) is invertible Definition: Let M be spin, g R + (M). Callρ c τ s (D/) the direct limit of s ρ c τ s (D/), s U ρ c τ s lim 0 U Maps(U C) Theorem: bordism invariance of ρ c τ s If (M, g 0 )and(m, g 1 ) are Γ-cobordant in R + (M), then ρ c τ (D/ g0 )=ρ c τ (D/ g1 ) Proof: immediate, applying a C -algebraic Atiyah Patodi Singer index theorem, and the Lemma. Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

26 Projectively invariant operators What are eta and rho? Theorem (with C. Wahl 2013) M closed spin, connected, π 1 (M) =Γ 1 Γ 2, dim M =4k +1 Γ 1 has torsion N, π 1 (N) =Γ 2 s.t. N Â(N) f N c 0, c H 2 (BΓ 2, Q) If R + (M) then {g j } j N R + (M) non cobordant ρ c τ s (D/ gi ) ρ c τ s (D/ gj ). Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

27 Projectively invariant operators What are eta and rho? Theorem (with C. Wahl 2013) M closed spin, connected, π 1 (M) =Γ 1 Γ 2, dim M =4k +1 Γ 1 has torsion N, π 1 (N) =Γ 2 s.t. N Â(N) f N c 0, c H 2 (BΓ 2, Q) If R + (M) then {g j } j N R + (M) non cobordant ρ c τ s (D/ gi ) ρ c τ s (D/ gj ). Idea of the proof (after Botvinnik Gilkey, Piazza Schick, Piazza Leichtnam): Start with g 0 R + (M). Apply a machine to generate new non-cobordant metrics on M: a) Bordism theorem (by Gromov Lawson, Rosenberg Stolz, Botvinnik Gilkey) b) X 4k+1 closed spin, π 1(X )=Γ, ρ c τ s (D/ X ) 0, j [X ]=0 Ω spin,+ k (BΓ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15

28 The machine: Projectively invariant operators What are eta and rho? a) Bordism theorem (Gromov Lawson): M, N spin Γ-cobordant manifold, M connected. If g R + (M), then G R + (W ) (of product structure near the boundary) b) Basic case: Γ = Z n (Botvinnik Gilkey): dim M = 7. There exists N 1 = S 7 /Z n lens space, g N1 R(N 1 ), and ρ Γ (D/ N1,g N1 ) 0. Moreover: Ω spin,+ k (BZ n )isfinite j [N 1 ]=0 Ω spin,+ k (BΓ) (here it is pure bordism, no PSC!) then k: ρ Γ (M, g k )=ρ Γ ((M, g 0 ) k(jn 1, g N1 )) = ρ Γ (M, g 0 )+kjρ Γ (N 1, g N1 ) ρ Γ (M, g k ) ρ Γ (M, g h ) k, h. Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, / 15