Secondary invariants for two-cocycle twists Sara Azzali (joint work - PowerPoint PPT Presentation

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

Outline and keywords Overview context: index theory of elliptic operators ◮ primary: index, index class ◮ secondary: eta, rho, analytic torsion... Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 2 / 15

Outline and keywords Overview context: index theory of elliptic operators ◮ primary: index, index class ◮ secondary: eta, rho, analytic torsion... 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, 2014 2 / 15

Outline and keywords Overview context: index theory of elliptic operators ◮ primary: index, index class ◮ secondary: eta, rho, analytic torsion... on the universal covering ˜ M of a closed manifold projectively invariant operators 2-cocycle twists in physics: magnetic fields, quantum Hall e ff ect in geometry: main ideas (Gromov, Mathai) ◮ c ∈ H 2 ( B Γ , 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, 2014 2 / 15

Introduction Spectral invariants of elliptic operators Primary: the index D elliptic, Dirac type on M , closed manifold in particular: d + d ∗ on a Riemannian manifold M 1 D / “the Dirac”on a spin manifold M . 2 spec D = { λ j } j ∈ N ind D := dim Ker D − dim Coker D ∈ Z primary Theorem (Atiyah–Singer 1963) � ind D + = � A ( M ) ch E / S M Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 3 / 15

Introduction Spectral invariants of elliptic operators Primary: the index D elliptic, Dirac type on M , closed manifold in particular: d + d ∗ on a Riemannian manifold M 1 D / “the Dirac”on a spin manifold M . 2 spec D = { λ j } j ∈ N ind D := dim Ker D − dim Coker D ∈ Z primary Theorem (Atiyah–Singer 1963) � ind D + = � A ( M ) ch E / S M � d + d ∗ on Λ + / − ind ( D + ) = sign ( M ) = L ( M ) Hirzebruch’s theorem 1 M � / + = � / on spinors A ( M ) D ind D 2 M Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 3 / 15

Introduction Spectral asymmetry Secondary: the eta invariant Atiyah–Patodi–Singer, 1974: for D = D ∗ ( dim M = odd) � sign ( λ j ) η ( D , s ) = | λ j | s 0 � = λ j ∈ spec D � η ( D ) := η ( D , s ) | s =0 “ = ” sign ( λ j ) 0 � = λ j ∈ spec D Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 4 / 15

Introduction Spectral asymmetry Secondary: the eta invariant Atiyah–Patodi–Singer, 1974: for D = D ∗ ( dim M = odd) � sign ( λ j ) η ( D , s ) = | λ j | s 0 � = λ j ∈ spec D � η ( D ) := η ( D , s ) | s =0 “ = ” sign ( λ j ) 0 � = λ j ∈ spec D M = ∂ W Atiyah–Patodi–Singer theorem (1974) � A ( W ) ch E / S − 1 ˆ ind D W = 2 ( η ( D ∂ W ) + dim Ker D ∂ W ) W η ( D ) spectral contribution, non-local! � 0 , a ∈ Z M = S 1 , η ( − i d dx + a ) = 2 a − 1 , a ∈ (0 , 1) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 4 / 15

Introduction rho invariants and classification of positive scalar curvature metrics Secondary: rho invariants Atiyah–Patodi–Singer: α : Γ → U ( k ) flat bundle 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, 2014 5 / 15

Introduction rho invariants and classification of positive scalar curvature metrics Secondary: rho invariants Atiyah–Patodi–Singer: α : Γ → U ( k ) flat bundle 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 g = ∇ ∗ ∇ + 1 / 2 then scal g > 0 ⇒ D / g invertible D 4 scal g Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 5 / 15

Introduction rho invariants and classification of positive scalar curvature metrics Secondary: rho invariants Atiyah–Patodi–Singer: α : Γ → U ( k ) flat bundle 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 g = ∇ ∗ ∇ + 1 / 2 then scal g > 0 ⇒ D / g invertible D 4 scal g ( g t ) t ∈ [0 , 1] ∈ R + ( M ) := { g metric on TM | scal g > 0 } � W ˆ A ( W ) + 1 / g 0 ) − 1 0 = 2 η ( D 2 η ( D / g 1 ) � A ( W ) ch ∇ α + 1 W ˆ g 0 ) − 1 / α / α 0 = 2 η ( D 2 η ( D g 1 ) / ): π 0 ( R + ( M )) → R ⇒ it gives a map ρ ( D Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 5 / 15

Introduction Role of primary vs. secondary invariants Role of index and rho: positive scalar curvature g = ∇ ∗ ∇ + 1 / 2 M closed spin manifold D 4 scal g . R + ( M ) = { g metric on TM | scal g > 0 } R + ( M ) = ∅ ) ind D / is an obstruction ( ind D / � = 0 ⇒ Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 6 / 15

Introduction Role of primary vs. secondary invariants Role of index and rho: positive scalar curvature g = ∇ ∗ ∇ + 1 / 2 M closed spin manifold D 4 scal g . R + ( M ) = { g metric on TM | scal g > 0 } R + ( M ) = ∅ ) ind D / is an obstruction ( ind D / � = 0 ⇒ / ) can distinguish non-cobordant metrics, assuming R + ( M ) � = ∅ , ρ ( D Theorem: (Piazza–Schick, Botvinnik–Gilkey) ∃ infinitely many non-cobordant ◮ R + ( M ) � = ∅ { g j } j ∈ N ⊂ R + ( M ) ⇒ ◮ dim M = 4 k + 3 , k > 0 ρ ( D / g i ) � = ρ ( D / g j ) ∀ i � = j ◮ π 1 ( M ) has torsion Theorem: (Piazza–Schick) / g ) = 0 for g ∈ R + ( M ) ◮ π 1 ( M ) torsion free, and satisfies Baum–Connes ⇒ ρ ( D Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 6 / 15

Projectively invariant operators Projective actions Exemple: π : ˜ M → M universal covering, Γ = π 1 ( M ) R n → R n / Z n = T n Γ -invariant: ˜ D γ = γ ˜ ∀ γ ∈ Γ D Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 7 / 15

Projectively invariant operators Projective actions Exemple: π : ˜ M → M universal covering, Γ = π 1 ( M ) R n → R n / Z n = T n Γ -invariant: ˜ D γ = γ ˜ ∀ γ ∈ Γ D 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, 2014 7 / 15

Projectively invariant operators Example: the magnetic Laplacian Typical construction π : ˜ M → M universal covering, Γ = π 1 ( M ). On the trivial line L = ˜ M × C → ˜ M consider ∇ = d + iA , where dA ∈ Ω 2 ( ˜ M , R ) is Γ -invariant: dA = π ∗ ω , ω ∈ Ω 2 ( M , R ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 8 / 15

Projectively invariant operators Example: the magnetic Laplacian Typical construction π : ˜ M → M universal covering, Γ = π 1 ( M ). On the trivial line L = ˜ M × C → ˜ M consider ∇ = d + iA , where dA ∈ Ω 2 ( ˜ M , R ) is Γ -invariant: dA = π ∗ ω , ω ∈ Ω 2 ( M , R ) Then: γ ∗ A − A = d ψ γ σ ( γ , γ ′ ) = exp( i ψ 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 − i ψ γ u , u ∈ L 2 ( ˜ M ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 8 / 15

Projectively invariant operators Example: the magnetic Laplacian Typical construction π : ˜ M → M universal covering, Γ = π 1 ( M ). On the trivial line L = ˜ M × C → ˜ M consider ∇ = d + iA , where dA ∈ Ω 2 ( ˜ M , R ) is Γ -invariant: dA = π ∗ ω , ω ∈ Ω 2 ( M , R ) Then: γ ∗ A − A = d ψ γ σ ( γ , γ ′ ) = exp( i ψ 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 − i ψ γ u , u ∈ L 2 ( ˜ M ) Remark: the construction can be done starting from c ∈ H 2 ( B Γ , R ) Sara Azzali (Potsdam) Secondary invariants for two-cocycle twists Villa Mondragone, June 17, 2014 8 / 15